The collected papers of Stephen Smale

X, and let V be the vector space generated by m, then it is easy to see that d(V\) ^ X» for any V\ of dimension 25 wi, which with lemma 4b contradicts X„+i > X, . Therefore, dim V = m and hence
Furthermore, since BL(
where n0 is the first integer with X„. = X, . Therefore,
-> C"(E)

540

S. SMALE

1054

be as in lemma 1. Then there eodsts e > 0 with the following properly. If M' is a domain of M with smooth boundary, and measure < e, then the restriction U : CT-i(tfO -» C"(E) has only positive eigenvalues, where E' is the bundle over M'. Proof. In the following estimates u t C"(E'), and u has compact support in interior M'. One variant of Gardings inequality [8], asserts

I M I 1 . „ . , £ c||tt||L.»(»', - c , ||«||i.(Jr.>. If M' is such that Ci can be taken zero, then it is clear that the conclusion of Lemma 7 follows. Thus, it is sufficient to probe given ti > 0, there exists S > 0 such that IMU«c*'> < «i 11«|!//■*(*')

for measure M' < 8 (i.e., take «i = C/Cx in Gardings inequality). Choose r > 2 such that 1. 1 2fe r ~ 2 dim M Then by the Sobelov Imbedding Theorem (see e.g., [2], lemma 5),

I ML' ^ c, ||tt|L.». Holders inequaUty implies ||«||i. 2 ||tt||I-(Vol.Jlf)*, where 2/r + 7 = 1. Putting these together, we obtain ||i«IL. ^ c,(yoi.M)yn

||t«|U.i,

which we have seen gives us lemma 7. The main theorem follows directly from lemmas 1, 2, 7, the conjugate t being these with X{ = 0 for some i. We complement the main theorem by some final remarks. Theorem. Let Mbeas in the main theorem, e > 0. Then there exists a deforma­ tion M, of trtype. The proof is an exercise in differential topology. For example, one could use elements of handlebody theory (see [11] and its references) to smoothly deform M onto a neighborhood of some cells of dimension less than dimension M. Also, the proof of the main theorem yields immediately. Theorem. Let L : Ct-i(E) —> C(E) be a self-adjoint strongly elliptic operator of order 2k with uniqueness in the Cauchy problem. Then there exists e > 0 such that if M, , a £ t ^ b is any deformation (not necessarily smooth) of e-type, then 00 > fi(L) ^

£

a(t).

541

MORSE INDEX THEOREM

1055

REFERENCES

[1] F. BBOWDZR, Functional analysis and partial differential equations. II, Math. Annales, 145 (1962) 81-226. [2] F. BBOWDER, On the spectral theory of elliptic differential operators. I*, Math. Annales, 142 (1961) 22-130. [3] COUKANT-HILBEBT, Methods of Mathematical Physics, Vol. I, New York, 1953. [4] H. EDWARDS, A generalized Sturm Theorem, Annals of Math., 80 (1964) 22-57. [5] L. HttBMANDEB, Linear Partial Differential Equations, New York, 1963. [6] J. MILNOB, Morse Theory, Princeton, 1963. [7] M. MORSE, Introduction to Analysis in the Large (mimeographed), The Institute for Advanced Study, Princeton, 1947. [8] L. NIBDNBERO, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa, 13 (1959) 115-162. [9] R. PALAIS, Differential Operators on Vector Bundles (Seminar on the Atiyah-Singer Index Theorem), (mimeographed) Institute for Advanced Study, Princeton, 1964. [10] RTJBSX-NAGY, Functional Analysis, New York, 1955. [11] S. SMALK, On the structure of manifolds, Amer. Jour, of Math., LZXXIV (1962) 387-399. [12] M. I. VISIT, On strongly elliptic systems of differential equations, Mat. Sbornik, 29 (1951) 615-676. Columbia University

542

Corrigendum On the Morse Index Theorem By S. SMALE J. Math, and Mech., 14 (1965) 1049-1056. The following selection of letters between Richard S. Palais and me is selfexplanatory. From Palais to Smale—April 27, 1966 . . . but there seems to me to be a gap in the proof. I doubt that it is very serious, but I have not been able to fill it in. The problem is this: suppose M' is an open submanifold of M with smooth boundary. If V is a f.d. subspace of Hh0(£) = H then you speak of the orthogonal projection wV of F o n f f ' = #£(£ | M') (relative to the L2(£) = H°(£) inner product of course). Now the trouble is that, while you showed that H' is closed in H relative to the L2 topology, it is not closed in H°(£) so the best you can do is define an orthogonal projection T of Z/*(£) onto R', the closure of H' in fl°(|), and take irV. One might hope that in fact ir maps #£(£) into Hk0(£ \ M') but unfortunately it only maps it into H"iX | M'). In fact clearly # ' = L2(£ | M') and ^ _f/

in M> lO in M -

M'.

Thus TV $ H' and I don't see how to proceed with the proof that \'n ^ X, . From Smale to Palais—May 3, 1966 . . . Of course you are right. It seems to me (at first glance) as if the following is true and will repair the trouble. Lemma 5'. Under conditions of lemmas 4 and 5, define e(V) = max BL on V C\ S, for a finite dimensional subspace V of H. Then X, = min«iim r s . e(V). This (I guess!) follows from Lemma 1 and seems to imply \'n ^ X, . From Palais to Smale—May 5, 1966 . . . What you say seems to be quite right. Lemma 1 should say that if / e U(E) and / = 2 < c<*>» ^ i*3 orthogonal expansion in the eigenvectors of L, 1069 Journal of Mathematics and Mechanics, VoL 16. No. 9 (1967).

543

1070

S. SMALE

then / e Hk0(E) => ]£, a,p< converges to / in Hl(E)—this is of course well-known and must be in Browder's paper. Then if En = linear space of \„ aiVi] it follows that Hk0(E) = En@E\ so 2?t has codimension n in Hk0(E). Then if 7 is any subspace of Hk0(E) of dimension ^ n it follows that V H ££_, ^ 0 hence 3 P t S C\ V C\ Et-i and clearly BL(p) ^ X„ so max„ s n r J3t(p) ^ X» so mindim F i , max„srvv BL{p) 2= X, and since the other way is clear we have V C Hk0{E), equality as you state, and the rest of the proof is the same. Actually the version of the minmax theorem seems to make the whole proof look simpler. Lemma 5 it seems to me is unnecessary. Institute for Advanced Study Date Communicated:

NOVEMBER

26,1966

544

WHAT IS GLOBAL ANALYSIS? S. SMALE, University of California, Berkeley

There has recently been a lot of activity in that branch of mathematics now referred to as "global analysis." For example, the subject of the 1968 Summer Institute of the American Mathematical Society was global analysis. My definition of global analysis is simply the study of differential equations, both ordinary and partial, on manifolds and vector space bundles. Thus one might consider global analysis as differential equations from a global, or topological point of view. Even the earliest studies of differential equations contained an element of global analysis; this element had become quite important for example in the work of Poincare" on ordinary differential equations. G. D. Birkhoff's develop­ ment of dynamical systems and especially M. Morse's theory of geodesies are both excellent examples of global analysis. After the rapid recent progress in topology, the subject of our exposition has been moving especially fast. After mentioning a couple of references in partial differential equations, I shall devote the rest of my article to an account of a theorem in dynamical systems to illus­ trate the global analysis point of view. Recently there have been nice results in the topology of linear elliptic differ­ ential operators, especially in the work of Atiyah, Singer, and Bott (see for example [2] and [4]). One cannot expect to have a satisfactory framework for nonlinear partial differential equations with linear function spaces. Thus it is important that nonlinear partial differential equations are beginning to be attacked by a syste­ matic use of infinite dimensional manifolds of maps. A good survey of this is Eells [3]. The work of Andronov, Pontryagin [l ] and Peixoto [5] in dynamical systems (or ordinary differential equations), on one hand can be explained in relatively simple terms and on the other hand gives a real insight into this modern way of looking at differential equations. I shall try to give a brief account of their theory now. Consider an ordinary differential equation (1st order, autonomous) defined on a domain D in the x, y-plane: dx

dy

— PC*,?)

J-G(*,?).

Stephen Smale received his Ph.D. at the University of Michigan in 1956. He has occupied various positions at the University of Chicago, the Institute for Advanced Study, Columbia Uni­ versity, and his present location, the University of California at Berkeley. For his outstanding research in differential topology and in global analysis, Professor Smale was awarded the Fields Medal of the International Mathematical Union in 1966 and the Veblen Prize of the American Mathematical Society in 1964. Editor

4

545 1969]

WHAT IS GLOBAL ANALYSIS?

5

We shall assume that these functions P, Q defined on D are continuously differen­ tiate (or of class C1). Now the fundamental existence theorem of ordinary dif­ ferential equations yields for each (x* yt) in D and real t sufficiently small in absolute value, |/| <e, functions f(x0, yt, t), g(xt, yt, t) which satisfy the initial conditions /(*», y», 0) —*», g(x», yt, 0) -y» and the differential equation

W(**

y l)

*

~■ p tf ( * 0 ■ y " *>' g(-x*

y f))

"

(—J(xo, y», 0 - Q(J(xt, yt, t), *(*•, yt, <))• Let us look at this phenomenon from a more geometric point of view and in fact get away from the particular choice of *, ^-coordinates. To each (*, y) in D associate the vector (P(«, y), Q(x, y)) of the x, y-plane with the initial point at (x, y). This gives us what is called a Cl vector field on D. For each point p of D, we will call the associated vector for short X(p). Then the existence theorem we just stated may be interpreted to yield a system of plane curves $t(p) with $»(£) •■£, and with the property that the tangent of the curve at a point q of D will be the vector X(g) (see Figure 1).

Fio.1.

The right context for the study of this differential equation becomes clearer now. More generally than a domain of the Euclidean plane, consider a 2-dimensional smooth manifold M. Roughly speaking, one can think of this as a surface in 3-dimensional Euclidean space E? or better abstractly as a space on which differentiation makes sense and a neighborhood of each point is a domain in the plane. To each point p of M there is associated a 2-dimensional vector space

546

6

WHAT

is GLOBAL ANALTSIS?

[January

T,(M), the tangent space of M at p. If M is a surface in E* then TP(M) is the plane tangent to M at p. A vectorfieldX on M is an assignment, continuously differentiable, p-*X(p) for p in M to ^(p), a "vector" in T,(M). The vectorfieldon D defined previously from the differential equation given by the functions P, Q on D is now a vector field on the 2-manifold D in this sense. To define the basic idea of this article, structural stability of a differential equation, we need to develop two things: one, the space of differential equations on M, x(-M)i and two, an equivalence relation on x(AO» the phase portrait. We have seen that the kind of differential equations on M we are studying (which are really pretty general except for the low dimension) correspond to vector fields on M. We call the set of all vector fields (C1 as usual) on M, x(-M). Now x(Af) has the structure of a vector space, using the fact that for each pEM, the values of all vector fields lie in the same linear space T,(M). That is if X, Y belong to x (Af), (X+Y)(p)-X(p) + Y(p). This space x(AO will be basic in what follows. The solution curves $»(p) of a vector field X on M, defined earlier, may be "pieced together " so that for each p, fa(P) will be defined for all at of transformations on M. Thus for each real /, tf>, is a C1 transformation of M, tftr. M—*M, with a C1 inverse, 0o is the identity and <£<(#.) =■#<+.• In short i is a dynamical system.

FIG.

2a.

To abstract the qualitative features of a differential equation on M, the concept of a phase portrait becomes important. Usually the phase portrait means the picture of the solution curves of the differential equation. For example, Figure 2a is the phase portrait of a differential equation in the plane. To give a precise mathematical content to "phase portrait," we proceed as

547

1969]

WHAT IS GLOBAL ANALYSIS?

7

follows. Say X, Y in xC-^O *"* topologically equivalent when there is a homeomorphism h: M—*M taking solution curves of X into those of Y. Thus the differ­ ential equation in Figure 2a is topologically equivalent to that described in Figure 2 b.

FIG.

2b.

Then two differential equations on M have the same phase portrait if they are topologically equivalent. A definition of phase portrait is thus a topological equivalence class of differential equations on M. A main goal of the qualitative study of ordinary differential equations is to obtain information on the phase portrait of differential equations. To make progress in this direction, one soon sees the need to avoid "degener­ ate" cases. For example a differential equation that is zero on all of M, or even on some nonempty open set of M should be considered degenerate and excluded from most considerations. I think that engineers and physicists will agree with this statement. To aid in discussing the question of degeneracy, a topology or metric on xC*0 is useful. To simplify matters in defining this metric, in the rest of our article, we will assume M compact. This excludes many or even most interesting examples, but on the other hand the main features are not lost. Assuming M compact define a norm || || on x(-W) as follows. Let U\, • • • , Uk be a covering of M, T7 with each V< a plane domain. Then on each V{, X in \{M) is represented by Pt(x, y), Qt(x, y) as at the beginning. Then ||.XJ| is defined as the maximum of the following finite set of numbers: sup

| P(x, y) |

1, • • • , *

sup

| (?(*, y) |

I,---,*

548

8

WHAT is GLOBAL

sup

ANALYSIS?

[January

— (*» y)

dx

sup

— (*. y) dy

1, • • • , *

Slip

— (*» y) dx

1, ■ ■ • , *

sup

— (*» y) dy This gives x(if) the structure of a complete normed space or a Banach space. A metric on x(Af) is then defined by d(X, Y) -\\X- Y\\. With this metric on x(-&0 it is possible to say when differential equations are "close." In terms of local coordinate representations, two differential equa­ tions are close when the P and Q are uniformly close, with their first derivatives uniformly close as well. With this background, we say that X in x(<&0 is structurally stable when there is a neighborhood N(X) in xC^O with the property that every Y in N(X) is topologically equivalent to X. Thus X is structurally stable when nearby differential equations have the same phase portrait. A little thought will indicate that this excludes degeneracy; a structurally stable X cannot be degenerate (in some senses at least). It is an important concept for the engineer who studies qualitative differential equations, since in engineering the differential equations one works with are only approximations of the real equations. The engineer wants the qualitative conclusion he makes to be valid for the actual differential equation which describes his world. In fact the original idea of structural stabil­ ity was the joint work of an engineer, A. Andronov, and a mathematician, L. Pontryagin. Thus it becomes important to know if most differential equations are struc­ turally stable. (M. Peixoto) If M is a compact 2-dimensional manifold, then the structurally stable differential equations in x(2f) form an open and dense set. THEOREM.

This theorem is an excellent theorem in global analysis. One sees in two ways how it is global. First the differential equation is defined over a whole manifold, and structural stability depends on its behavior everywhere. Second, the theo­ rem makes a conclusion about the space of all differential equations on M. The proof gives much information on the structure of differential equations on 2-manifolds. We state the main lemma which indicates how this is so. The nonwandering set Q(X) of X is defined as the set of x in M such that for every neighborhood U of * and t», there is a t>h with +,(U)P\ Ui*0.

549 1969]

WHAT IS GLOBAL ANALYSIS?

9

If M is a compact 2-manifold and X is in xt^O, *&** x « structurally stable if and only if the following conditions are met: (a) Each closed orbit and each singular point of X is "nondegenerate." This nondegeneracy is defined in terms of derivatives associated to the closed orbits and singular points. (b) The separatrices of saddle points don't meet. (c) Q(X) consists of thefiniteunion of closed orbits and singular points. MAIN LEMMA.

[Separatrices are the trajectories which come to and leave from the saddle points.] If a is a singular point or closed orbit, let W'(a) be the set of x in M with $,(x)—*a as *-*<». Then if AT is structurally stable, it provides for a decomposi­ tion of Af as the finite union of W'(a) as a ranges over the closed orbits and singular points. This decomposition gives a good practical understanding of the differential equation X. A survey of this subject with many references is [6]. This article i» baaed on an address before the Mathematical Association of America, San Frandaco, 26 January, 1968. References 1. A. Andronov and L. Pontryagin, Systemes grossiers, DoH. Akad. Nauk. SSSR, 14 (1937) 247-2S1. 2. M. Atiyah and R. Bott, A Lefachets find point formula for elliptic differential operators, Bull. Amer. Math. Sot, 72 (1966) 245-250. 3. J. Eelta, A setting for global analysis, Bull. Amer. Math. Sot, 72 (1966) 751-807. 4. R. Palais et at., Seminar on the Atiyah-Singer index theorem, Ann. of Math., Study No. 57, (1966). 5. M. Peixoto, Structural stability on 2-dlmenaional manifolds, Topology 1 (1962) 101-120. 6. S. Smale, DifferentiaMe dynamical systems, BulL Amer. Math. Sot, 73 (1967) 747-817-

550 BOOK REVIEWS

683

tULLETIN OF THE AMERICAN MATHEMATICAL SOCIETY Volum II. Number 4. July 1971

Global variational analysis: Weierstrass integrals on a Riemannian manifold, by Marston Morse, Mathematical Notes, Princeton University Press, Prince­ ton, New Jersey, 1976, ix + 255 pp., $6.50. The first thing that comes to mind in reviewing a new book by Marston Morse on the calculus of variations is that he wrote a book, The calculus of variations in the large, forty years ago. The early book gave the foundations of what is now called Morse theory. The publication of a new book by Morse on the same subject presents an occasion to give some personal perspectives on how this mathematics has developed in the last few decades. I say "personal perspectives" and indeed, I, myself, have been involved in, and inspired by. Morse's mathematics. For example, three of my papers contain the word Morse in the title. Another mathematician much influenced by Morse, Raoul Bott, was my adviser, and even work of Morse (but not variational theory) suggested to Bott the thesis problem he gave me (leading eventually to my work in immersion theory). Another factor in writing a review like this is that, today, global analysis is very much alive, both in mathematics and other disciplines. It may give us some perspective to trace the development of one of the main roots of the subject. Let us see what Morse, in 1934, had to say about global analysis (he used the word macro-analysis, then). I quote the full first paragraph of the Foreword of his book. "For several years the research of the writer has been oriented by a conception of what might be termed macro-analysis. It seems probable to the author that many of the objectively important problems in mathematical physics, geometry, and analysis cannot be solved without radical additions to

551 684

BOOK REVIEWS

the methods of what is now strictly regarded as pure analysis. Any problem which is nonlinear in character, which involves more than one coordinate system or more than one variable, or whose structure is initially defined in the large, is likely to require considerations of topology and group theory in order to arrive at its meaning and its solution. In the solution of such problems classical analysis will frequently appear as an instrument in the small, integrated over the whole problem with the aid of group theory or topology. Such conceptions are not due to the author. It will be sufficient to say that Henri Poincare was among the first to have a conscious theory of macroanalysis, and of all mathematicians was doubtless the one who most effectively put such a theory into practice." Note Morse's acknowledgment of the role of Poincare in this subject Already in 1885, Poincare knew of Morse inequalities for a surface. Note also how Morse feels that problems (of analysis) nonlinear in character are likely to require considerations of topology and group theory. I would like to echo this point, which even today, forty years later, has still not been digested by some analysts, analysts, for example, who are not willing to relinquish their linear spaces and linear space methods to confront nonlinear problems. Suspicions of geometry and the uses of geometry in analysis have indeed deep roots. Even G. D. Birkhoff wrote in 1938, in Fifty years of American mathematics, of his " . . . disturbing secret fear that geometry may ultimately turn out to be no more than the glittering intuitional trappings of analysis." He used the word geometry to include topology or "analysis situs" as it was called then. The simplest case of Morse theory is just the phenomenon that a differentiable function on an interval with two local minima must have a local maximum between the local minima. This "minimax principle", while very old, received a big push by G. D. Birkhoff in 1917. It is interesting to see what Morse had to say about some of the origins of the global calculus of variations. In his obituary of Birkhoff, he wrote (on the minimax principle): "In Birkhoffs applications this principle reduces to an existence theorem for critical points of an analytic function F(x) of n-variables. If one supposes for the sake of definiteness that F is denned over a regular, compact, analytic manifold, then, suitably counted, there exist at least /?, + MQ - 1 generalized saddle points, where /?, is the linear connectivity of the manifold and M0 the number of points (supposed isolated) of relative minima of F. In similar or related forms this principle was known and applied by Poincare, Maxwell, and Kronecker, and has an origin even more remote in the past. Birkhoffs bold step was to conceive of its application to functions of curves such as the integral J. He applied it in the billiard ball problem [26] (motion on a convex table) and to obtain closed geodesies on a convex surface." Thus, one has here a relation between the analysis, saddle points or geodesies, and the topology, "linear connectivities", or more generally, Betti numbers.

552 BOOK REVIEWS

68S

Morse's central contribution was to take these ideas and apply them systematically to a functional on the "manifold" of curves joining two points to deduce the existence of extremals, especially geodesies. In doing so, he developed relations between the critical points and the topology for any (smooth, nondegenerate) function on a compact manifold. In particular, it was a great accomplishment of Morse, in the years 1925-1930, to have given a global geometric abstract base for the variational calculus. To that base we proceed. A fundamental insight here deals with the problem: How does the topology change as a function passes a nondegenerate critical point? To explicate matters, let / : Q -* R be a C00 real valued function denned on a compact manifold, ft. We will suppose that all maps are C x and use the symbol Q for a manifold for reasons that will become more natural later. A point x in Q is called a critical point of J if it has zero derivative, i.e., if DJ(x) = 0. In that case the second derivative D2J(x) is an invariantly defined bilinear symmetric form Hx on the tangent space Tx of vectors tangent to Q at jr. This form is called the Hessian and plays an important role in Morse theory. Recall that for any bilinear symmetric form Wona linear space E, the index is the maximum dimension of a subspace on which H is negative definite. The nullity of H is the dimension of the null space, i.e., the set of v in E such that H(u,v) = 0 for all u in E. Then H is nondegenerate if its nullity is zero. If H is nondegenerate and E is R", then there are linear coordinates u on R" such that

i-i

<-*+i

Here k is the index. All of these definitions pass over to a critical point. Thus the index of a critical point is the index of its Hessian; the critical point is called nondegener­ ate if its Hessian is nondegenerate, etc. A nondegenerate critical point is necessarily isolated. One knows more. The Morse lemma asserts that if a critical point x of a function J is nondegnerate, then there are coordinates u near x to make J quadratic, i.e., so that J(u) =■ Hx(u,u). My own belief is that the Morse lemma, while nice to know, is not vital, and in fact Morse theory develops more naturally, more conceptually, without it. To understand the topology of a function on a manifold, one uses a Riemannian metric to define gradient lines of the function. The choice of coordinates in the Morse lemma will not respect this metric. On the other hand, simply Taylor's formula already gives sufficient local information to do the "handle attaching". Perhaps the preceding remark will become clearer as we proceed with our story of what happens to the topology as a nondegenerate critical point is passed. For any real number a let Ja be the set of all x in Q with J(x) < a. If there is no critical value in the interval [b,c], then /"'[&,c] is differentiably

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isomorphic to a product, J (b)x [b,c\. (A critical value is the value of a critical point.) Now suppose there is exactly one critical point JC* with value c, and that JC* is nondegenerate with index k. How is the topology of Jc+l related to that of 4 - e f o r s m a " enough c? The fundamental result is that Jc+e is Jc_t together with a cell (or "handle") of dimension k attached. This handle attaching statement has three versions, which relate to three periods in the development of Morse's critical point theory; and the last two relate to the application of this theory to problems of topology. The, first of these versions is on the homology level and states the relative homology (over the rationals) result:

dimffk(Jc+t,Jc_t)

= 1.

Recall k is the index of the critical point. This result is central in Morse's 1934 book and is used to deduce the existence of critical points. These critical points correspond to solutions of variational problems, in particular, to geodesies as we shall see later. Thus it is used to make a passage from topology to analysis and geometry. The second version is a sharpening, explicated by Bott in 1959, which puts the theorem on a homotopy level. This result was used by Bott to study the homotopy of Lie groups. In particular, he obtained the first proof of the Bott periodicity theorems this way. Specifically, the stable homotopy groups of the unitary group U and the orthogonal group O satisfy «i(U) = *I+2(U),

»,(0) - » < + l (0)

alii.

Here for example one may think of U as the union of (/(«) over n = 1, 2, 3, . . . and then rr( is the ordinary homotopy group. This result was the starting point of tf-theory. Bott's version of handle attaching is the statement that 7c+t is homotopically equivalent to Je_t with a cell Dk of dimension k attached by a homeomorphism from the boundary 3D* of the cell into the level surface J~x{c — e). The homotopy statement yields the homology statement as a corollary. The third version of handle attaching is on the diffeomorphism level which I believe I was the first to explicate. In fact, this was one very important ingredient in my solution of the "higher dimensional Poincare conjecture", work on "handle body theory", and the structure of manifolds. To work on the level of differentiable isomorphism, one must thicken /)* to bring the dimension up to the dimension of the manifold, say n. Thus, let a khandle be D"~k X Dk. Then the strongest version of handle attaching asserts that Jc+t is diffeomorphic to Jc_t with D"~k x Dk attached by an imbedding of D"~k x9Dk into J~l(c - e). The attaching process involves a smoothing at the corners. This statement yields the homotopy version as a corollary. For my favorite proof of these theorems, one supposes that the manifold has

554 BOOK REVIEWS

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a Riemannian metric (by imposition, if necessary) and uses the flow defined by the negative of the gradient of J. At the critical point x*, the linearized flow has two invariant subspaces in the tangent space 7^». One of these is contracting under theflow,say £""*, and one is expanding, say £* (with the dimension of Ek equal to k, the index of x*). Use these spaces to define a local product structure in the manifold near x*; then take small disks D"~k, Z)* about the origins in E"~k, Ek, respectively. Using Taylor's formula to expand J about x*, one can show that theflowon the boundary of D"~k X Dk has the requisite properties to give the attaching statements. The same proof works at the homotopy and diffeomorphism levels, but requires slightly more checking in the latter case. The passage from the homology version of handle attaching to the Morse inequalities proceeds most simply via the exact homology sequence of a pair. Suppose that there are real numbers, c0 < c, < c2 < • • • < cm, so that each interval (Cj,cj+l) contains the value of exactly one critical point of J: Q -* R and all the critical values are in such intervals. For each./, one has the exact homology sequence of vector spaces over the rational numbers, writing ^ f o r / ,

From linear algebra, summing from / = 0 to k yields for each it: 2 (-l)*-'dimfljC^i)" 2

(-l)*-'dimtf,(/)

+ 2 (-l)*~'dim#,(/,./ ,) > 0. Summing overy and evaluating by the handle attaching theorem gives - 2 (-l)*"'dimflj(yw) + 2 (-1)*''", > 0 where M( is the Morse type number or the number of critical points of index i. Let B; be the ith Betti number, or dim/^(S), and since Jm — B, we have Morse inequalities. 2 ? - 0 (-1)*"'^ > 2 f - o ( - 1 ) * " ' ^ . each k = 0, 1, 2, By adding these inequalities for k and k - 1, we get Simple Morse inequalities. Mk > Bk, k = 0, 1, 2, . . . . In the preceding discussion we have assumed that different critical points took different values; a minor extension in the handle attaching argument can remove that hypothesis. We have also assumed that all of the critical points of J were nondegenerate. Such functions are called Morse functions. How general are Morse functions? Morse has in his 1934 book a prototype of the theorem that Morse functions form an open and dense set among all C°° functions. This result is now centrally imbedded in transversality theory and the theory of singularities of maps a la Whitney, Thorn and Mather.

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Sard's theorem that with enough differentiability the values of critical points form a set of measure zero is basic in this development and it is no accident that Sard was a student of Morse. It is important to remark that a complement to Morse's work was developed, already by 1930, by Lustemik and Schnirelmann in the Soviet Union. These mathematicians gave a global existence theory for critical points without making use of nondegeneracy hypotheses. What we have described above is still in the finite dimensional and abstract realm, thus twice removed from Morse's contributions to the study of geodesies. But before we turn to that study, we give a couple of ways this finite dimensional theory has made itself felt in analysis in the domain of our own experience. One way is in dynamical systems, where starting from a Morse function / on a Riemannian manifold, one obtains a differential equation with a rather simple structure. The negative of the gradient vector field of/ has the property that along 'solutions (of dx/dt — - grad J(x)), J is never increasing. Thus the dynamical behavior permits no nontrivial periodicity or recurrence. Further­ more, the zeroes of this differential equation, coming from a nondegenerate critical point, possess a certain local robust character. If one adds a second condition of transversality, that the asymptotic sets (the "stable and unstable manifolds") of these zeroes intersect transversally, one can obtain such properties globally. Via this route is the result I obtained with Jacob Palis, that every compact manifold supports a structurally stable dynamical system (so that the "phase portrait" or qualitative structure persists under perturbations). Moreover, it was the interplay between dynamical problems and topology that helped lead me to the handlebody results mentioned earlier. In this vein, it is worth remarking that the catastrophes of Thorn deal with bifurcations of gradient dynamical systems. A second example is in celestial mechanics where one can show that the relative equilibria in the Newtonian fl-body problem correspond to critical points of the Newtonian potential function on complex projective space, properly interpreted. Morse theory of this function suggested to Julian Palmore the existence of new relative equilibria in the 4-body problem which he found. Let us turn now to the variational theory of Morse, the ultimate object of the abstract theory previously discussed and the subject proper of the book under review. The prinicpal example is the global study of geodesies on a manifold. Let us see how this goes. Let M be a Riemannian manifold. Recall that this equips each tangent space Tx = TX(M), x in A/, with a norm written || H, or sometimes simply || ||, defined by an inner product (, )x in Tx. One can define the length 1(a) of a curve a: [a,b\ -* M by /(a) = j

||d(/)||
where d = - £ .

556 BOOK REVIEWS

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From this, define a metric on M by letting d(p.q) equal the infimum of 1(a) over all curves a joining p to q. This is indeed a metric. We will assume that M is complete for this metric. On M, let points P, Q be given. Denote by 8 = QPQ(M) the loop space, or set of all (C 00 ) curves on M from P to (?. That is, 2 = {a: [0,1] -► M\a(0) = P,a(l) = Q). The e/iCTxy is the map J: Q -> R given by J(a) = /J ||d(/)l|2
^(A/XTKO)

=

O.TKO

= 0}.

One can think of Ta(Q) as the space of vector fields along the curve a. One can think of geodesies as being like "critical points" of J: ft -* R. It would be natural to define the derivative of J at a G 8, DJ{a): Ta(Q) -* R as follows. In case M is R", then the tangent bundle T(M) is M x R" and one can proceed by letting F: T(M) -> R be defined by F(x,x) = Hill, for x G Tx. Then for 17 G 7^(8) let rl 3yr

3/r al^= (aF/aJc)(o(f),d(0) is a linear map on /?" for each /. Then if

J0 83T»

where 3 / / 3 J C DJ(a) = 0,

+

WWW = /o' (If" Sff)1' " 0 f ° r a11 * e ^^^ So Euler's equation for a is satisfied or: Euler's equation: d/dtdF/dx(a(t),a(t)) - dF/dx(a(t),a(t)) = 0. If M is not /?", then one has the argument and equation valid in each coordinate chart, or one could use the covariant derivative. Recall that a curve o: [a, b] -» M is a geodesic if it locally minimizes length. Then the above kind of derivation shows that a in 8 is a geodesic if and only if a is a solution of Euler's equation in each coordinate chart. Thus the geodesies are indeed like critical points of J. The earliest part of our review motivates the question as to whether there are the concepts of index and nullity for a geodesic a. In the "second derivative" of J at a, as before, one can find an answer. In a coordinate chart of M, it is natural to write for D2J(a), the second derivative of J at a, the symmetric bilinear form on ^ ( 8 ) defined by D2J(a)(r,,t)

= Ha(V,S) = /Q' Fxx(r,,i) + FXX(U) + Fxx(r,.S) +

Fxx(r,4)dt.

557 690

BOOK REVIEWS

Here, 17, £ e Ta(2) and Fxx = Fxx(a(t),a(t)) is the second partial derivative of Fas a bilinear form on T^(M) = R", etc. One defines the index and nullity of a simply as the index and nullity of Ha on Ta(Q). Furthermore, a is nondegenerate if Ha is. Define an inner product on Ta(Q) via that on M: i.e.,

(TJ. £ )a = J0' WO. <(/))„(„ *,

r», * e r a (n).

Using integration by parts, one obtains that for all ij, £ in Ta(fl), Ha(V,£) = (Lij.O where Lv

=

'Jt^***

+ F

^

+ F

**v ~

F iil

"

is the Jacobi (linear) differential operator. One may express L invariantly in terms of the Riemann curvature tensor. Say that P and Q are conjugate along a if LTJ = 0 for some nonzero 17 e ^(fipp). The multiplicity of this conjugacy is the dimension of the linear space of such 17. The Morse index theorem asserts that the index of a is equal to the number of points a(t), 0 < / < 1, such that a(t) is conjugate to o(0) along a, counting conjugate points with multiplicity. I like to think of this result as belonging to the spectral theory of differential operators. To prepare us for his main result, Morse shows that for prescribed P in M, if Q is excluded from a set of measure zero in M, then all the geodesies in Qp Q will be nondegenerate. Thus for such Q, J is like the Morse functions defined earlier. We may call such a pair (P, Q) a nondegenerate pair. Now we may state the following basic theorem of Morse in the calculus of variations. THEOREM. Let (P,Q) be a nondegenerate pair on a complete Riemannian manifold M. Let Bi denote the dimension of the homology group // ( (Qpo(^)) (over the rationals) of the loop space. Let Mi denote the number of geodesies joining P to Q (in Q) of index i. Then the Bt, M- satisfy the Morse inequalities M0 > B0,

M] - M0 > B, - fl0, etc.

as before. Morse proves the theorem by taking a sequence of finite dimensional manifolds which approximate Q and applying the earlier abstract theory. These approximating manifolds are manifolds of piecewise geodesic curves. As a particular case of this theorem, Morse in his first book takes M to be homeomorphic to the n-sphere S". By applying the same Morse inequalities to the standard Riemannian w-sphere in Rn+ . he is able to compute the Bt, Betti

558

BOOK REVIEW

691

numbers of the loop space, which are given by (say for n > 2), B, = 1 for / = 0, n - 1, 2(n - 1), 3(w - 2 ) , . . . and zero otherwise. Therefore, he con­ cludes that for an arbitrary Riemannian structure on S", the existence of geodesies of index 0, n - 1, 2(/i - 1), . . . joining P to Q. Applications of this basic theorem to a more general class of manifolds were hampered by lack of knowledge of the homology of loop spaces. In fact, the breakthrough on this problem didn't come until 1951 with Serre's thesis. Using the Leray spectral sequence, Serre showed that //((J2(A/)) # 0 for a sequence of i going to infinity for a compact manifold M with finite fundamental group. From the Morse theorem, this is enough to conclude that on any compact Riemannian manifold, any nondegenerate pair (P, Q) is joined by an infinite number of geodesies. The history of the problem of closed geodesies on a manifold homeomprphic to a sphere Sm is one with fine achievements; it is also a subject over which many mathematicians have stumbled. For example, in the foreword of his 1934 book, when discussing Poincare's work on existence of closed geodesies on a convex surface, Morse says the validity of Poincare's reasoning "has been questioned". Explicit objections are presented by Morse in his Chapter 9. These last two chapters of his book in fact are devoted to showing the existence of m(m + l)/2 closed geodesies of a special kind on a Riemannian manifold homeomorphic to S m . Yet this work of Morse is in error, as was pointed out by A. S. Svarc in 1960. Bott in 1954 gave a new proof which again was in error as Svarc noted. Some successes in this line have been accepted. Lusternik and Schnirelmann (1929) showed the existence of three closed geodesies without self-intersection on a surface homeomorphic to S2. G. D. Birkhoff in 1927 showed that a manifold homeomorphic to 5" had at least one closed geodesic. In the last two decades, these questions have "been pursued with very substantial success, especially by Soviet and German mathematicians. Klingenberg in a recently printed set of notes (Bonn, 1976) gives an account of the subject of closed geodesies complete with history. These notes include a proof of his newest theorem: On every simply connected manifold there exist infinitely many closed (prime) geodesies. (Let us hope ....) At this point, we add that two expositions of Morse theory have probably been much more widely read than Morse's original treatise. These are the books of Seifert and Threlfall and of Milnor. Especially for those who have learned their mathematics in recent decades. Milnor's book is to be recom­ mended. The language and concepts in the calculus of variations since 1736 (Euler) have suggested that extremals could be thought of as critical points of the functional, length, area, energy, etc. Our review emphasizes this analogy. In fact, this analogy is actual. The geodesies are critical points of the energy. One can put a manifold structure on Q in such a way that the energy J becomes a differentiable map on fi and an abstract Morse theory on infinite dimensional manifolds can be developed which yields Morse's theorems for geodesies. This

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is what Palais and I did in 1962-1964. One obtains a unification of the first and second parts of the material of our review. For example, a geodesic is literally a critical point, the definition of Hessian and index of an abstract critical point apply directly to give these notions for geodesies as a special case. One needs no longer to approximate ft by finite dimensional manifolds of broken geodesies. The abstract theory already applies to Q. Two contributions had made the way easier for us. First, Eells had put an infinite dimensional manifold structure (locally Hilbert or locally Banach) on certain function spaces in 1958. Secondly, Lang did the foundations of differential topology for Banach manifolds in 1962. Then Ralph Abraham, Palais, and I, in 1962, worked out systematically some theory of manifolds of function spaces and properties of manifolds of function spaces. The idea for this way of doing Morse theory is to replace the compactness condition of ft by a compactness condition on the map J, a condition which Palais and I called Condition C. More precisely, let ft be a complete Riemannian manifold, no longer compact or even finite dimensional, but defined as before with coordinate charts as open sets in Hilbert space. Let J be a (smooth) function on ft which is bounded below, has nondegenerate critical points (of finite index for simplicity) and satisfies: CONDITION C. If a( in ft is a sequence with J(o ( ) bounded and such that ||£>7(a,)|| tends to zero, then a, has a convergent subsequence. The result is that for such a function, the Morse inequalities are true, relating the type numbers defined by critical points of J and the Betti numbers of ft. The proof is essentially that given above via handle attaching. Now for the Morse theory of geodesies, one only has to show that the energy J on the loop space ft satisfies the above properties, with an appropriate manifold structure on ft. Palais and I used the Sobolev completion of ft, with norm defined by I? and first derivatives in L2. Actually it was Palais who wrote out the theory for this case in his article in Topology in 1963. Some of the new results on closed geodesies mentioned earlier were proved using Condition C as above. Recently, Tony Tromba seems to have found a drastic modification of Condition C, and developed a Morse theory for geodesies using infinite dimensional manifolds with a space of much smoother curves. Now I would like to spend the last few words of this review on the global variational calculus for more than one independent variable. It was in fact just this problem that led me into infinite dimensional manifolds in the early sixties. It remains a problem; although very recently, especially through the work of Tromba and Karen Uhlenbeck, there seem to be signs of progress. The most interesting case for more than one independent variable is minimal surfaces. !n the theory of Plateau's problem, I had been intrigued by a result of Morse-Tompkins and Shiftman in 1939. Their theorem asserted that if a Jordan curve in R3 spans two stable minimal surfaces, then it spans a third of unstable type. This was suggestive of a Morse theory for Plateau's problem.

560 BOOK REVIEWS

693

In the sixties I tried without success to find such a theory, or to imbed the Morse-Tompkins-Shiifman result in a conceptual general setting. Tromba and Uhlenbeck may now have succeeded in initiating a development of calculus of variations in the large for more than one independent variable. REFERENCES 1. C D . Birkhoff, Collected mathematical papers. Vote. I, II, III, Amer. Math. Soc, Providence, R.I., 1950. 2. W. KJingenberg, Lectures on closed geodesies, 2nd rev. ed., Univ. of Bonn, 1976. 3. J. W. Milnor, Morse theory, Ann. of Math. Studies, no. 51, Princeton Univ. Press, Princeton, N J., 1963. MR 29 #634. 4. M. Morse, The calculus of variations in the large, Amer. Math. Soc. Colloq. Publ., vol. 18, Amer. Math. Soc., Providence, R.I., 1934. 5. , George David Birkhoff and his mathematical work, Bull. Amer. Math. Soc. 52 (1946), 357-391; reprinted in Collected Mathematical Papers, Vol. I, Amer. Math. Soc., Providence, R.I., 1950, pp. xxiii-lvii. 6. H. Poincare, Sur les courbes definies par les iquation differentielles, part 3, Liouville J. (4) 1 (1885), 167-244. 7. H. Seifert and W. Threlfall, Variationsrechnung im Grossen (Morsesche Theorie), Teubner, Leipzig, 1938. STEPHEN SMALE

561 Comment. Math. Helvetia 55 (1980) 1-12

Birkhiuscr Verlag, Basel

Smooth solutions of die heat and wave equations

By

STEPHEN SMALE

Section 1 The motivation for this work was to try to give proofs for the existence of C* solutions of the heat and wave equations on bounded domains by Fourier methods. I wanted to show that the Fourier series (i.e., eigenfunction expansion) of solutions would converge not just in L2, but smoothly to smooth solutions. In contrast to more abstract methods, eigenfunction methods bring the existence theory closer to the practice of physics, and also to ordinary differential equations and numerical methods as well. In fact I found that by the addition of an extra term generated by the boundary of the domain, one could obtain this smooth convergence. For this proof one needs no significant estimates beyond those needed for the elliptic theory. And in general, our proof below gives sharp results by simple conceptual arguments. The difficulty with Fourier expansions can be seen in the problem: (Bujdt)(dzu/dx2) = f satisfying u(0, x) = v(x), u(t, 0) = u(t, 1) = 0. Here the data /:JR + x[0,1]->R and t>:[0,1]-*R are given, and u:l? + x[0, l ] - * R is to be found. If /(f, x) = L,,eZ* a„(t) sin nmt is a Fourier expansion which converges in C^O, 1], then fit, 0) = fit, 1) = 0. This is a special condition on /. We state now our problem in general for the heat equation. Let Q be a closed bounded set of R" with smooth (i.e., C°) boundary dO and let R + = [0, *>). Let L= -A, A the usual Laplacian, or more generally any self-adjoint real elliptic (smooth) operator on C°°(ft) with no eigenvalue equal to 0 (see Section 2). Suppose the following C data are given: f:R*x{l—*R, initial condition Mo: O -* R and Dirichlet boundary data g: JR+ x il -*■ R with g(0, x) = 0. We seek

I would like to acknowledge partial support for this work from the NSF (No. MCS77-17907), and hospitality from MIT, IHES (Paris) and the University of Geneva. 1

562 2

STEPHEN SMALE

a solution, a C~ function u: R* x {1 —*■ R such that —+Lu = f dt

on

R+x£l,

'

u(0, x) = u0(x),

all xeil

and

(1)

u(t, x) = g(r, x) all xedO. We may incorporate the data into /. More precisely let v = u — u0 - g and h = f-(dg/dt)-Lg-LUQ. The main problem becomes: Find Cv:R+ x(1 —► R such that (dv/dt) + Lv = h, u(0,x) = 0, xefl and v(t,x) = 0, xedO. Thus we may take u o >0, g « 0 in (1) and ask: Given / : J?+ x tl -* R, C, when is there a du C function u: R+ x Q -*■ R such that —+Lu = f dt ' on R* x ft, u(t, x) = 0 if f = 0 or x e dft?

(2)

For the answer define a sequence of polynomials in 2 variables by: P k ( L , T ) = t (-lYL'-'T

for

k = 0,l,2,....

i-o

Thus P 0 =l,

p, = L - T ,

P ^ I ^ - L T + T 2 , etc.

Main tkeoren A necessary and sufficient condition for the P k (L,T)/]^" n = 0, all k where T = (d/dt). Similarly, NASC for the existence of a C function u:Rx(l-*R f on Rx(l with u(0, x) = (d/dt)u(0, x) = 0 all x and u(f,

solution of (2) is that for the wave equation. A satisfying (d2ujdt2) + Lu = x) = 0, all x e dfl is that

PkO, T)f , . 0o= 0 and P k (L,T)f , . 0 = 0 for all k = 0,1, where T = (d2/dt2). Here /' denotes (a//df).

563

Smooth solutions of the heat and wave equations

3

The condition of f in this theorem is a kind of compatibility condition which can be translated to non-trivial initial data via the previously defined function h. While the necessity of the condition comes out of the proof, one can test directly for the necessity as follows. Suppose u is a solution, given / as in the first part of the main theorem. Then Pk(L, T)(L + T)u = Pk(L, T)f so (Lk+1±Tk+i)u

= Pk(L,T)f.

(3)

But by the boundary conditions, if xedft, then l**1^^ x) = 0 all t Similarly if r = 0, Lk+1u(t, x) = 0 all x. Thus if both t = 0 and x e 3/2, the left hand side of (3) vanishes and so does Pk(L, T)f. The same argument works for the second part noting first that (T+£.)«' = /'. Thus only the sufficiency has to be proved. One can reasonably ask: to what extent is our main theorem a new result in partial differential equations (apart from the methodology introduced here)? I have not seen it explicitly in the literature and the mathematicians in partial differential equations I've talked to were unaware of it. However, it overlaps and is close to, e.g., the work of Solonnikov in [5] and Rauch-Massey [9]. On the other hand Solonnikov doesn't discuss the wave equation and has a different generalization of the classic heat equation so that his compatibility conditions don't come out so neatly; they are only given by a recurrence relation. Rauch-Massey treat only the hyperbolic case, first order hyperbolic systems explicitly, and again these conditions are given by a recurrence relation. Also they suppose 155 0 in contrast to our treatment (in the hyperbolic case) where R x Q has no corners. They state that their methods can be applied to hyperbolic equations of higher order than one. In texts where heat and wave equations on bounded domains are treated, e.g., Friedman [2], [3], Lions [8], Treves [10], the results presented are not so sharp and the proofs seem more complicated. Treves does use eigenfunction expansions, but only to obtain weaker solutions. Also some of the PDE literature is not very clear as to what are natural initial value problems for the heat and wave equation on bounded domains. For example, in the well-known paper of Lax and Milgram [7], p. 182, it is stated: "if the initial function U0 is sufficiently differentiate, u(t) approaches u0 as t tends to zero not only in the Lj sense but pointwise." But later, p. 184, " . . . if UQ is sufficiently smooth, i.e., belongs to the domain of Am..." The domain of Am is basically one of our H+. And u0 can be even C°° and not in H*. Section 2 is devoted to the elliptic theory and section 3 gives the proof of the main theorem. Extensions and generalizations of the main theorem are discussed in section 4.

564 4

STEPHEN SMALE

Finally I wish to acknowledge brief but useful discussions with Vic Guillemin, Dick Palais and Bob Seeley among others. Section 2 Our methods depend heavily on the Sobolev spaces H'(fl) = H', s = 0 , 1 , 2 , . . . With fl<^Rn as in section 1, recall that H' consists of all real valued functions on 12 with (generalized) derivatives up through order s in L2(il). A complete norm on H* is given by

l « & - I f \Dau\2 0«|o|«« J "

where a is a multi-index, a = ( a „ . . . , a„), a, is a non-negative integer, £a, = |ot|, and Dau = (d^/tai) (d""/dx„). The norm is induced by a inner product and H° coincides with L2(fi). The Sobolev imbedding theorem asserts that H'+kcC(il) if fc>n/2 (where n - dim fl) and the inclusion is continuous for all $ 3s 0. Here C(fl) is the Banach space of C functions on ft, natural norm. See e.g. [3] or [10] for this and other background on Sobolev spaces. The Rellich theorem states that the inclusion H' -* H'~l is compact. Let Hi be the closure of C£ in H1 where C£ is the subset of C(C1) of functions which are zero on dfl. Let J: H " -► Hl be the natural inclusion and ff" n Hj = J~\Hl). Since Hi is a closed linear subspace of H1, and J a continuous linear map, f T O H j is a closed linear subspace of Hm. This space Hm H Hi is the set of all functions in H " which are essentially zero on dfl. It is a natural space for the Dirichlet boundary conditions for second order elliptic operators that we will consider, with m independent of the order of the operator or the dimension of il. These elliptic operators are linear maps L: C"(fl) -* C~(Q) of the form (Lu)(x)=

I

aa(x)Dau(x)

0«|a|«k

where a is a multi-index, k is the order and aa: fi —*■ R are C functions (all functions are real valued here). We will assume k = 2, for our notation. Our standing hypotheses on L are L is elliptic.

(1)

585 Smooth solutions of the heat and wave equations

S

For each x e f l the polynomial £ w _ k aa(x)^a + 0 all | e R " , if f # 0 , where L is self-adjoint.

(2)

I.e., (Lu, v) = (u, Lv) for all u,veC% where (u, v) denotes the L2 inner product. L:C£->C°°(Q) is injective. (no "eigenvalue" is zero)

(3)

Condition (3) just make things go more simply. If L satisfies (1) and (2) it can be "translated" to satisfy (3). As we remarked above, we use second order notation for L throughout. This comes into the boundary conditions in particular. But the proofs go over im­ mediately to arbitrary order. Thus we suppose L is second order and so 32v

(Lu)(x)« I

H

aii(x)T±:+l

l«i.j«n

OXi0Xi

flu

K(x)—+c(x)u(x) k

_,

dXfc

where (a^x)) is a negative definite matrix for each x, negative definite rather than positive definite by our convention. The map L:CZ^> C°(il) extends naturally to L: tf" D Hj -* H"" 2 . Fudaweatal theorem of elliptic theory For each m = 2,3,...L:HmC\Hl-*Hm~2 is an isomorphism. That is, Lhasa bounded linear (2-sided) inverse G : f T " 2 - » f f " n H j . This could be considered as a regularity theorem, including boundary regular­ ity. For a proof see e.g. [3]. The maps, L, G and inclusions J described above make sense with various domains; sometimes will use them without specifying this domain if the context makes it clear. A second theorem from the elliptic theory is that providing L with eigenfunctions. Etfeafuction theorem Suppose given an elliptic self-adjoint operation L: Co*—» C" as above. Then there exist a non-decreasing sequence of real numbers Alt A 2 ,... called eigenvalues,

566 6

STEPHEN SMALE

with Aj -* oo as i —* <», and a sequence of elements fa of C£ called eigenfunctions so that Lfa = ktfa. Furthermore the fa constitute a Hilbert basis for L2(0) = H°. We sketch how the proof of the eigenfunction theorem follows from the Fundamental theorem. Consider

tfnHj-^f^nHj

Then G0 = GJ is compact using Rellich and self-adjoint relative to a Hilbert structure on H 2 n H 0 induced from that on H° via L. Apply the spectral theorem for compact self-adjoint operators (the simplest spectral theorem, see e.g. [6]) to G0 to obtain real m, ifc e H 2 n Ho with G0ift = tt.^. Take A; = 1/m indexed so that the Aj are non-decreasing, and fa = Ajift. The tfa are a Hilbert basis for H 2 fl H 0 and Lipi = kipt = ^ a basis for H°. Finally, the repeated use of the Fundamental theorem applied to Lfa = A ^ implies that fa e ff" n H 0 every m and thus faeC°° by the Sobolev theorem. Define HJ, m = 0 , 1 , . . . as the closure of the subspace of H™ spanned by the eigenfunctions fa. For example it follows from the above that H j = H°, H i = Hj, Hl = H2C\Hl0 and that H J <= / T n H 0 for m > 1. But H j is a proper subspace of H3DHl since H 0 is a proper subspace of H 1 and L : H 3 n H 0 - + H 1 is an isomorphism. In fact H j = L"'(Hj). Since in general H J is not all of H m n H 0 , the simple expansion by eigenfunctions is not sufficient to give smooth solutions for the heat and wave equation. It follows from the eigenfunction theorem that (the restriction) LiH+^HZ'2 is an isomorphism with inverse G:H^~2-*H^, ms*2. Actually one may define HJJ? without the use of eigenfunctions by H2?+2 = Gm{H2t) = L-m(Hl),

Hi m + I = Gm(Hi),

Consider the composition [Go: H m n H 0 -» H m n H 0 , I H m D H 0 — ^ Hm'2 - ^ + H™ n H 0

m>0.

567 Smooth solutions of the heat and wave equations

7

PROPOSITION. The image of G'0 is contained in H£ for s3*[m-1/2], the largest integer in (m -1/2). 77tis is false for s <[m -1/2]. The proposition is a kind of spectral theorem for the operator G 0 : H m H H 0 -* H D H\. It implies for example if m > 2, there is no Hilbert structure on Hm n H 0 so that G0 is self-adjoint. On the other hand modulo HJ, G0 is nilpotent, and on HJ, G0 has the spectral theory defined by G0d>i = (1/A,)d>,. The proof of the proposition can perhaps best be seen by studying the following diagram for m even (m odd goes similarly): m

H6 U H* <-i-H*nH, U

/'

u

tf^-tfDHi*

F

u X

u

u

HJ «

H$

f^DH^Hj*

Here F = L " ' ( H 4 n H i ) with L : H 6 n H j >H4 etc. Now G o ^ ' O H o 1 - * 4 l H D H 0 is of form G0=GJ = L J. So im(G0)^Hi. Similarly G 0 : H 6 n H 0 - » H^DH'o is given by G0 = GJGJ=G2J2 and im(G 0 )c im(G2Hl) or H$,. Con­ tinue in the same way to finish the proof.

Section 3 The goal of this section is to prove the main theorem of section 1. We do that first for the case that the data can be expanded in a Fourier series. More precisely: PROPOSITION. Suppose t -► w„ 13=0 is a C°° curve in HJ andve HJ. Let I satisfy m -2/3= 2, / > 0 . Then there is a unique C curve t-*v,in H+~2' such that J,(d«,/df) + Lv, = Jj/w, with v0 = Jv. Here J,:HJ- 2 , -»- HJT2<'+1) and J: HJ? -+ H y 2 1 are all inclusion maps. One may relax the C°° condition on t -* w, as the proof shows.

568 8

STEPHEN SMALE

COROLLARY. Under the same hypotheses, there exists a unique C' curve t-*v,in HZ'21 such that v0 = Jt5 and (1+ G0T)vt = Jw,. Here I: HJ~21 -*■ HS~2t is the identity and T=d/dt. For the corollary simply apply G to the equation of the proposition. The main part of the proposition is contained in the following lemma. LEMMA 1 (of Fourier type). Under the hypotheses of the proposition, there is a unique C1 curve v, in H+~2 such that v0 = Jlv and J, — t>, + Lu, = J,w„

(1)

at

where Jt:HS-* H+~2 is the inclusion. Postponing momentarily the proof of the lemma, we see how the proposition is a consequence via a simple induction. Say v, is a Ck curve in H+~2k satisfying (1), / , the appropriate inclusion. Apply / 0 :}^" ! k -»f^""" 2 to both sides to obtain that v, is c k+1 in HJ"" - 2 . For the proof of the Lemma, first examine just what convergence in HJ means. Say m = 2k (we only use these results for m even; and for m odd, the proofs are similar). Then since Lk :H+—*H% is an isomorphism, £"., Cj^, con­ verges in H ; if and only if ICjAfcft converges in H5, = L 2 or equivalently I|c,Ak|2<=o. Now expand the data of the lemma in a Fourier series, i.e., we may write v = Ei°-i c,^ and w; = £°1, a,(t)t in HJ. Hence the c, are constants and the a,(f) are real valued functions of t. In fact, Oj(f) is C°° since it is the projection of a C° function. For u, =TT-i t>,(t)„ the equation of the lemma is

or for each i, b{(i) + AA(r) = a,(i),

M0) = c,.

The unique solution is (see practically any book on ordinary differential equa­ tions)

M0 = «" A '{[ a,(S)eA'd* + c,].

569 Smooth solutions of the heat and wave equations

9

We claim that the curve u, defined above in terms of the b,(t) converges in HJ and satisfies the properties in the lemma. Since for each fs»0, e - x , '«l except for a finite number of i, and Ec,d>( converges in HJ, it follows that £ Cje"A''d>j also converges in HJ for each t. The continuity in t of this sum is an easy check which we leave to the reader. Next we show that X4(')4>, converges to a continuous function of t in HJ where m = 2k and 4 ( 0 = f ai(s)e^") ds. Jo First estimate by Cauchy's inequality, \d,(t)\*=£\ai(s)\2ds£e2K>*-»ds = ('|aj(S)|2dS^[l-e-"''] Thus

I !4(0|2A2k« I [faisWds + K A,»l/2

A,»l/2 J0

where K=max|Lkw,|JI<. 0«sj«r

Thus £ 4(0^i converges in HJ and so does u, = £ M0R satisfying zero boundary conditions such that Tu + Lu = f on R+xil

(2)

where T = d/df. Let f,(x) = f(.t,x); then it is easily seen that the map R+-^Hk given by t-*f, is a T curve in Hk any k. Let J:H k + 2 nH5-*H k be the inclusion and consider the following version of (2). TJu + Lu, = f„ u, a curve in H k + 2 nHj,

u„ = 0.

(3)

570 10

STEPHEN SMALE

Let m = k + 2 and apply G: Hk -► f f n Hj to both sides of (3) to obtain (I+TG0)Mt = g„

M.eHTlHi,

Uo = 0

(4)

where the datum g, = Gf, is now a curve in H"" n Hj. This form suggests trying to invert 1+ TG0 or to look at: u, = [7-(G 0 T) + (G0T)2 (I+TG0)v, = (-G0T)'

M

+(-l)'(G0T)']g, g, = w,

+ vt

(5a) (5b)

For s large enough w, e HJf by the proposition of section 1. We may apply the above Corollary to solve (5b) for u„ and with an appropriate boundary condition at r = 0, put this in (5a) to obtain our desired solution u,. Motivated by the above, we proceed more formally. Set m = 2k = 41,1 some positive integer. The data / define a curve t -* f, in m 2 H ~ . Let g, = Gf, be the corresponding curve in / T n H i , C°° in t. Define y, = [I-(G0T)+ ■ ■ ■ +(-G 0 T) k - 2 ]g„ which is a C~ curve in H T I H j for 0 « t < 00.

LEMMA 2. y0€H% Proof. Denote by C„(/) the condition of the theorem Pq(L, T ) / | , _ 0 G C ^ , q = 0 , 1 , . . . . . We will show that if Cq(f) for q *£ k - 2, then I ^ 2 ( - TG0)'g, |,_0 e HJ. Let J:H' -* H'~2 be the inclusion for various / and suppose /, is the curve in FT"2 defined by the inclusion C^H***- 2 . Define R, =I?_o(-TJ) i L q -'/ ( |,_0 and note Rq = (-T/) q /t |«-o + i-Rq-i- This latter could be used as an inductive definition of Rq starting with R_, = 0. Rq lies a priori in Hm"2(q+1), but Cq(/) implies that Rq lies in H m_2(,, ' , " l) nHj. Now suppose C,(/) is true for q*zk-2. Then J? k _ 2 eH 2 nHi = H3„ so Gk_IJRl[_2€JF^. By the inductive definition of R, above, the L used in the definition of Rk-2 have domain some H'HHo so GL = identity. Thus G k - I R k - 2 = lS- 2 (-TG 0 )'g, |,. 0 eH*. The curve w, = (-G 0 T) k_1 g, lies in H£ by the proposition of section 2. Let J:HT\Ho and •/:/*£->/*£ denote the inclusion. Apply the Corollary of the proposition in this section to obtain v, in H* of class C in f such that (/ + G0T)v, = Jwn v0= - Jy0. Now define u, in Hk n H j by u, = Jy, + u,; so u, is C' in t, uo = Jyo + vo = 0 and (/+GoT)^ = /g, in H k DH£. If different /, say /,, l2, above are chosen, the corresponding u defined by the above process agree in Hk n Hj where k = 21, 1 = min (/lf l2) using the uniqueness in the Corollary. Thus we obtain a u„ which lies in each Hk n Hj. Thus by the Sobolev theorem, u, is C°° and we have proved thefirsthalf of the main theorem.

571 Smooth solutions of the heat and wave equations

11

For the second part, the above proof, with a couple of modifications which we state, is applicable. The first modification is in lemma 1 and its consequences. One obtains a different ordinary differential equation, namely fc7(i) + AA(i)«fl,(i),

i = l,2,...

with b,(0) = c„ fr{(0) = 4 where u, = £ M0i = "o. Efli(0 0 ,,s

,

,

, sin tyJX, [' A

v i

Jo

. , sin (t-2)J A, vAt

The finite number of equations with Aj<0 are handled as easily. Now one proceeds as before, with similar estimates to get convergence of £ b,(t)4>, = u,. The other modification relates to Lemma 2; but here just apply that construc­ tion of / and /' as well.

Section 4 This section is a series of remarks on extensions and relations to other problems of the above. Section 3 of this paper could be considered as a theory of separation of variables for boundary value problems in PDE. It works well for problems which are the product of understood problems. Thus the evolution problems above are the product of space and time problems. We give more examples to illustrate this point. Consider Au = f on the rectangle Q = flaxQh where Oa=[0, a], A, = [0, b]. Given C~/:fi-*i?, find a C~ solution u:O^R such that u = 0 on dO. Write (t,x)eilaxilb and Au = (ti?u/dt2) + (di2u/dx2)= Tu-Lu and proceed as before to obtain NASC on / for the existence of a solution u. The Fourier lemma and proposition at the beginning of section 3 must be replaced by a simple spectral analysis of T (similar to that of L). A second example is the wave equation on the same domain, (T+L)u = f on O with Dirichlet boundary conditions u = 0 on dO(\). This problem has been considered By Fritz John [4], V. Arnold [1] and others. Now the above analysis

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STEPHEN SMALE

applies. Besides the compatibility condition on /, one needs in general that the ratio a/b be not rational (or not even close to rational?). Finally we list some ways in which the main theorem might be extended. (A) If condition (3) on the elliptic operator is dropped, i.e., some eigenvalues are allowed to be zero, the methods extend easily to yield similar results. (B) If L is not self-adjoint, one could no doubt replace the Fourier lemma of section 3 by a different existence proof, the rest being the same as before. (C) The extension to complex coefficients or systems should not require substantial changes. (D) The operator T in the theorem of section 1 could be replaced by any ordinary linear differential operator with leading coefficient 1. Then the results would have to be modified at the boundary condition t = 0. Schrddinger's equa­ tion on bounded spatial domains thus can be included. (E) il could be a compact manifold with boundary (F) Perhaps one could obtain C solutions to Navier-Stokes on compact (1 <= R", dfl smooth, for small time via eigenfunction expansions this way. (G) In the main theorem of section 1, L is time independent. I am not sure how the extension of this result to the case of time dependent L should go.

REFERENCES [1] ARNOLD, V. I., Small denominators I, Translations A.M.S., Prov., R.I., 2e series, 46, 213-284. [2] FRIEDMAN, A., Partial Differential Equations of Parabolic type, Englewood Cliffs, N. J., Prentice Hall, 1964. [3] , Partial Differential Equation, New York, Holt, Rinehart and Winston, Inc., 1969. [4] JOHN, F., The Dirichlet problem for a hyperbolic equation, Amer. Jour. Math. 63 (1941), 141-154. [5] LADY2ENSKAJA, O., SOLONNIKOV, V., and URAL'CEVA, N., Linear and Quasi-linear Equations of

Parabolic Type, Vol. 23, Translations of Mathematical Monographs, A.M.S., Prov., R. I., 1968. [6] LANG, S., Analysis II, Addison-Wesley, Reading, Mass., 1969. [7] LAX, P. and MILGRAM, A. Parabolic Equations in Contributions to the theory of Partial Differential Equations (Ed. by Bers, Bochner, John) Annals of Math. Studies #33, Princeton, 1954. [8] LIONS, J. L., Equations Differentielles OperationnelUs, Springer, Berlin, 1961. [9] RAUCH, J. and MASSEY, F., Differentiability of solutions to Hyperbolic Initial-Boundary Value Problems, Trans, of A.M.S., l t 9 (1974), 303-318. [10] TREVES, F., Basic Linear Partial Differential Equations, Academic Press, New York, 1975. Dept. of Mathematics Unitiersity of California Berkeley, Calif. 94720 U.S.A. Received February 2, 1979

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17

On the Contribution of Smale to Dynamical Systems J. PALIS

It is rare, and, in fact, one may not ever be so lucky in one's lifetime, to follow rather closely a dramatic change, an absolute upgrading of a branch of Sci­ ence. In our case, Mathematics, and, more specifically, Dynamical Systems, whose foundations were set by Poincare about 100 years ago and enriched in the next decades by Birkhoff, Lyapunov, Hadamard, Perron, Morse, Fatou, Julia, and others: One aims at the description of the asymptotic behavior of solutions of differential equations (or trajectories of iterated maps) without necessarily integrating them so as to find explicit expressions for the solu­ tions. Also, in the appropriate setting, one looks for the behavior of solu­ tions of "typical" equations and of "typical" solutions. After a noble infancy, an upturn in the subject took place in the 1960s with two main developments: the construction of the hyperbolic theory for general systems, led by Smale, and the KAM theory (after Kolmogorov, Arnold, and Moser) for conservative systems, which incidentally is mentioned here only briefly. The hyperbolic theory stands today as a very solid and rather well-under­ stood region of the large world of dynamical systems. It has, in the tradition of the founders Poincare and Birkhoff and almost by conception, a global character: After all, one searchs for the behavior at large of many trajectories of a system. But perhaps due to his topology background, Smale excelled at that: His vector fields were always defined on manifolds (their solutions often wrapping around in the phase space), his diffeomorphisms were global Poin­ care maps of vector fields defined on one higher dimensional manifolds, his stable and unstable manifolds (of points, later of sets of points or "foliations") were always globally defined. Of course, the hyperbolic theory, and par­ ticularly Smale's contribution, goes much beyond its global character. Yet, one may ask nowadays, how could dynamical systems be otherwise studied? Well, if one looks at the literature of the 1950s, one may not see this global setting that often or that sharply as offered by Smale from the beginning, even when he was just a newcomer to the subject, and yet dared to propose what he thought should be some of the main goals of the theory. Indeed, 165

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sometimes one has the feeling that if this perspective had been present, some of the good work developed at that period could have been at least better focused or better integrated with other current research. But let us go back to the hyperbolic theory and how its construction in the 1960s was orchestrated by Smale with the help of, and in fact with contribu­ tions of fundamental importance from, colleagues, the Russian School, his students, students of his students, and so on. In 1937, Andronov and Pontryagin introduced the notion of topological equivalence (same dynamical behavior) for vector fields: One is equivalent to the other if there is a homeomorphism of the ambient space which sends trajectories of the first onto trajectories of the second. A vector field is struc­ turally stable if it is equivalent to any Ck-small perturbation of it (the vector fields are taken to be & with k £ 1). Similar definitions can be given for maps of a manifold into itself: f is stable if it is topologically conjugate to Ck-small perturbations of it, i.e., for any g which is Ck-close to f there is a homeo­ morphism h of the ambient manifold such that hf — gh. In that early work, Andronov and Pontryagin considered vector fields on the two-dimensional disk, say transversal to the boundary, from the point of view of (structural) stability. They stated that such a vector field is stable if and only if —it has only finitely many singularities and periodic orbits, all of them hyperbolic, —it exhibits no saddle-connections. Recall that a singularity of a vector field is hyperbolic if no eigenvalue of its linear part (Lyapunov exponent) is pure imaginary. When some eigenvalue has positive real part and some negative real part, the singularity is called a saddle, otherwise a source or a sink. A saddle-connection means an orbit joining two saddles, i.e., having one of them as its co-limit set and the other as its a-limit set. Notice that on the 2-disk or the 2-sphere, the two conditions of Andronov and Pontryagin imply a third one: every orbit has a singularity (fixed point) or a periodic orbit as its (o-limit set and as its a-limit set. This is a consequence of the Poincare-Bendixson Theorem. Adding this third condition, Peixoto generalized the statement of Andronov and Pontryagin to any two-dimensional compact orientable sur­ face, proving in 1962 that any vector field on the surface can be approximated by one of these. The relevant case of vector fields on the 2-torus without singularities was independently done by Pliss and by Arnold. Back in 1959, when he got acquainted with the work on two-dimensional flows that Peixoto was then developing, and inspired by it, Smale imme­ diately defined a higher-dimensional version of the set first introduced by Andronov and Pontryagin on the 2-disk: the limit set is formed by a finite number of hyperbolic singularities and periodic orbits and their stable and unstable manifolds intersect transversally. This was quite a step: Instead of no saddle connections, in two dimensions, he was talking about transversality between stable and unstable manifolds in a much more general setting. Also, he

577

17. On the Contnbution of Smale to Dynamical Systems

167

defined a corresponding set of diffeomorphisms with similar dynamic condi­ tions. Note that instead of the limit set we can define these systems using the (usually larger) nonwandering set: A point is nonwandering if any neighbor­ hood of it intersects some of its positive or negative iterates. It is a bit puzzl­ ing why Smale changed from one concept (limit set) to the other (nonwander­ ing set) to define these systems. My guess is that he became more acquainted with the work of Birkhoff who often deals with the nonwandering set. Also, Smale played on the safe side from the point of view of stability when he imposed more structure to the system (the nonwandering set is usually larger than the limit set). Anyhow, he never really proved that the two sets of systems using the two different concepts are the same (the reader may check that it is not that simple). For such a vector field or diffeomorphism, the ambient manifold become decomposed in a finite number of stable manifolds, i.e., embedded submanifolds of several dimensions (indices) from possibly zero (point repellers) to the top one (attracting orbits). To Smale, this cried out for the topologically fundamental Morse inequalities, done classically for nondegenerate gradient flows and, indeed, this was his first result in dynamics! Thorn, from whom Smale had learned transversality not long before, christened the set of flows or diffeomorphisms as above Morse-Smale systems. Perhaps Andronov and Pontryagin should also be remembered in connection with these systems, but the grandiose view about their place in dynamics was due to Smale. In fact, by 1959 he was conjecturing that on any closed (compact, boundaryless) manifold the Morse-Smale systems formed an open and dense subset of all systems (flows, diffeomorphisms) of class Ck, k ^ 1, in the C* topology. Moreover, each of its elements should be structurally stable (dynamically robust). Since these systems are dynamically rather simple, he was just pro­ posing the world (of dynamics) to be mostly simple: Any system could be approximated by a Morse-Smale one (and thus with a limit set consisting only of finitely many singularities and periodic orbits) and any Morse-Smale system would be dynamically robust. In his own words, he was at the time extremely naive about differential equations and also extremely presumptuous and "if I had been at all familiar with the work of Poincare, Birkhoff, Cartwright and Littlewood I would have seen how crazy this idea was," refering to the possibility of approximating any system by a simple one. Yet he was on the threshold of his fundamental contribution to Dynamical Systems. First of all, although the robustness (or stability) of Morse-Smale systems was only proved about 7 years later, it was clear to him that his program made a lot of sense for gradient systems. In this context, he was even doing some bifurcation theory by cancelling singularities (handles) with indices dif­ fering by 1 through what we call saddle-node bifurcations. In particular, he showed that every gradient field is isotopic to one exhibiting only one source (point repeller) and one sink (point attractor). And this was a step toward his famous solution in 1960 of the generalized Poincare Conjecture in dimen-

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sions greater than foun A smooth closed manifold of the homotopy type of the sphere is homeomorphic to the sphere. At this point, he was visiting the Instituto de Matematica Pura e Aplicada (IMPA) at Rio de Janeiro in 1960, when he got a (dramatic for him) letter of Levinson. There were certainly robust examples of flows and diffeomorphisms in three and two dimensions, respectively, for which the limit set contained infinitely many coexisting periodic orbitsl This was clear from the existence of transversal homoclinic orbits in the work of Levinson and that of Cartwright and Littlewood on the Van der Pol's equation with friction ("rear world) and, coming straight to the point, a result of BirkhofF in 193S stating that a transversal homoclinic point of a surface diffeomorphism is an accu­ mulation point of periodic orbits. Amazingly enough, 70 years before Poin­ care was receiving a dramatic letter saying that there was something wrong concerning homoclinic orbits in the last part of his prize memoir (King Oscar of Sweden) on the stability of the solar system. Before the final publication of his essay, in fact that issue of the Acta Mathematica was already printed and about to be distributed, he got it straightened up: A second corrected printing was made! Of course, Poincare still deserved that and all other possible prizes. This incident may have been in his mind when he wrote in Les M&thodes Nouvelles de la Micanique Cileste: "On sera frappe de la complexity de cette figure, que je ne cherche memc pas a tracer. Rien n'cst plus propre a nous donner une idee de la complication du probleme des trois corps et en general de tous les problemes de Dynamique — " Homoclinic orbits, that puzzled both Poincare in 1890 and Smale in 1960, are orbits of intersection of the stable and unstable manifolds of saddles (excluding the saddles themselves). Again curiously (symptomatically?), Lit-

g(R)

\ZJ FIGURE 1

g2(R)

579

17. On the Contribution of Smale to Dynamical Systems

169

tlewood refers to his work on the subject, done partially with Cartwright, as the "monster." Studying Levinson's work, Smale considered the Poincare transformation, say / on a section to the flow. He abstracted from a transver­ sal homociinic orbit of this transformation the horseshoe map: a map g send­ ing a rectangle R into a horseshoe as in Fig. 1. The horseshoe map can be taken as (or it is conjugate to) a high power of the original transformation f restricted to a rectangle R* containing the local sta­ ble manifold of the associated saddle. See Fig. 2. So he was indeed showing that if a diffeomorphism exhibits a transversal homociinic orbit, then this orbit is contained in an invariant hyperbolic Cantor set, like A = f]mt z g"(R\ and a power of the original map restricted to the Cantor set is conjugate to the shift on two symbols. The ^-invariant set A is hyperbolic in the sense that there are two continuous bundles over A in which the derivative dg acts as a contrac­ tion and as an expansion, respectively. The corollaries were remarkable: In any dimension, a transversal homoclinic orbit implies a rich dynamics, including nontrivial minimal sets, in­ finitely many coexisting periodic saddles, infinitely many coexisting trans­ versal homociinic orbits, all "living together" densely in an invariant Cantor set. And, as it is clear from the previous statement, this dynamic phenomenon was not pathological; much to the contrary, it was robust (open) and structur­ ally stable. Moreover, it lived in any manifold of dimension two or larger in the case of diffeomorphisms or, via suspension, for flows in dimension three or larger. It was also realized later that this structure (the horseshoe) also implied positive topological entropy.

f n (R*)

FIGURE 2

580

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J. Palis

Let me make a little disgresion as to how often this phenomenon occurs. / have been conjecturing in the past 5 years or so, that the systems exhibiting a horseshoe (or a transversal homoclinic orbit) together with the Morse-Smale ones form an open and dense subset of all C* systems, fc ;> 1. (Actually, this conjucture is implied by one stating that the hyperbolic systems, or Axiom A systems to be defined later, together with the ones exhibiting a homoclinic bifurcation form a Ck-dense subset). Noteworthily, the (stronger) conjecture has been proved for C1 surface diffeomorphisms by Aranjo and Mane. It can be interpreted as a bit of vindication for Smale's original proposal (MorseSmale systems are dense) or its later modifications (stable or limit set-stable systems are dense) as we shall discuss in the sequel. Still, nowadays we are much focused on attractors, i.e., sets attracting an open or at least a positive Lebesgue measure set of orbits. And, worth noting, the set of orbits attracted to a horseshoe has measure zero, at least if the system is differentiable of class C2. (Ten years later, Bowen provided examples of C 1 horseshoes with posi­ tive Lebesgue measure.) Concerning the horseshoe, another important fact is that every point of the Cantor set has stable and unstable manifolds (locally embedded Euclidean spaces), defined as the set of points whose orbits are positively and negatively asymptotic to the orbit of the initial point. They form a net: The Cantor set is locally the product of two Cantor sets, which are the closure of (transversal) homoclinic points with the same local stable and unstable manifolds, respec­ tively. By abus de langage, we refer to them as the stable and unstable folia­ tions of the Cantor set: The leaves are, respectively, the stable and unstable manifolds of the points of the Cantor set. Now armed with Morse-Smale systems, which were still only conjecturally stable, and the horseshoe, a prototype of a large set of stable systems with a much richer limit set, Smale began an admirable battle to have a global description (even if only conjecturally) of "most" of the world of dynamics, still hoping that the stable systems formed an open and dense subset of it. He then went to Moscow and Kiev, where he met Kolmogorov and, in his words, an extraordinarily gifted group of young mathematicians: Anosov, Arnold, Novikov, and Sinai. Recently, I have spoken to Arnold and Anosov about this encounter with Smale, and both (and probably the others too) have a very vivid image of it. In their intense conversations, Smale presented the horseshoe and discussed other possibilities for stable systems. In particu­ lar he conjectured that the geodesic flows on negatively curved closed manifolds were stable and, following Hadamard, the closed (periodic) geodesies would be dense in the whole manifold. The world of possible stable systems was growing from the ones with finitely many orbits in their limit set (Morse-Smale sys­ tems) to some for which the limit set was the whole manifold, as in the geodesic flows above. And somewhat in between, lay the surely stable horse­ shoe. Thorn was also asking about the hyperbolic linear toral diffeomor­ phisms: They were globally hyperbolic, their periodic orbits being dense in the whole manifold and these properties were persistent under perturbations.

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17. On the Contnbution of Smale to Dynamical Systems

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Were these maps stable1! The visit of Smale to Russia and Ukraine was most fruitful: Not long in the future, Anosov would announce his fundamental results answering positively the last questions, as we shall mention in the sequel. That was perhaps the most visible mathematical consequence of the trip, and a remarkable one. There is a paper of Smale that I particularly like to discuss: his address at the International Congress of Mathematicians at Stockholm in 1962. When I first read it about 3 years later, I was initially amazed and then taken by it. It looked to me like an unfinished painting with several superposed sketches: Was that allowed in Mathmatics? It was certainly very inspiring then as well as three (and many more) years later. Some new results were announced in the paper, but mostly it made transparent his struggle at the time to reach the right concept of hyperbolic systems (to be called later Axiom A systems by him), still dreaming the semi-impossible dream that they could be both {structurally) stable and form a dense subset of all systems. It is curious to remark that at that very moment, a remarkable work that would remain unknown to us for the next 10 years was being developed: Lorenz was proposing the impossibility of the second half of the dream through his still experimental robust (open) strange attractors (unstable). Unaware of that, Smale himself would show in 1965 that the stable systems are not dense. And, in the seventies, we became definitively aware that the world of dynamics is much bigger (and fascinating and per­ haps not so indomitable) than the set of hyperbolic systems and its boundary. Back to Smale's address at the International Congress of 1962. Smale offered a possible set of axioms and a variation of it to be satisfied by the "hyperbolic systems," but the definite concept would be achieved only a bit later. Among the novelties, he announced that for the elements of a residual subset of C* systems, any k ^ 1, the fixed and periodic orbits are hyperbolic and their stable and unstable manifolds all mutually transverse. This was also and independently proved by Kupka. Smale then suggested the question whether such a diffeomorphism could exist on the 2-sphere without displaying a sink or a source. The question has been considered by a string of mathe­ maticians in the span of 25 years: Bowen and Franks, Franks and Young, Gambaudo and Tresser and van Strien, with a positive answer for Cl, C2, and C°° diffeomorphisms, respectively. Is there an analytic example? As before, let us call an invariant set for a dynamical system an attractor if it contains the co-limit set of all points in an open (or a positive Lebesgue measure) set in the phase space. Another question that one may say is cer­ tainly inspired by the same paper is: Can a diffeomorphism on S2 exhibiting some attractor always be approximated by one exhibiting a sink? An important result of Plykin in the mid-seventies showed that this was not so: There are robust or fully persistent (under C* small perturbations) attractors for diffeo­ morphism on S2. The Plykin attractors are, in fact, "hyperbolic," as we shall define later, and (structurally) stable. Smale reminds the reader that the stability of the Morse-Smale systems still

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remained an unsolved problem, stressing that they exist on every manifold and that they satisfy the Morse inequalities. He also hints at the centralizer problem: Given a diffeomorphism, say of a closed manifold, how "large" is the group (under composition) of diffeomorphisms that commute with it"! In particular, is it common for a diffeomorphism to embed in a flow, i.e., to be the time-one map of a flow? One of the first results in this direction, a rigidity result, is due to Bochner and Montgomery and concerns the differentiability of a flow, i.e., a one-parameter group of C* diffeomorphisms depending C on the parameter, r <,k: The result states that r = 0 implies r = k. Again, a number of mathematicians have since worked on the problem; for example, Kopell, Lam, Mather (responsible for synthesiz­ ing an obstruction invariant to imbeddability in a flow), myself, Brin, Ander­ son, Walters, Morimoto, Sad, Yoccoz and myself, Rocha, showing in several contexts the rigidity of centralize?s: They are commonly (open and dense or residual subsets) very small, consisting just of integer powers of the initial map (or constant multiples of the initial vector field). Rigidity of group actions has become a rather large topic of research (Katok, Hurder, Zimmer, Ghys, etc.). Smale also asks if any diffeomorphism (flow) can be C* approximated by one with a periodic orbit for k ^ 1, a form of the difficult and important "closing lemma" proved by Pugh 4 years later in the Cl category: Residually the nonwandering set is the closure of the periodic orbits. The paper mentions the horseshoe and that the stability of the hyperbolic linear automorphisms has been resolved by him (unpublished), by Arnold and Sinai ("my only paper with a wrong proof," Arnold tells me), and, it announces the fundamental results of Anosov in that line without any further detail. The last paragraph of the 1962 paper shows his enthusiasm: "Certainly the main problems stated here are very difficult. On the other hand, it seems quite possible to us that this field may develop rapidly and already as indicated here, there have been some initial steps in this direction. By 1965, Smale cleverly combined two diffeomorphisms with a very differ­ ent dynamical behavior, that he knew were persistent under small perturba­ tions and, in fact, stable, to provide an open set of unstable diffeomorphismsl As a consequence, his dream that stability would be a property displayed by the elements of a dense subset of systems was shattered (was it?) and shat­ tered by himself. He wrote at the end of the paper T h e example in this paper surely reduces the importance of the notion of structural stability." Still he also manifested his usual tenacity in creating new and more complex sce­ narios as needed, promising a paper "based on an axiom, which we consider to be of central importance, axiom A "In fact, he was still hoping that some kind of stability, say limit set-stability or nonwandering set-stability, was a property displayed by a dense (and open) subset of dynamical systems. More importantly, he had finally the set of axioms, now synthesized in the letter A, he was so much looking for The concept of hyperbolic systems. Thus, the hyperbolic theory was about to get its contour.

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The example introduced by Smale was based on the product of the north pole-south pole diffeomorphism on the circle S1 (a source and a sink) and the linear hyperbolic authomorphism of the torus T 2 mentioned above. Thus, the example was constructed just on T 3 . He then produced a "bump" to provide a persistent lack of transversality (sometimes a tangency) between stable and unstable manifolds. That is, he provided an open set of C* diffeomorphisms on T 3 , it > 1, such that the elements of a dense subset of it exhibit an orbit of nontransversal intersection between stable and unstable mani­ folds of periodic orbits. (Recall that residually on the space of diffeomor­ phisms, stable and unstable manifolds of periodic orbits are in general position.) This fact was clearly an obstruction to stability: Topological con­ jugates must send periodic orbits and their invariant manifolds (and thus their intersection) onto similar orbits and invariant manifolds (and their in­ tersection). Later, Williams and Robinson used the same principle to provide an open set of unstable two-dimensional diffeomorphisms on the bitorus: It was like a "north pole -south pole" example, the attractor and repeller being the (nontrivial) ones constructed by Smale in what he called diffeomorphisms derived from Anosov (in particular, derived from hyperbolic torus automor­ phisms), to be again mentioned in the sequel. And that is not all: Again and again the same principle would be used to shatter the "modified dream" that the limit set-stability or nonwandering set-stability would be a dense property for diffeomorphisms in two or more dimensions (Abraham and Smale, Newhouse, Simon, Shub, Mane, and so on). But let us not invert history and come back to this point later. A last word here on the "con­ structive" side: Several of these "shattering" examples have recently inspired some insights in the dark realm of dynamics (complement of the hyperbolic systems). We are now arriving at a masterpiece of the mathematical literature: Smale's celebrated paper of 1967 "Differentiable Dynamical Systems." Noth­ ing else synthesizes better his permanent legacy to dynamics. Like the 1962 paper, this was exuberant in new ideas, new problems, new or unfinished paths of research and, in fact, so much more. Yet it was also much more solid, coherent, overwhelming: a theory, the theory of hyperbolic systems. Some of us were much aware that he was preparing an extensive paper, but the result surpassed the expectations for the varied scenes and the global scenario of dynamics it presented. Still dreaming that nonwandering set-stability could be shared by a dense subset of systems (he called it O-stability because in his writings Cl always represented the nonwandering set). But this was not emphasized as much any more: The theory of hyperbolic systems was now standing by itself, solid and fundamental. (And just a year later, the dream was finally shattered by Abraham and himself in higher dimensions (^4): again an example displaying a persistent orbit of nontransversal intersection of stable and unstable manifolds was provided, but this time the phenome­ non occurring inside the nonwandering set!). A diffeomorphism f is called hyperbolic or satisfies Axiom A if its non-

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wandering set Sl(f) is hyperbolic and its set of periodic orbits is dense in *}(/)• We say that an invariant set is hyperbolic if the tangent bundle of M restricted to it splits into two subbundles, in which the derivative of the map operates as a uniform contraction and expansion, respectively. In particular, the hyperbolic torus automorphisms as well as the Morse-Smale diffeomorphisms satisfy Axiom A. One can also construct an Axiom A diffeomorphism of the 2-sphere with its nonwandering set consisting of one source, one sink and a horseshoe. In the first case, the whole manifold is a hyperbolic set for the diffeomorphism: We call such a diffeomorphism globally hyperbolic. In the other "extreme," the Morse-Smale diffeomorphisms have their nonwan­ dering set consisting of finitely many hyperbolic periodic orbits. It is to be noted that in the first case the indices (dimension of the stable manifold) are the same for all periodic orbits, whereas in the other cases they vary: In the Morse-Smale case, they go from zero to the dimension of the ambient manifold. The previously mentioned result of Anosov states that globally hyperbolic diffeomorphisms are structurally stable. Also, the periodic orbits form a dense subset of the nonwandering set. In proving these results, Anosov constructed stable and unstable manifolds for all points in the ambient: They consist of the set of points whose orbits are, respectively, positively or negatively asymptotic to the positive or negative orbit of the initial point. By 1966, Moser provided a different and elegant proof of Anosov's Theo­ rem, and this was the theme of Mather's Appendix to Smale's paper. Smale was much inspired by the work of Anosov to finally achieve the concept of a hyperbolic diffeomorphism. Thus, he named the globally hy­ perbolic diffeomorphisms after Anosov. He now saw the nonwandering set for an Axiom A diffeomorphism decomposed into "Anosov pieces" assem­ bled together somewhat as in the Morse-Smale case. Each of these pieces being transitive (dense orbit) is called a basic set. This is the content of Smale's Spectral Decomposition Theorem: The nonwandering set of an Ax­ iom A diffeomorphism can be decomposed into a finite number of basic sets. It was now clear to him that by imposing some global condition on the basic sets, to avoid "fi-explosions," the diffeomorphism would be fl-stable, i.e., its restriction to the nonwandering set would be structurally stable. He introduced another axiom, Axiom B: If for a pair of basic sets A, and A 2 , the stable manifold W^x,) intersects the unstable manifold W*(x2), for some xt e A, and x2 6 A 2 , then W{x\) has an orbit of transversal intersection with W{x*)for some x j e A, and xf e A 2 . He then announced: Diffeomorphisms satisfying Axioms A and B are Q-stable. Not so untypical of Smale at the time (somehow his lectures now seem much more "organized"), he was once speaking on this theorem to a rather large audience in Berkeley and in a kind of noisy atmosphere with Moe Hirsch going often to the blackboard to help him out. Terry Wall was quietly attending the lecture, when Smale declared: In the presence of Axiom B the basic sets are partially ordered, i.e., if W^x,) intersects W{x2) and W'fxf)

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intersects W(x3), for x t e A,, x 2 , xf e A2 and x3 e A3, for basic sets Aj, A2, and A3, then W(x^) does not intersect W*(x\) for any xj e A3 and x? e A^ At this point, Wall asked (his only question): Is it reasonable to assume just the partial ordering of the basic sets? Smale shrugged his shoulders and went on lecturing. Before the end of the exposition, I was suggesting that the no-cycle condition, as defined above, should be adopted instead of Axiom B because this was necessary to avoid ^.-explosions under small perturbations of the map (Rosenberg had 10 years earlier suggested the no-cycle condition in the context of the Morse inequalities). And so the O-Stability Theorem of Smale when published in 1970 in the Proceedings of the 1968 Global Anal­ ysis meeting at Berkeley states: an Axiom A diffeomorphism satisfying the no-cycle condition is Q-stable. We also knew that in the presence of Axiom A, the no-cycle was a necessary condition for the result. In another development, as his Ph. D. student at the same time as Nancy Kopell and Michael Shub (the first three except for Moe Hirsch, an "infor­ mal" student back in their Chicago time) I had proved that the Morse-Smale systems are stable in low dimensions (less than four). Kopell proved very nice results concerning centralizers of diffeomorphisms, especially for maps of the circle. And Shub proved definitive results about the stability of expanding maps (derivative with norm bigger than one everywhere) and initiated their classification, which was finally and beautifully done by Gromov many years later, showing in particular that they only exist on infranilmanifolds. After many discussions with Smale (always good humored and loud—once an office neighbor asked us to be quieter!), we succeed in proving: The MorseSmale systems are stable. In particular, an open and dense set of gradient systems are stable (a restricted true piece of the "dream"). We then dared to state the Stability Conjecture: A system is Ck stable, k^. I, if and only if it is hyperbolic (Axiom A) and satisfies the transversality condition (all stable and unstable manifolds are in general position). For il-stability one imposes the no-cycle instead of the transversality condition. Notice that if the limit set of a diffeomorphism is hyperbolic, then the periodic orbits are automatically dense on it. So the Spectral Decomposition Theorem is valid for the limit set in this case (Newhouse). And if the no-cycle condition is also valid, the limit set is the same as the nonwandering set, and the diffeomorphism satisfies Axiom A. On the other hand, the nonwander­ ing set may be hyperbolic without the periodic points being dense on it (Dankner); such a phenomenon cannot occur for surface diffeomorphisms as shown by Newhouse and myself. Let me say that these concepts and results concerning diffeomorphisms were also established for flows (or differential equations as Smale liked to stress). The hyperbolicity of a flow 0. However, about the Q-Stability Theorem for flows, Smale declared that it was very likely to be

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true using the same methods of proof. Well, this was only done a few years later by Pugh and Shub, the methods being a bit more involved. In the direction of the Stability Conjecture, Robbin for C 2 diffeomorphisms, de Melo for C 1 surface diffeomorphisms, and Robinson for C 1 diffeomorphisms in general proved in the early seventies the direct implica­ tion: Hyperbolicity and transversality implies stability. Robinson also proved the corresponding result for flows. Around 1980, Liao, Mane, and Sannami proved the converse for C1 surface diffeomorphisms. Finally, in a remarkable paper, Mane, 20 years after it was stated, proved the C1 Stability Conjecture for diffeomorphisms in all dimensions. I was able to extend slightly his formi­ dable work to prove the C 1 fl-Stability Conjecture. The question for flows remains essentially unsolved. Smale also asked for the classification of Anosov diffeomorphisms and flows. For instance, we still do not know if the periodic orbits of an Anosov diffeomorphism are dense in the ambient manifold except in the codimension-one case (Newhouse), i.e., when the dimension of the unstable (or the stable) manifold is 1. Ten years later, Franks and Williams showed that this is not the case for Anosov flows even in three dimensions. On the positive side, there are the results of Franks, especially in the codimension-one case, and Manning for Anosov diffeomorphisms on tori, nil and infranilmanifolds. About Anosov flows, there are the works of Tomter, Plante, and Verjovsky (codimension-one case in dimension larger than three). Notably, Smale constructed hyperbolic attractors which were neither sinks nor the whole manifold (Anosov diffeomorphisms). The main ones were the solenoid and, perhaps more unexpected, the DA (derived from Anosov) attractors, the last ones being obtained through a saddle-node bifurcation of, say, a hyperbolic torus automorphism. These and the Anosov ones can also be "multiplied" by strong normal contractions to generate nontrivial attractors in higher dimensions. A very relevant contribution to the study of hyperbolic attractors is the work of Williams. Hyperbolic attractors were used a bit later to construct examples of persistent lack of transversality (in particular, persis­ tent tangency) inside the nonwandering set or even the limit set, as first done by Abraham and Smale. As mentioned before, these examples proved that the Q-stable or limit set-stable systems are not C* dense in dimensions greater than two and any k ^ 1. It remained the case of diffeomorphisms on surfaces. This was settled with the same answer and around the same time for C 2 surface diffeomorphisms by Newhouse (also a student of Smale, like Franks) in a much more subtle way: he used the arithmetic difference of subsets of the line and a fractal dimension (thickness) to show that often two Cantor sets in the line intersect each other and do so in a robust way when we slightly change the Cantor sets metrically (this corresponds to persistent tangencies in the creation and unfolding of a two-dimensional homoclinic tangency). Still we do not know the answer for C 1 surface diffeomorphisms. Smale talks again about the centralizer problem, now in more precise terms: For an open and dense set of hyperbolic diffeomorphisms or even for

587

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177

general diffeomorphisms they should be trivial (conjecture). As mentioned before, this was solved by Kopell for general diffeomorphisms of S1 and for smooth hyperbolic diffeomorphisms in higher dimensions by Yoccoz and myself (Rocha in the analytic case). Smale also discusses hyperbolicity and stability of group actions and this was the theme of Camacho's thesis, also his student. Similarly, he posed the question of the rationality of the zeta function for a residual set of diffeomorphisms (which turned out to be false by Simon's work) and for hyperbolic diffeomorphisms (which is true by Williams and Guckenheimer). He also pointed in the direction of ergodic theory to be brillantly followed by Bowen, who worked very closely with Ruelle and, although far apart, with Sinai (he also did some initial work close to Lanford). A high point was the construction of the Bowen-Ruelle-Sinai measure for hyperbolic attractors about 10 years later. I could go on mentioning other topics and the work of other of his students at the time or a bit afterward: Fried, Guckenheimer, Nitecki, Schneider, Shahshahani, and so on. But when the atmosphere in Berkeley seemed most extraordinarily excit­ ing, Smale was preparing to leave the subject! He became more interested in Mechanics, then Economics, then Theory of Computation (Complexity of Algorithms), and then Numerical Analysis, and so on. I particularly like to detect some of the same old spirit of dynamics in his work in these different areas, but maybe it is just my passion for dynamics! I also like to remember the many discussions we had about dynamics and mathematics in general, almost never about details and almost always about ideas, directions, theo­ ries and this in a concrete way, nothing like abstract supermachines. He certainly influenced many of us in these ways.

FIGURE 3

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But the pace of research in dynamics continued unabated for us and still to a large extent for him up to the Global Analysis meeting that he has organized with Chern in the Summer of 1968. Yet somehow there seemed to be a sense of hurry to "complete" the hyperbolic theory. The decade was closing and Smale was leaving. We still had to fix a piece of his Q-Stability Theorem, concerning the stable set and the union of the stable manifolds of the points in a basic set. Hirsch, Pugh, Shub, and myself were quite busy doing that in the Warwick meeting of 1969. And Smale calmly but frequently would come around to see how that was going. The seventies would reserve a big surprise for us, happy builders of the hyperbolic theory of the sixties. The Dark Realm of dynamics was much "bigger" than we thought, as indicated to us mostly by physicists, biologists, astronomers, and so on doing experimental work using dynamical systems models. Bigger and also very challenging and exciting. In particular, Feigenbaum made his celebrated discovery on cascades of period doubling bifurca­ tions (Coullet and Tresser are to be mentioned in this context too) after listening to a lecture of Smale on quadratic maps of the interval. We are indeed lucky, as I mentioned in my first sentence, to be part of one revolu­ tion in the subject: Imagine two! It is my belief that it is just happening Of course, the hyperbolic theory stands as solid as ever and is the starting point to look across its boundaries. Neverthless, the methods and views are different, more analytic perhaps, the sense of prevalence of a dynamical phe­ nomenon being more of a measure theoretical sense in a parametrized form (positive measure, positive density, total measure) than residuality. But this is a topic of another discussion and I shall finish by just presenting my pictorial view (Fig. 3) of how hyperbolicity stands vis-a-vis its complement (the Dark Realm) in the World of Dynamical Systems in the sixties and nineties.

References 1. S. Smale, Dynamical systems and the topological conjugacy problem for difleomorphisms. In: Proceedings of the International Congress of Mathematican 1962. Institut Mittag-Leffler, Djursholm Sweden, 1963. Editor V. Stenstrom, pp. 490-496. 2. S. Smale, Differentiable dynamical systems, Bull Amer. Math. Soc. 73 (1967), 747817. 3. S. Chem and S. Smale, cds.. Proceedings of the Symposium on Pure Mathematics, Vol. XIV, American Mathematical Society, Providence, RI, 1970. 4. S. Smale, The Mathematics of Time, Springer-Verlag, New York, 1980.

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18 Discussion S. NEWHOUSE, R.F. WILLIAMS, AND AUDIENCE MEMBERS

R.F. (Bob) Williams: Sheldon and I have been thinking about this discussion for about 45 minutes. It's not well prepared, but we agreed a moment ago that I would talk a little bit about the work of Rufus Bowen, John Franks, John Guckenheimer, and others on zeta functions, since that was something that Jacob did not have a chance to talk about. It was, of course, one impor­ tant thing about the 1967 paper which many of us were working hard on in 1966, because it came out in preprint. It was very much an outline. In it, Smale did things that he's done so many times that have been so helpful to us. He's laid out ideas with strong statements, conjectures if you wish, which he didn't feel all that secure about, but he's brave enough to make the conjec­ tures, so we got to play with them. Two kinds of zeta functions were defined. Some proofs were given, some computations were made. I remember very well correcting a slight error of his in this paper, and pointing it out, and of course when it came out, not only was the error corrected but he used my own notations .... Anyway, he was brave enough to conjecture something that took years to prove by John Guckenheimer—first Mane—the ratio­ nality of the zeta functions. Jacob Palis: It's Manning. Bob Williams: Did I say Manning? Jacob Palis: You said Mane. Bob Williams: Yes, the first proof of the rationality was given by John Guckenheimer. and then Manning in England gave the second proof. Man­ ning proved this slightly stronger theorem. Jacob Palis: You're being shy about your own work. Bob Williams: Well, that's all right. I haven't always been. Jacob Palis: Their work was just an improvement of yours. 179

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Bob Williams: It's true that I was very struck by this strange idea that showed the zeta function was rational. I worked very hard on this myself. An offshoot of my work was a seminar I had where we went through the '67 paper. And one of the things that wasn't hinted at, I think, was Paul Schweitzer's work. He showed the Seifert conjecture was false, for C1 vector fields. Paul was part of the seminar that I ran and therefore his work was very much an offshoot of Smale's ideas. Schweitzer produced a counterexample to the Seifert con­ jecture. In addition, it also related to my own work. It's the work on symbolic dynamics. As I understand it—and I should have, during this 45 minutes, looked up the literature—there are two definitions of what we now call subshifts of finite type. One was given earlier by Parry, but the definition given by Smale is in some sense the proper one. In particular, the name that is now used, the name that Smale chose, is "subshifts of finite type." Bill Parry called them "Topological Markov Chains." This was an important idea. It came from the knowledge of horseshoes and related systems, and it immedi­ ately was taken up by students of his, Bowen and Lanford. They showed that the zeta functions of subshifts of finite type were rational. They introduced the whole terminology to make that easy to see. Rufus proceeded to use this concept, symbolic dynamics, to set up the matrix theory. In addition, there's a slightly tighter collection of work that stems from this; that is to say, a more geometric aspect of subshifts of finite type pursued very effectively by me and many young people, and I can still call John Franks young since I'm so old. John Franks, Mike Boyle, Bruce Kitchens, a large collection of peo­ ple, have made a very pretty theory of this. In large part, one can say this stems from Smale's ideas about horseshoes. There's something kindly called a Williams conjecture. I thought I proved it It still occupies a lot of thought. John Franks: Do you call it the Williams conjecture or the Williams problem? Bob Williams: Well, I certainly have a problem with it. John Franks: Would you care to venture a prediction on whether or not it's true? Bob Williams: I still think it's true. John Franks: Would you say it's true? Bob Williams: Uh, yes. Possibly. Let's see, it concerns matrices A and B whose entries are non-negative integers. One defines two equivalence rela­ tions, one of which is very natural, and one of which is correct The natu­ ral one, one says A is shift equivalent to B, provided there are three diagrams, A takes place across the top and B on the bottom, and there are matrices R

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which form a commutative diagram. If you think of these as matrices, you hardly need to say what their corners are:

That's one diagram. There's a similar diagram with a matrix S going up: A

Then, finally, if these were to be conjugated, one would want to know that R was the inverse of S. R and 5 put together is the identity. But one has the weaker assumption that R followed by S is A to somepower, and 5 followed by R is B to some power: SR = A",

RS = Bm.

The point is, that's very natural. It would take a while to show how natural it is, but it is natural. The other notation is sort of crazy, but it's correct. And it says that A is equivalent to B provided you can proceed by a finite string. A is equal to RlSl.Al is equal to S1K1. But Ax is also R2S2. Then A2 is S2R2, etc. In other words, you proceed by finitely many steps. A = R1S1, A\ — K 2 S 2 ,

= R.S.

5,/?, = ^ , ; ^2^2

=

^2>

S.R. = B.

Eventually you get to B. Now the difficulty with this problem, assuming these are equal, is that the non-negativeness of the integers involved. If one does this over any other category, it's easy to see these are essentially equivalent; very easy arguments have been given that work in any other sort of category. Well, that's certainly enough about me. Did I miss something about Rufus? John Franks: There are many names that deserve more praise than you and I can give them today. But the thing that hasn't been mentioned is the rela­ tionship between this and the other things we've been talking about today, in particular hyperbolic dynamics. The theorem on Markov partitions basically says all of the hyperbolic systems that Jacob told us about today can be understood in terms of symbolic dynamics. And there's a great deal of infor-

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mation to be gleaned about these hyperbolic systems from this symbolic dynamics. The existence of that connection is the existence of Markov parti­ tions, in the development of which Rufus was perhaps the most important figure. Bob Williams: Well, I've left to Sheldon the enormous task of describing developments in ergodic theory. Moe Hirsch: Before we get to that, I'd like to give a comment about the hyperbolic theory, and that is that it's had an enormous influence in applied dynamics. Now, we don't talk much about applied dynamics in this confer­ ence, but anyone who deals with applied dynamics would know the extreme importance Smale's work on hyperbolicity has had in giving rigorous exam­ ples of chaotic systems. There, I said the V word; nobody has said "chaos" yet. But as you know from reading the popular magazines and the press, chaos is a very popular word, and we all appreciate—but a lot of people who are novices don't appreciate—how important the hyperbolic systems are in rigorously demonstrating the existence of chaotic systems and the impor­ tance of horseshoes. I don't know if Jacob said that one of the things Smale proved is that when you have a horseshoe, you can model it with symbolic dynamics. Jacob Palis: I did. Moe Hirsch: Did you say that? Jacob Palis Why, of course. Moe Hirsch: Okay. It's extremely important. And in a sense, this responds to Poincare's despair in understanding horseshoes, because you can understand them now in the sense that we have labeled them by symbols from a finite alphabet, and we can understand how that goes. So, this is a far-reaching offshoot with all sorts of people who could not tell you what a nonwandering point is, but have to deal everyday with applied systems. They can understand, and you can do the same. A very important influence in applied mathematics. Jacob Palis: Yes, I did not use the word chaos, because I was afraid of Rene Thorn. Mike Shuk It is important to say that the first complicated structurally stable attractors were introduced in the 1967 paper. Up to that point the only structurally stable attractors were points and periodic solutions. The 1967 paper introduced a whole new class of structurally stable attractors. Takens and Ruelle adopted those attractors as an alternative model of turbulence and chaotic behavior.

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Bob Williams: Yeah, I claim even the word "attractor" was invented by these guys, Thorn and Smale. Each says the other invented it. Mike Shub: It is also important to stress that the whole thing is expressed in terms of topological conditions—there are no equations. It had the effect of restructuring the entire subject. Sheldon Newhouse: Yeah, let me just add—I think Mike hit on the funda­ mental thing, at least from my point of view—one of the great influences that Steve had, at least on me and I think many others, aside from excitement, was the whole idea that one might get a structure theory for general dynamical systems. That such a general structure theory for most systems could exist I think was a goal of Poincare and Birkhoff, and somehow in the intervening years, it was lost sight of by people working on specialized problems. The idea of getting a global structure of all systems has spawned a lot of develop­ ment in the past, and it is still going on. I think Jacob did an excellent job of describing the atmosphere that existed in the sixties, with all the excitement and developments happening virtually every other week, in the seminar, out of the seminar. I want to give a little personal perspective of what it was like to be a rather ignorant graduate student at that time, with all of this stuff going on, not understanding much of anything. In particular, I remember one afternoon in a seminar, John Franks was doing what you would call nowadays a thesis defense. So you know what a graduate student feels like at his first big lecture. He was talking about some theorems in his thesis and then Moe asked him a question, "Well, how do you prove that?" And John said, "Well, that's in Steve's paper." And then Steve piped up, "Yeah, but I don't think I know how to prove that anymore!" John just sort of didn't know what to say. I think he worked out a proof. How long did it take? John Franks: About two days. Sheldon Newhouse: About two days. So, there was a very kind of cavalier idea about trying to develop a subject, and the idea always was "What are the general sort of structures? And let's not worry about the details." And that got even to the level of the graduate student. So, another historical thing that Rene Thorn was just telling me today which I think is quite interesting is that the

CO map actually occurred in response to a question that Steve asked about whether every diffeomorphism in the 2-sphere had an attracting periodic orbit. Well, obviously, it couldn't be that statement, right? Because an areapreserving diffeomorphism doesn't have it. He was obviously thinking about

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gradient systems and dissipative dynamical systems. That question in itself led to the

C!) map on the torus, which was not a counterexample for the 2-sphere; it was a counterexample to the general kind of question about whether one always had attracting periodic orbits. John Franks: I don't know the details, but I would like to make a comment about your historical remarks about being a graduate student. It was a strange period, even in retrospect. The people who were there as graduate students include many of the people who are speaking at this conference and in this room today: Jacob certainly, Mike Shub, Nancy Kopell, Sheldon was here, John Guckenheimer, Rufus Bowen, Oscar Lanford was an instruc­ tor, I believe. Anyway, with this large group of people, mostly graduate stu­ dents, it was a very heady time, but we didn't really realize that, I think. Being naive graduate students, as he said, we assumed this is the way it always was for students in graduate school. It was a period when we learned a lot; we started with a clean slate, in a way. I think many of us—certainly I did— learned dynamics from Steve's '67 paper. If you knew nothing and you took that paper, you could become an expert by reading through it. At least, that's the way we viewed it in our naive way. I think that Steve's success as a thesis advisor is illustrated in that group and the 30 other people who will go at the top of Steve's family tree. But I'd like to make one further comment While I think Steve's done an amazing job as a thesis advisor, he shouldn't get all the credit. A great deal of praise goes to two people, Moe Hirsch and Charles Pugh, who haven't gotten mentioned very prominently yet in this meeting, but were a real essential part of my education and I think the education of many of the rest of us here today. They were the people creating a lot of the ferment and a lot of the talk, a lot of the chaos, if you will, that was flowing around. It was being churned up by them. Sheldon Newhouse: I would second that thought. Thank you. Nancy Kopell: I'd like to comment also on Moe's point about Steve's influ­ ence on applied mathematics, which I think has been enormous. It's not just in chaos or horseshoe dynamics, or particular things that he's done, but rather it's the whole qualitative point of view. And this is extremely impor­ tant, especially in subjects unlike physics where you don't know these exact equations, but instead you have to reason with qualitative properties. In particular, mathematical biology has been strongly influenced by this point of view precisely because one has only some qualitative properties to begin with, and one has to reason about the consequence of those qualitative prop-

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erties. And it turns out that dynamical systems is a wonderful framework in which to do that. John Guckenheimen I'd like to add to that Steve's influence even in the realm of physics, and the attention drawn to chaos in recent years. One was able to draw very specific conclusions from qualitative models. Feigenbaum says that he was motivated to begin looking at the dynamics of maps of the interval by a lecture that Steve gave in Aspen at the Center for Physics, I think in 1975, and that this was oriented to properties of physics and popula­ tion models. I think that the one event which, in some sense, was the key to catching the attention of physicists that this was real stuff was in the experi­ ment of Libchaber, in which he took this phenomenon of period doubling and saw it in a real fluid experiment. It is totally inconceivable to me that anyone could have predicted, by staring at the Navier-Stokes equations, that something like this would be found in the real world. Christopher Zeeman: It's extremely plausible that you could stare at them forever. Sheldon Newhouse: Well, perhaps we ought to have some refreshments.

596

MORSE INEQUALITIES FOR A DYNAMICAL SYSTEM BY STEPHEN SMALE1 Communicated by S. Eilenberg, October 24, 1959

1. Introduction. We consider dynamical systems (X, M), where X is a C" vector field on a £> closed manifold M satisfying the following conditions. (1) There are a finite number of singular points of X, say Pu • • • . Pk, each of simple type. This means that at each Pi, the matrix of first partial derivatives of X in local coordinates has eigen­ values with real part nonzero. (2) There are a finite number of closed orbits (i.e., integral curves) of X, say /3*+i, • • • , Pm, each of simple type. This means that no characteristic exponent (see, e.g., [2]) of Pi, i > * , has absolute value 1. (3) The limit points of all the orbits of X as /—*± <» lie on the Pi. In other words, denote by 4>t the 1-parameter group of transforma­ tions generated by X (as we do throughout this paper). Let o(y) = limit set^(y),

u(y) — limit set^«(y),

y £ M.

Then for each y, a(y) and w(y) are contained in the union of the Pi. (4) The stable and unstable manifolds of the Pi (see §2 for the definition) have normal intersection with each other. More precisely for each i let W, be the unstable manifold and W? the stable manifold of Pi and for x £ W.- (or Wf) let WiM (or W£) be the tangent space of Wi (or Wf) at x. Then for each i, j if 1 6 WtC\ Wj, dim Wi + dim W* - n = dim (Wi. C\ W*t). See [5 ] for example for more details. (5) If Pi is a closed orbit there is no y£M with a(y) =o)(y) =/3,. First we remark that systems satisfying (l)-(5) may be very im­ portant because of the following possibilities. (A) It seems at least plausible that systems satisfying (l)-(5) form an open dense set in the space (with the C1 topology) of all vector fields on M. (B) It seems likely that conditions (l)-(S) are necessary and suffi­ cient for X to be structurally stable in the sense of Andronov and Pontrjagin [l]. See also [6]. (A) and (B) have been proved for the case M is a 2-disk, [3 ] and

H

1

Supported by a National Science Foundation Postdoctoral Fellowship.

43

597 STEPHEN SMALE

44

(January

We expect to have more to say about this subject at another time. It is true that conditions (l)-(5) are independent. With (X, M) as above let k, k, with
Mt-

if, £ R0, JA - M, £ U, - Rt, Mt + M,Z R,- Ri + R,,

E(-i)w» = (-i)-x where dim M=n and x « the Euler characteristic of M with respect toK. Theorem 1.1 contains the Theorem of £l'sgol'c [4] which excludes closed orbits. It also contains Reeb's theorem [ l l ] which excludes singular points. However, both fil'sgol'c and Reeb made the highly restrictive assumption* that no orbit joined saddle points (i.e., p\, i£k withffiT^O,n) or saddle type closed orbits (i.e., p\, i>k with
2. Construction of the stable and unstable manifolds. 2.1. Suppose P is a singular point of simple type of the C" system (X, M). Let k be the number of eigenvalues associated to fi with real part positive. Then (e.g., [2, p. 330]) there is a k dimensional 0° submanifold W of M passing through 0 such that if x£W then a(x)=/3. If k = 0, let * Reeb hms asked me to note that his footnote 3,2nd paragraph, of [ll, p. 62] (that this assumption is unnecessary) is incorrect

598 I9&>1

MORSE INEQUALITIES FOR A DYNAMICAL SYSTEM

45

W=(i. Then W is tangent at 0 to the linear subspace of the tangent space Mf of M at /3 denned by these k eigenvalues [2, p. 333]. W is called the unstable manifold of AT at /9. Let /?* denote Euclidean kspace considered as a vector space. We will show that W is the image of a continuous 1-1 onto map / : /?*—*W, with /(0)=/3, and / is C with Jacobian of rank k except at 0. Consider the new system X* obtained by reversing the direction of each vector of X on M. Then /3 is a simple singularity of A'* and the above applies to yield the un­ stable (n-/fe)-dimensional manifold W* of X* at 0. Call W* the stable manifold of X at /3. Note W and W* have normal intersection at/3. 2.2. Suppose /3 is a closed orbit of (AT, M) of simple type. Let k — 1 be the number of characteristic exponents of /3 with absolute value greater than one. Then ([7] or [13]) there is a A-dimensional O* submanifold W of M passing through /3 such that if x £ W then a(x) =p\ If k = 1 let W=/3. Also W is tangent at each point y of /3 to the linear subspace of Af, defined by these k— 1 characteristic ex­ ponents and the tangent vector of /3 at y. Call W the unstable mani­ fold of X at 0. We will show there is a continuous 1-1 onto map / : -R*- I XS , -»W, with/(OX 5') = 0 , and except along 0 X 5 ' is C" with Jacobian of rank k. Similarly to 2.1, one defines the stable manifold W* of X at /3 whose dimension in this case is n — k + l. We now construct the m a p / o f 2.1. There exists* a differentiably imbedded (* —l)-sphere K in W, which is everywhere transversal to X. Let 5 0 be the unit sphere of Rk and h: S9—*K be a diffeomorphism. (A diffeomorphism is C* homeomorphism with a differentiate inverse.) Let ^/t be the 1parameter group of transformations of Rk generated by the vector field Y(x)=x on Rk. For x£./?*, x?*0, let t(x) be the unique / such that x/IMI =*«,)(*)GSo. Then l e t / ( 0 ) = / 3 and /(*) =*-«(x)W<(.)(*). It is easy to check that / : Rk—*W thus defined has the desired prop­ erties. To construct the map / : Rk~lXS1-^W of 2.2, first let Y be the vector field {x, 1) on Rk~lXSl. Then if $t is the 1-parameter group of transformations generated by Y we have^»(x, 0) = (xe1, t mod 2x). Let Rk~l = Rk~lXOCR"-1 XS\ and C be the unit ball in Rk~\ dC=50. Define j : /?*-1—»/?*_1 by q(x) =xeu and let qiSo = Si for each integer ». Let Q be a surface of section (i.e., transversal to X, see [6]) locally about a point of /5 in W, diffeomorphic to a (k — l)-cell. Then [6] the orbits of X define a diffeomorphism h: Q—*Q in a neighborhood of fi(~\Q leaving (Hr\Q fixed. There is' a closed Jb-cell B differentiably * By Liaponov theory for example.

599

46

STEPHEN SMALE

(January

imbedded in Q, dB = Ft such that h~l(Fa) is contained in the interior of B. Let h^Ft) = F„ i£0. Let / be an orientation preserving diffeomorphism of a neighbor­ hood Vt of St in Rk~* into a neighborhood of Ft in (?. Then extend / to a neighborhood of U, s 0 Si in Rk~l into a neighborhood of U, s o Fi in Q by the formula (2.3)

/(x) = A-'Mx),

x G nbd. K« of 5,-.

This makes sense for an appropriate choice of the K.'s. Now consider the closed region U in Rk~l bounded by S0 and 5_i. We have defined / in a neighborhood of the boundary dU of U. After restricting/ to a smaller neighborhood of dU.fc&n be extended to a diffeomorphism of all of U into the region of Q bounded by Ft and F_i. This fact fol­ lows from arguments which are now standard in differential topology. We won't include them here. Then as in 2.3 we can extend/ to a map of all of C into B which is a diffeomorphism except at f(0)=Br\Q. Next d e f i n e / on P = f^«(*)|x£C, <<0} by the following: Let T(X, 9) be the smallest positive number such that 4>r{*.t)fl>-*{x, 9) has 9 as its second coordinate in a fixed product structure QXB,{x, 0 ) E P . Then let/(x, 9) =,il.r>f4>-t(x, 9). Define/: OXS l -*B by /(OX0) =9. Consider now the surface of section S - i X S ^ - d in Rk~lXSx and its image under/. Restrict/to the closure of the bounded component K of A. Finally extend / to all of Rk~xXSi as follows. For yERk~l XSl-K let t{y) be the unique t such that ^«(»)(y)EA. Then let /(?) =#-«»>#«<»>toO- After a change of parameter near A, f will have our desired properties. 3. Implications of (l)-{5). Assume throughout this section that (X, M) is given as in §1. If B{ is a singular point then /,•: Rk—*Wi is as in 2.1. If p\ is a closed orbit then/,-: Rk~lXSl-^Wi is as in 2.2. LEMMA 3.1. 7/xEAf, a(x) =p\, co(x) =/S/, 0 , there is a y £ Wt with d(x, y)<$ and if dim W,=dim Wit there is a subcell K of Wt such that H and K are within 8 in a C1 metric.

600 I9&>l

MORSE INEQUALITIES FOR A DYNAMICAL SYSTEM

47

Define dWj = {lim*-„/,(**) |*» any sequence in Rk with no lps. j . Then let d1Wi = d{dWi), etc. Note C\ W.= W.UdW,. LEMMA

3.3. / / W,C\ W**0,

dW^Wj.

This follows from 3.2. LEMMA 3.4. Suppose dim W, = dim W* = dim Wj. If and WkC\ Wj *■ 0 then W,C\ Wj * 0.

^nW

t

V0

PROOF. Let x£Wkr\W*\ apply 3.2 using the fact that WfC\ W* 7*0. Since Wk and W* have normal intersection at x, it follows from 3.2 that WiC\W**0. LEMMA

3.5. Suppose

WikC\ W?k^j*0,

k = l, ■ ■ ■ , m. Then Wik

PROOF. First note by 3.1, dim W^,t+1^dim Wik and equality occurs only if /3, 4M is a closed orbit. This implies we can restrict ourselves to the case of the lemma where all the Wik's are of the same dimension. Then if Wik=Wijt k*j, 3.4 implies that WitC\ W*.*0. This con­ tradicts condition (5). LEMMA 3.6. IfdWyC\ Wtj*0, then there is a sequence Wiv • • • , Wim such that WikT\ W ^ + , ^ 0 . Wy-Wix, and W,= W^. PROOF. Let a(W7) = l i m « _ . Wf. Then it follows that Cl Wy r\a(Wi)*0. L e t f r e C l WyT\a{Wl). Then WjC\W^0. If j*y, similarly let /3*GCI Wy(~\a{W*). Induction and 3.5 yield 3.6. LEMMA 3.7. / / dWf\W^0, then dW.DWy and either dim W* >dim Wj or dim W; = dim Wit W , n Wf^0, and 0, is a closed orbit

This follows from 3.6, 3.5, 3.3, 3.1, and 3.4. LEMMA

3.8. Each W, is an imbedded R" or

R'~lXSl.

This follows from §2, 3.7 and (5). LEMMA

3.9. dkWi?*0,

any i for large enough k.

If not there isa Wjsuch that a - W / W / ^ 0 . By 3.7 t h e n a W / W y 9*0, contradicting 3.8. 4. On Morse theory. A version of one of the standard theorems of Morse theory is stated in this section. The proof is a short wellknown argument using the exact cohomology sequence of a pair and for example can be found in [lO]. THEOREM 4.1. Let M be an n-dimensional topological space with closed subspace L* for each integer p such that L"DL*-1, and there

601 STEPHEN SMALE

48

(Jimiarjr

exist integers a, b with L* = 0 and L* = M. Using any fixed cohomology theory and coefficient field, assume dimension //«{£.', L" - 1 ) is finite for each p and q. Let B f = dim H'(M) and M,= ]£,*-. d i m #*(£", I / - 1 ) . TA«n M f and B f satisfy the Morse relations Mt - M, £ 5 , - B0> AT» - Mi + Mo ^ ft - Bi + So,

*-o

t-o

5. Proof of the main theorem. Define K' = \Jumw,Sp Wi. Thorn in [14] considers subspaces related to K* to prove the classical Morse inequalities. By 3.7 it follows that the K* are closed sets. However, examples show that K' has the following bad property. It may be that for WiC.K", dWt is not contained in K'~K To avoid this we define a new structure on M. We define by induction a sequence of closed subsets L,- of M with I , O i i - i and L, = M for large enough i. Define Lt = 0, and if L<_i has been defined, let L,- be the union of all the W> whose boundary lies in £.,_». It is immediate that L.O-^-.-i, that L, is closed in M and that L{ — L,_i consists of a disjoint union of W}. It follows from 3.9 that there is an integer b such that !.»= M. One can construct an example to show that the Z..- need not be locally connected and that for WtQLi — Li-x, WiC^-i-i. dim Wt = d i m W,. LEMMA

5.1. Using Cech theory if M, is as in 1.1 then

M, = JT dim H<(L<, ! . _ , ) . PROOF. As noted previously i<—L<_i consists of a disjoint union of the W> and as i ranges from 0 to b all the Wj are obtained. Denoting cohomology with compact carriers by K*K, since H\{P—Q) = H*(P, Q) for Cech theory, we have

£ dim B\Lit •-0

Lt-x) -

D

dimff*(W,).

all Wj

Using 3.8 and Poincar6 duality #X(W,) = #<"-. w,-,(W,). Further­ more dim Ht{Wj) = 1 for all j , dim #i(H0) = 1 if 0, is a closed orbit and dim H,(Wj)=0 otherwise. The lemma follows.

602

i960]

MORSE INEQUALITIES FOR A DYNAMICAL SYSTEM

49

Theorem 1.1 follows from 4.1 and 5.1. 6. An analogue of the main theorem. Suppose instead of a vector field X on M, we just have given a 0° diffeomorphism h on M which satisfies certain conditions analogous to ( l ) - ( 5 ) . (1') There are a finite number of periodic points (i.e., ;c£Jlf such that h"(x)=x for some integer p) of h of simple type (i.e., the differ­ ential of h at p has no eigenvalue of absolute value 1). (3') The limit points of all the orbits of h (i.e., {/tp(x) | all integers p) = orbit of x) are periodic points. (4') The "stable" and "unstable" manifolds of the periodic points have normal intersection. The previous theory extends to cover this case. In particular if M% is the number of periodic points with q eigenvalues having abso­ lute value greater than one, the Morse relations of 1.1 hold. One can ask the corresponding questions of (A) and (B) of §1 for the above situation. REFERENCES

1. A. A. Andronov and L. S. Pontrjagin, Syslimes grossiers, Dolcl. Akad. Nauk vol. 14 (1937) pp. 247-250. 2. E. A. Coddington and N. Levinson, Theory of ordinary differential equations, New York, McGraw-Hill, 1955. 3. H. De Baggis, Dynamical systems with stable structure. Contributions to the Theory of Nonlinear Oscillations, Princeton University Press, vol. 2, 1952, pp. 37-59 (Annals of Mathematics Studies, no. 29). 4. L. E. El'sgol'c, An estimate for the number of singular points of a dynamical sys­ tem defined on a manifold, Amer. Math. Soc. Translations, no. 68, 1952. Translated from Mat. Sb. (N.S.) vol. 26 (68) (1950) pp. 215-223. 5. R. K. Lashof and S. Smale, Self-intersections of immersed manifolds, J. Math. Mech. vol. 8 (1959) pp. 143-158. 6. S. Lefschetz, Differential equations: geometric theory. New York, Interscience Publishers, 1957. 7. D. C. Lewis, Invariant manifolds near an invariant point of unstable type, Amer. J. Math. vol. 60 (1938) pp. 577-587. 8. M. Morse, Calculus of variations in the large, Amer. Math. Soc. Colloquium Publications, vol. 18, 1934. 9. M. Peixoto, On structural stability, Ann. of Math. vol. 69 (1959) pp. 199-222. 10. E. Pitcher, Inequalities of critical point theory, Bull. Amer. Math. Soc. vol. 64 (1958) pp. 1-30. 11. G. Reeb, Sur certaines propriitfs topologiques des trajectoires des syslimes dynamiques, Acad. Roy. Belg. Cl. Sci. Mem. Coll. in 8* 27 no. 9 (1952). 12. S. Smale, On structural stability, to appear. 13. S. Sternberg, Local contractions and a theorem of Poincarl, Amer. J. Math. vol. 79 (1957) pp. 809-724. 14. R. Thorn, Sur urn partition en cellules associee & une fonction sur une varittt, C. R. Acad. Sci. Paris vol. 228 (1949) pp. 973-975. INSTITUTE FOR ADVANCED STUDY

603

ON DYNAMICAL SYSTEMS* B T STEPHEN SMALE

1. General considerations We consider here systems (X, M) where X is a C" vector field o n a C " closed manifold M. The vector field X generates a global 1-paramcter group k, has absolute value 1. (3) The limit points of all the orbits of X as t —* ± » lie on the ft . (4) The stable and unstable manifolds have normal intersection with each other. This can be explained as follows. Let ft , i ^ fc be one of the singular points of X, and let h be the number of eigenvalues associated to ft with real part positive. Then [5] there is an A-dimensional C" sub-manifold W, of M pass­ ing through ft such that if a: € Wit lim«,_. A:, where the ft are closed orbits. Thus for each i, * This research was supported by a National Science Foundation Post-doctoral Fellow­ ship. 195

604

1%

STEPHEN' SMALE

1 :£ i' ^ m, we have Wt and Wt, the unstable and stable manifolds of X at #,. For i € Wi (or W*) let WtI (or H'*r) be the tangent space of W, (or H'*) at x. Then by the normal intersection condition \vc mean for each i, j ,

if x € w,n w*. dim Wi + dim W* - n = dim (W„ D W*,). (5) If 0, is a closed orbit then there is no y 6 M with lim«^._oV>i(y) = 0, and lim,^.v'«(y) = 0. • It can be shown that these five conditions are independent. I do not think it would be difficult to show that C is open in B. However the question as to whether C is dense in B seems to be very difficult. Work of Peixoto [7] implies this is true where M is the 2 disk. The following theorem solves the corresponding approximation problem for gradient fields. (Detailed proofs of the theorems stated here will be given elsewhere.) 1.1. / / X = grad f,faC function on M, then X can be Cl approxi­ mated by aC field Y on M such that Y satisfies (1)—(5) with no closed orbits. THEOREM

Since there are a great variety of C* functions on every closed manifold, Theorem 1.1 guarantees that for every M, C is far from being empty. The idea of the proof of l.l is as follows. Given /, a theorem of Morse [6] implies there is an approximating function g on M such that g has only non-degenerate critical points. Then Z — grad g is a C° vector field on M satisfying conditions (1), (2) and (3) with no closed orbits. Furthermore any C1 approximation of Z will have the same properties. This last fact is not true for Z in general and uses strongly the fact that Z is a gradient field. It remains to show that Z can be Cl approximated by a field satisfying (4). To do this one changes Z using Sard's theorem [10] so that the stable and un­ stable manifolds fall into general position with each other. Sections 2 and 3 contain evidence that C is amenable to classification. 2. Morse relations for X in C For X in C let a, be the number of 0,, t' £ k with dim W, = q, and 6, the number of /3<, i > k with dim W, = q. THEOREM 2.1. Let X € C, K be any field, R, be the rank of H*(M; K) and Mt — o, + 6f + 6 f+ i. Then Mt and ftt satisfy the Morse relations,

Mt ^ ft, M, - Af0 £ Rt - ft. E ( - l ) M / , = (-l)"x where dim M — n and x is the Euler characteristic of M with respect to K. Theorem 2.1 contains theorems of El'sgoPc [3] and Reeb [9J and by 1.1 the

605 DYNAMICAL SYSTEMS

l<»7 1<»7

classical theorem of Morse (61. In dimension 2, Theorem 2.1 is contained in Haas [4]. We give a short sketch of the proof of 2.1 now. Consider the sequence of closed sets Lt of M defined by IJO = 0, and induc­ tively Li = union of W, such that dW, c Li-, . Then strongly using conditions ( 3 ) - ( 5 ) it can be proved there is an r such that L, = M. This is the hardest part, of the proof, and we do not go into it here. The more obvious sequence Kp = \iWj, dim Wj g p docs not work. Next 2, dim //«(L,, L,_i) is evaluated in Cech cohomology to be M , . Then by a standard argument from Morse theory the theorem follows. A problem connected with the above is the following. Let X € C with no closed orbits. In this case one can use the sequence K, mentioned above in the proof of 2.1. Then Kp is a union of cells. Does Kr have a corresponding CW structure? If this could lie shown, then probably it would lead to an intrinsic proof of the theorem that every differentiable manifold could be triangulated. 3. On structural stability We say an equivalence is an t-equivalence if it is pointwisc within t of the identity; let d denote a C' metric on B. Then X € B is structurally stable (ac­ cording to Andronov-Pont.riagen (1)) if given t > 0, there is a S > 0 such that if X' g B, d(X, X') < S, there is an (-equivalence between X and X'. Andronov and Pontriagen stated the theorem that if M = 2 disk, X is structurally stable if and only if X has (1') at most a finite number of critical points all elementary and none a center, (2') at most a finite number of closed paths each a limit-cycle with a non­ zero characteristic number, (3') no separatrix joining 2 saddle points. A proof was published by DeBaggis ([2]; see also (5]). It is easy to see that for the 2-disk these conditions coincide with ( l ) - ( o ) . Peixoto and Peixoto have extended this work to 2-manifolds and have corrected a mistake of DeBaggis. It seems likely to us that the n-dimensional structurally stable systems are exactly the elements of C. The following problem has been considered by several people without success. Does a structurally stable system have a finite number of closed orbits? If X has only a finite number of closed orbits and is structurally stable then it must satisfy ( l ) - ( 5 ) . THEOREM 3.1 (L. Marcus). If X is structurally stable and has a finite number of closed orbits then it satisfies (1), (2), and (3).

3.2. If X is structurally stable and has a finite number of closed orbits then it satisfies (4) and (5). THEOREM

The methods used to prove that X satisfies (4) do not differ very much from those used in the proof of 1.1. If X did not satisfy (4) then an arbitrarily small change of X could be made so that new intersections of stable and unstable

606 198

STEPHEN SHALE

manifolds are introduced. This is used to show that X could not have been struc­ turally stable. To prove that .V satisfies (5) one shows that if (5) is violated one can introduce new closed orbits without changing the old closed orbits by arbi­ trarily small changes in X. This of course is impossible if X is structurally stable. At this time we have made a little progress on the problem as to whether con­ ditions (l)-(5) are sufficient for structural stability. THEOREM 3.3. / / X satisfies (1 )-(5), there are no closed orbits, and dim M ^ 3, then X is structurally stable.

The idea of the proof is to define a new structure on M depending on X. Each point of M belongs to exactly one stable manifold and one unstable manifold; hence the submanifolds <x„ = W, fl W> as i, j range from 1 to m give a decompo­ sition of M. Let 2* = { g k\. ForX' sufficiently close to X the corresponding 2 is related to 2* by an isomorphism preserving the boundary operation. Then the desired homeomorphism is defined first on 2°, then 21, etc., by induction. The induction step poses considerable difficulties however, especially as the dimension increases. It is not known if there exist any structurally stable systems on a given mani­ fold. There are however examples on 2-manifolds and the n-spheres [8]. THEOREM

3.4. There exist structurally stable systems on every closed 3-manifold.

This follows from 1.1 and 3.3. T H E INSTITUTE FOR ADVANCED STUDY, PRINCETON, N E W JERSEY BIBLIOGRAPHY

ll| A. A. ANDRONOV and h. S. PONTRIAOEN, Systbnes grossiers, Doklady Akad. Nauk, 14 (1937), 247-251. [2] H. DEBAOOIS, Dynamical system* tvith stable structure, Contributions to the theory of nonlinear oscillations, vol. 2, Princeton University Press, 1952, (Annals of Math. Studies, no. 29), pp. 37-59. (31 L. E. EL'SOOL', An estimate for the number of singular points of a dynamical system de­ fined on a manifold. Amer. Math. Soc. Translation no. 68 (1952). Translated from Mat. Sbornik (N.S.) 26 (68) (1950), 215-223. [41 F. HAAS, On the total number of singular points and limit cycles of a differential equation, Contributions to the theory of nonlinear oscillations, vol. 3 , Princeton Uni­ versity Press, 1956, (Annals of Math. Studies, no. 36), pp. 137-172. [51 S. LErscHETz, Differential equations: geometric theory, Interscience Publishers, New York, 1957. [61 M. MORSE, Calculus of variations in the large, Amer. Math. Soc. Colloquium Publica­ tions, vol. 18, 1934. [71 M. PEIXOTO, On structural stability, Ann. Math., 69 (1959), 199-222. [81 M. PEIXOTO, Some examples on n-dimensional structural stability, Proc. Nat. Acad. Sci., 45 (1959), 633-636. [91 G. R E B B , Sur certaines propritlts lopologiques des trajectories des syslemet dynamique, Acad. Roy. Belgique Cl. Sci. Mem. Coll. in 8° 27 no. 9 (1952). [10| A. SARD, The measure of the critical values of differenliable maps. Bull. Amer. Math. Soc., 48 (1942), 883-890.

607

DYNAMICAL SYSTEMS AND THE TOPOLOGICAL CONJUGACY PROBLEM FOR DIFFEOMORPHISMS By S. SMALE

For simplicity we consider an ordinary differential equation (or a dynamical system) to be a C vector field o n a ( 7 manifold which generates a 1-para­ meter group of diffeomorphisms. An equivalence (or topological equivalence) between two differential equations is a homeomorphism preserving sensed trajectories (or orbits). The qualitative problem of differential equations is to obtain information on equivalence classes of differential equations on a given manifold. This motivates us to consider the topological conjugacy problem for diffeomorphisms (a diffeomorphism is a differentiable homeomorphism with a differentiable inverse). More precisely, we say two diffeomorphisms Tlt T^-.M-^-M (say C diffeomorphisms of a C manifold, r always positive) are topologically conjugate if there exists a homeomorphism h:M-*-M such that AT, = TjA. The problem then is to study the topological conjugacy classes of diffeomorphisms of a given manifold. There are several reasons for studying the latter problem, the most important being the following. I t appears that usually a qualitative problem in differential equations has an analogue in the conjugacy problem. This analogue is a little simpler than the original, and if solved, its solution seems to give a way of doing the original problem. In any case, everything said in what follows on the conjugacy problem can be translated into statements about differential equations. At the end of our survey we indicate how this can be done. It should also be noted that the above problems may be viewed as special cases in the study of a non-compact Lie Group 0 acting differentiably on a manifold, corresponding to 0 = R and 0=Z. As enunciated in [7] for differential equations, the main conjugacy problem as we see it is the following. Given compact M, let T)M be the space of C diffeomorphisms of M in the C topology (diffeomorphisms are C close if they are point wise close together with their first r derivatives). Then one seeks an open dense subset C of D*, somehow amenable to classifica­ tion (say by numerical and algebraic invariants). A fruitful notion relative to this problem is that of a structurally stable diffeomorphism (the analogous definition for a differential equation was given by Andronov and Pontrjagin in 1937; see [3]). The h in the definition of topologically conjugate is called an equivalence between Tt and Tt, and if A is pointwise within e of the identity (in some fixed metric on M) it is called an e-equivalence. Then a diffeomorphism T^.M-^M, compact M, is structurally stable if given e >0, there exists 6 > 0 such that if d^(Tv Tt)<8 for some diffeomorphism Tt:M-*M, then Tx and Tt are e-equivalent. Here dci is a C1 metric on D*. The problem of structural stability (for diffeomorphisms) is: given M, are

608 DYNAMICAL 8Y8TKM8

491

the structurally stable diffeomorphisms dense in PM? If dim M>\, this is an open and difficult problem. In fact it is not known if there exists even one structurally stable diffeomorphism on a given manifold. In any case it is clear that the periodic points (i.e. points xBM such that Tmx=x, m=t=0, T™ the composition T:M-*-M with itself m times) and as­ sociated global stable and unstable manifolds will play a basic role in the topologies! conjugacy problem. So at this point we give the "stable manifold theorem". For more details and history, see [10]. Let T: M^-M be a diffeomorphism with fixed point p G M. The derivative of T at p is a linear automorphism of the tangent space M„ of M at p. The point p will be called an elementary fixed point of T if this derivative has no eigenvalue of absolute value one. (A) STABLE MANIFOLD THEOREM. Let p be an elementary fixed point of a G°° diffeomorphism T:M-*-M and Ex the {eigen) subspace of Mv corresponding to the eigenvalues of the derivative of T at p of absolute value less than one. Then there is a C*° map R:EX-*M which is an immersion (i.e. xoith Jacobian of ranife=dim El everywhere), 1-1, and has the property TR = RTl for some contraction TX:EX-+EV Also R(p)=p and the derivative of R at p is the inclusion of Ex into Mp. By a contraction we mean a diffeomorphism Tx of Et onto itself such that there is a differentiably imbedded disk D in Ex with TxD0T\ D=origin of Elt\)iM be the stable manifold of T™ at p where m is the least period of p,p in ourperiodic orbit. Then R: E[->-M is defined by R = T*q> where 0
609

492

S. SMALE

Axiom 2'. Let &,&>, W\, W\ be as above. Then there exists a neighborhood 1i of ft with the following property. Each component of V. fl Wl is a cell which has a non-empty transversal intersection with TF*. The same is true with W?, Wi replaced by W\, W% respectively. Axiom 3. (a) Let Q be the closure of the set of periodic points of T in M. Then for every x£M, limit m->± oo T J c i J . (6) The union of the stable manifolds of all the periodic orbits of T is a dense subset of M. The same is true of the unstable manifolds. We pose two questions: (o) Is Cj# open, dense, in Dj,? (6) Is TtCu a necessary and sufficient condition that T be structurally stable? Although very possibly, in the final picture, CM will not be the struc­ turally stable diffeomorphisms, it seems that to date it is the best guess for such ((compare [7] or [8]!). One can study these problems from the following point of view: I. The approximation problem. Approximate a given diffeomorphism by a diffeomorphism in C*, and II. The regularity problem. Find regularity properties of elements of CMWe first discuss I. One can approximate an arbitrary T€DM by T satisfying Axiom 1. In fact, (B) THEOREM. Let E be the subspace of VM consisting of T with every perio­ dic point elementary. Then 8 is the countable intersection of open and dense sets in "D„. For a proof see [10]. Independently, R. Abraham has shown that this follows from a general transversality theorem [1]. The paper of L. Markus [4] is also in the direction of Theorem G. A similar situation holds for Axiom 2. (0) THEOREM. Let"3be the subspace of E {of the previous theorem) of diffeo­ morphisms with the normal intersection property. Then "3 is the countable intersection of open and dense sets in VuSee [10] for a proof. Unfortunately, there is no similar theorem known for Axiom 2'. In Axiom 3, parts (a) and (6) seem to be related in some fashion, but it is not clear how. Does (a) imply (6), or conversely? To approximate a given diffeomorphism by one satisfying 3 a or 3 b is a central problem, related to what is sometimes called the problem of the "closing Lemma". See Peixoto [5] for an account of this important problem. A special case of our approximation problem is the following. (D) Problem. Let T.M-+M be a diffeomorphism of a compact manifold. Is there a C approximation T' of T such that T' has a periodic point? The answer is not known for the 2-dimensional torus. In the discussion of the regularity problem stated above, we start with the case of diffeomorphisms satisfying a highly restrictive axiom in addition to Axioms 1, 2', 3. Axiom 4. T has a finite number of periodic points. The set of diffeomorphisms satisfying Axioms 1, 2', 3, 4 is denoted by CM.

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It is important to note that in assuming Axiom 4, an open set of diffeomorphisms of Vu is lost (see e.g. [11] or below). On the other hand one can say some substantial things about elements of CM(E) THEOREM . Diffeomorphisms in CM satisfy a form of the Morse inequalities relating the periodic points. The union of the stable manifolds associated to the periodic points is all of M. The boundary of a stable manifold of dimension p is the union of stable manifolds of dimension <j>. The proof is essentially contained in [81. A task which seems important and yet tractable is to show that the TECM are structurally stable. However this has not even been carried out under the additional assumption of Axiom 5. Axiom 5. Let W", W' be an unstable, stable manifold respectively of periodic orbits of T which have non-empty intersection. Then dim WT+dim JPJ>dim M. We say that the set of T satisfying all of our axioms is CM- The following can be proved along the lines of [8], [9]. (F) THEOREM. On every manifold there exist non-empty open sets of IDM which are contained in CM- If T€CM> the boundary of a stable manifold of T is the union of lower dimensional stable manifolds. The components of the stable manifolds generate the homohgy of M in a natural way and the corre­ spondence between the stable manifold of a periodic point and the unstable manifold induces Poincari duality. The Morse inequalities in the previous theorem can be interpreted to include the usual ones. This theorem shows that CM, and hence CM, CM a r e n o * empty for any M. Thus if it could be shown that every T£CM is structurally stable, we would have proved that there exist structurally stable diffeomorphisms on every compact manifold. Next the question comes up as to the existence of elements of CM which are not in CM, i.e. those with an infinite number of periodic points. The following is in [11]. (G) THEOREM. There exist open sets in Vu for M an aribitrary n-sphere, n > 1, with the following properties: (1) the diffeomorphisms are in CM', (2) they are structurally stable; (3) the diffeomorphisms have an infinite number of periodic points (and minimal sets homeomorphic to a Cantor set). This theorem answers the question as to whether a structurally stable diffeomorphism (differential equation) can have an infinite number of periodic points (closed orbits). It seems that this theorem and its proof are quite important. It shows that one can cope successfully with difficult phenomena present in differential equations of dimension greater than two and not present in two-dimensional differential equations. In the examples of Theorem 6, one has present homoclinio points. A homoclinic point associated to a periodic orbit /? of a diffeomorphism is a. point of intersection of the stable and unstable manifolds associated to /?. Homoclinic points were first discovered by Poincare' [6] in the restricted 3-body problem, and studied by Birkhoff [2]. Merely the existence of a homoclinic point implies considerable complications. At the end of his

611

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8. SHALE

three volumes on celestial mechanics [6, p. 389], referring to homoclinic points, Poineare* wrote: "On sera frappe" de la complexity de cette figure, que je ne cherche meme pas a tracer. Rien n'est plus propre a nous donner une idee de la complication du probleme des trois corps et en g6n6r&l de tous les problemes de Dynamique " The methods used in proving Theorem G not only completely describe the homoclinic situation in those examples, but can be applied to give some understanding of arbitrary homoclinic points as well. A different example which exhibits stable manifolds and homoclinic points has a simple description. In the plane E* let T0 be a linear automor­ phism given by a 2 x 2 matrix with integer entries, determinant + 1 , and an eigenvalue greater than one in absolute value. Then T0 induces a diffeomorphism T of the torus. The origin of E2 projects into a fixed point p of T and the eigendirections of T0 project into the stable and unstable mani­ folds of p. One easily sees that the homoclinic points associated to p are dense in the torus. (H) THEOREM. The T described above is a structurally stable diffeomorphism of the torus. After a meeting in September 1961 in Kiev on non-linear oscillations (where I announced Theorem G in dimension 2 [14]). I visited Moscow and spoke with mathematicians D. V. Anosov, V. I. Arnold, and Y. G. Sinai, among others. There I conjectured Theorem H and that geodesic flows on compact Riemannian manifolds of negative curvature were also structurally stable. Since then I and (as I have learned at this Congress), Arnold and Sinai have independently proved Theorem H, the latter proof having the advantage of being published [13]. In addition, Anosov has very recently proved (as I have learned also at this Congress) a beautiful theorem which settles the above conjecture for geodesic flows affirmatively and gives the n-dimensional generalization of Theorem H [12]. I t seems likely that if TtCM and TeCit, then T has homoclinic points. In the same vein one can ask if Axiom 5 implies Axiom 4 (certainly the converse is false). An elementary periodic point will be called elliptic if all the associated eigenvalues have absolute value less than one, or all greater. It seems resonable to expect that if T€CM, then T will have only a finite number of elliptic points. It would be nice to have a proof of this. Also if TZCM where M is the 2-sphere, must T have at least one elliptic point? We indicate briefly how the previous discussion goes over into the ana­ logous situation for differential equations. The definition of a structurally stable differential equation is exactly the same as for diffeomorphisms. The problem of structural stability for differential equations asks if the struc­ turally stable differential equations on a compact manifold M are dense in the Banach space "BM, C norm, of all C differential equations on M. This has been answered in the affirmative if dim M <2 by M. Peixoto, see Theo­ rem I, below. One constructs stable and unstable manifolds for differential equations associated to each singular point and each closed orbit of a general type (corresponding to elementary periodic points). For details see [10]. The analogues of the previous axioms can be stated for this case. The union of the singular points and closed orbits of the differential equation replace the periodic points of the diffeomorphism. Let B£ be the subspace of Bu of

612 DYNAMICAL SYSTEMS

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differential equations satisfying the analogues of axioms 1, 2', 3. The ele­ ments of B£ will have only a finite number of singular points but may have an infinite number of closed orbits if dim M >2. For the 2-dimensional case there is the following important theorem of Peixoto [5]. (I) THEOBEM. Let 31* be a compact 2-manifold. Then (a) X on Mis structur­ ally stable if and only if X e B i ; (6) B£ is open and dense in BMAlso the analogues of the previous theorems are all valid for the differen­ tial equations case. We comment on the problem of the existence of a first integral of a differ­ ential equation. A first integral of X on M in a C function f:M-+B such that f is constant on each orbit but not on any open set. Problem E has the following analogue. (E*) Problem. Can every non-singular vector field on a compact manifold be C approximated by one with a closed orbit? If Problem E* has an affirmative solution it follows that: (J) The subset of X 6B* with an elementary closed orbit is open and dense

inB*. Here elementary corresponds to elementary periodic point and the precise definition is in [10]. Putting J together with the analogue of Theorem B, as has been essentially observed by B. Thorn, we obtain: (J*) There exists a subset of X€B M with no first integral which is open and dense in B«. A relation between the topologies! conjugacy problem for diffeomorphisms and the equivalence problem of differential equations is given by crosssections, used by Poincare' and Birkhoff (e.g. see [2] or [10]). In a different direction, since a differential equation generates a 1-parameter group, one may ask under what conditions, is a diffeomorphism imbeddable in a flow? The following is not difficult. (K) THEOBEM. Let TeCu- If T can be imbeeded in a flow, then T satisfies Axioms 4, 5, every periodic point is fixed and in the neighborhood of every fixed point, T is imbeddable in a flow. It would be interesting to know if the converse is true. Certainly the main problems stated here are very difficult. On the other hand, .it seems quite possible to us that this field may develop rapidly and already as indicated here, there have been some initial steps in this direction.

REFERENCES [1]. ABRAHAM, R., Transvereality of manifolds of mappings. (To appear.) [2]. BIRKHOIT, G. D., Collected Mathematical Papers. New York, 1950. [3]. LxncHKTZ, L., Differential Equations: Geometric Theory. Interscience Publishers, New York, 1957. [4]. MABOTTS, L., Structurally stable differential systems. Ann. Math., 73 (1961), 1-19. [5]. PBIXOTO, M., Structural stability on 2-dimensional manifolds. Topology, 2 (1962), 101-121. 35 - 622038 Proceeding*

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[6]. POINCAKB, H., Les Mlthodta NouvelUs de la Micanique Cilette, Vols. I-III. Gauthier-Villars, Paris, 1899. (Reprinted New York, 1957.) [7]. SHALE, S., On dynamical systems. BoUtin de la Sociedad Matematica, 1960, 195-198. [8]. Morse inequalities for a dynamical system. Btdl. Amer. Math. Soc., 66 (1960), 43-49. [9]. On gradient dynamical systems. Ann. Math., 74 (1961), 199-206. [10]. Stable manifolds for differential equations and diffeomorphisms. (To appear.) [11]. Diffeomorphisms with many periodic points. (To appear.) [12]. ANOSOV, D. V., Structural stability and ergodicity of geodesic flows on compact, Riemannian manifolds of negative curvature. (To appear.) [13]. ABNOLD, V. I. & SINAI, Y. G., Dold. Ahad. Nauk S.SJS.R., 144, 4 (1962), 695. [14]. SMALE, S., Report on the Symposium on Non-linear Oscillations. Kiev Math. Institute, 1961.

614 EftttiiUo dagli Jnnali d»Ua Seuola Normal* Superior* di Seile ril. Vol. XVII. F M O . I-II (1963)

Pha

STABLE MANIFOLDS FOR DIFFERENTIAL EQUATIONS AND DIFFEOMORPHISMS (*) S. SMALK(*) (New York)

1. Preliminaries. A (first order) differential equation (« autonomous ») may be considered as a G°° vector field X on a C°° mauifold M (for simplicity, for the moment we take the C°° point of view; manifolds are assumed not to have a boun. dary, unless so stated). From the fundamental theorem of differential equa­ tions, there exist unique C°° solutions of X through each point of M. That is, if x 6 M, there is a curve 95, (ar), | t | < « such that,

d0 (x) = x, — (x)l

=

= x (

(b)


(') Leotures given at Urbino, Italy, July 1962, CIME. This work has been partially supported by the National Soienoe Foundation nnder Grant GP-24. («) The author was a Sloan fellow dnring part of this work.

615 98

S. SKALK : Stable manifold* for

Then for each t, i f is a diffeomorphism (a differentiable homeomorphism -with differentiable inverse). A differential equation on a com­ pact manifold defines or generate* a l-parameter group of transformations of M. We shall say more generally that a dynamical system on a manifold M is a l-parameter group of transformations of M. dw, v(x) I If (ft is a dynamical system on M, / = X(x) defines a C°° at \t _ o vector field on M -which in turn generates cpt. We also speak of X as the dynamical system. Let X, Y be dynamical systems on manifolds Mi, Mt respectively ge­ nerating 1 parameter groups • M2 with the property that h maps orbits of X into orbits of Y preserving orientation. The homeomorphism h: Ml —>■ Mt will be called an equivalence. Often Ml = M%. The qualitative study of (1st order) differential equations is the study of properties invariant under this notion of equivalence, and ultimately finding the equivalence classes of dynamical systems on a given manifold (s). In this paper we are concerned with the problem of topological equi­ valence. An especially fruitfull concept in this direction is that of structural stability due to Andronov and Fontrjagin, see [5]. The definition in our context is as follows. Assume a fixed manifold M, say compact for simplicity, has some fixed metric on it. An equivalence h: M —y M (between two dynamical systems on M) will be called an e-equivalence if it is pointwise within e of the identity. One may speak of two vector fields X and Y on M as being 0' close (or dc (X, Y) < d) if they are pointwise close and in addition, in some fixed finite covering of coordinate systems of M, the maximum of the difference of their 1st derivatives over all these coordinate systems is smalL (Similarly one can define a Gr topology, 1 < r < oo, see [7]). Then X is structurally stable if given e > 0, there exists d > 0 such that if a vector field Y on M satisfies dc (A', Y) < d, then X and Y are a-equivalent. The problem of structural stability i s : given M compact, are the struc­ turally stable vector fields on M, in the above C topology, dense in all vector fields. If the dimension of M is less than 3, the answer is yes by a theorem of Peixoto | 9 ] ; in higher dimensions it remains a fundamental and difficult problem.

(') For a survey of this problem see talk in the Proceedings of the International Congress of Mathematicians, Stockholm 1962.

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differential equation* and diffeomorphisms

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Although in this paper we are not concerned explicitly with structural stability, this concept lies behind the scenes. Attempts at solving the pro­ blem of structural stability, guide one toward the study of the generic or general dynamical systems in contrast to the exceptional ones. There seems to be no general reduction of the qualitative problems of differential equations. However, there is a problem which has some aspects of a reduction. This is the topological conjugacy problem for diffeomorphisms which we proceed to describe. Two diffeomorphisms T : Mt —> Mt, T': Mt —>• Mz are topologically (differentiably) conjugate if there exists a homeomorphism (diffeomorphism) h: Mt —>• Mt, such that T'h=hT. Often Mt = Mz . This topological conjugacy problem is to obtain information on the topological equivalence classes of diffeomorphisms of a single given manifold (*). When dim Jtf = 1, the problem is solved according to results of Poincar6, Denjoy and others, see [2]. For dim M > 1, there are very few general theo­ rems. We now explain the relevance of this problem to differential equations.

2. Cross-sections. Suppose X, or 0 with t (*) € 2. It is not difficult to prove that T-.2—+2 is a diffeomorphism, called the asso­ ciated diffeomorphism of 2. One can also easily prove that, if M is compact and connected, then conditions (c) and (d) in the definition of cross-section are consequences of (a) and (b). Suppose on the other hand T0: 20—>20 is a diffeomorphism of a ma­ nifold. Then on R x 20 let (t, x) be considered equivalent to the point (* -f 1, T (*)). The quotient space under this equivalence relation is a new differentiable manifold say M0. Let X0 be the dynamical system on MQ in­ duced by the constant vector field (1, 0) on B x 20, and n: B x 20 —> M0

(') See footnote (8).

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S. SMAMC : Stable manifolds for

the qaotient map. We say that X0 on M0 is the dynamical system determined by the diffeomorphism T0 : 20—*20. 2.1. LEMMA.

Let q>t be dynamical system generated by X on M, which admits a cross-section 2. Then by a C °° reparameterization «„ (t) of t, x £ M, one can obtain a 1-parameter group cp, of transformations of M such that if x £ 2, q>i (x) £ 2 and q>, (x)i2 for 0 < * < 1. We leave straightforward proof of 2.1 to the reader. 2.2.

THEOREM.

Suppose the dynamical system ■ 2. Then X0 on MQ is equivalent to X on M (by a diffeomorphism in fact). PROOF. First apply 2.1. Then the desired equivalence of 2.2 can be taken as induced by / : M —y R x 2, f(
If T0 : 20 —>- ^o is a diffeomorphism, the dynamical system it determines has a cross-section 2 with the property that the associated diffeomorphism is differentiably equivalent to T0. For the proof, of 2.3, one just takes n (0 x 20) for 2, and the equiva­ lence is induced by the map of 20—*Rx20 given by x —y(0,x). 2.4 THEOREM.

Let T0 : 20 —)- 20, Ti: 2i —>■ 2i be diffeomorphisms which determine respectively dynamical systems X0 on M0 and X, on M,. If T0 and T, are topologically (differentiably) equivalent then X0 and X t are topologically equivalent, (equivalent by a diffeomorphism). The proof is easy and will be left to the reader. The preceeding theorems show that if a dynamical system admits a cross section, then the problems we are concerned with admit a reduction to a diffeomorphism problem of one lower dimension. Furthermore every diffeomorphism is the associated diffeomorphism of a cross-section of some dynamical system. REMARK.

The existence of cross-sections in problems of classical mechanics first motivated Poincare* and Birkhoff [1] to study surface diffeomorphisms from the topological point of view.

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101

A local version of the proceeding ideas on cross-sections is especially useful. A closed or periodic orbit y of a dynamical system t (x) with the property q>t [x) = x for some t 4= 0. A periodic point of a diffeomorphism T: 2 —y 2 is a point p £ 2 such that there is an inte­ ger m 4= 0 with Tm(p)=p(Tm denotes the mth power of T as atransformation). The following is clear. 2.5 LEMMA.

Let (pi be a dynamical system on M with cross-section 2 and associated diffeomorphism T. Then p £ 2 is a periodic point of T if and only if the orbit of the dynamical system through p is closed. A local diffeomorphitm about p £ M is a diffeomorphism T: TJ -+M, U" a neighbourhood of p and T Q>) = J>. Two local diffeomorphisms about pl£Mi1p%£M2, Ti: CTj —)■ if,, Tt: TJ2—y M2 are topologioally (differentiably) equivalent if there exists a neighbourhood U of j>t in Uj and a homeomorphism (diffeomorphism) h: U—y U2 such that ft(j>t) = j?E and Tg A(x) = = ft Ti (a?) for x € 2T 1 (U")n U- The following is easily proved. 2.6 LEMMA.

If X is a vector field on a manifold Af, X(p) 4= 0, for some p £ M, there exists a submanifold 2 of codimension 1 of if containing p and transversal to X. Let y be a closed orbit of a dynamical system t (x) £ 21 where * is in some neighbourhood U of p in 2" and t is the first t > 0 with ■ .2, the local diffeomorphism associated to the closed orbit y, 2 a local cross-section. 2.7. LEMMA.

The differentiable equivalence class of T depends only on y and the vector field X. It is independent of p and 2. PROOF. Let pi, p% e y with local cross-sections 2t, 22 respectively. Assume first pi 4= p%. Then we can assume 2tn 2t= §}• Define ft : U—>■ 2S, for Z7 a sufflciently small neighbourhood of p in 2t, by ft (x) = q>t (x) for x 6 U by taking t > 0 the first t such that 9>< {x) 6 J-fj . Then ft acts as a dif­ ferentiable equivalence. If jp4 = j>2 , take p3 £ y, distinct from p{, and with local cross-section 23. Then apply the preceeding to show that the local diffeomorphism of 2t is differentiably equivalent to that of 23 and that of 22 is differentiably equivalent to that of 2t • Transitivity finishes the proof.

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S. SMAUC : Stable manifold* for

Now given a local diffeomorphism about p Z 2, T: U —> Z, one can construct a manifold MQ, with a vector field X 0 , containing a closed orbit y with Z as a local cross-section. The construction is the same as in the global case. Moreover, and this is a useful fact, the local analogues of 2.2-2.4 are valid.

3. Local Diffeomorphisms. 3.1. THEOREM.

Let A : En —> En be a linear transformation with eigenvalues satisfying 0 < | Xi | < 1. Then there exists a Banach space structure on E" such that

MH = a
Oy

and

yO 0y : - j J a

Here y can be taken arbitrarily small, and a -f- i/J, a —1/9, 6 are the eigenvalues. These canonical forms may be deduced from the usual Jordan canonical form, and the following two easy lemmas. 3.1a

LEMMA.

The linear transformations given by the following two matrices are equivalent, where a, /? are real. a A — /3 a)

/a + i/J 0 \ \ 0 a — ip)

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differential equation* and difftonxorjikunxt

103

3.1b LEMMA

The linear transformations given by the following two matrices are equivalent where y is nonzero, but otherwise arbitrary. X

\

IX 0

/X

\

I y X

••

'1XI

\

y X

The equivalence js given by / l

r

o

A linear transformation satisfying the conditions of 3.1 will be called a linear contraction. 3.2 THEOREM.

Let T be a local diffeomorphism about the origin 0 of En whose deri­ vative L at 0 is a linear contraction. Then there is an equivalence R between T and L which is 0 0 0 except at 0. In fact there is a global dif­ feomorphism 2" : E H —>■ E" which agrees with T in some neighbourhood of 0 and a (global) equivalence R between T' and L, C° except at 0. PBOOP.

_ By 3.1 we can assume // L // <9 <1, and that T(x) = Lx + / ( # ) where IJ1W-11-+0 as / / * / / - > 0. Choose r > 0 so that / / / ( x ) / / < (1 — 0 ) / / * / / IIx II for II x II < r. The following is well known. 3.2a

LEMMA.

Given r>Q, there exists a real G°° function

l/xl/>r. Let / (x) = q> (w)f(x) where *-, T0(x) = Lx. Define R-.En—*-B* by R (0) = 0 and Rx = T0W L~N x where If is large enough so that //£-**//>»-. It is easy to check that R is well-defined, has the equivalence property,

621

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S.

SMALI

: Stable manifold* for

and is a <7°° diffeomorphism except at 0. It remains to check that R is continuous at the origin, or that // R (x) // —■> 0 as // xjj —)• 0. First note that there exists h < 1, so that for all x 6 E ", // T0x // < lex. Also R (x) = = T ^ L~N x = T,,* y where y = XT* x and we can assume / / y // < .M. Then continuity follows from the fact that as // y // —»• 0, the JT of defintion of R(x) must go to infinity. A local diffeomorphism satisfying the condition of 3.2 is called a local contraction. A contraction of Ei is a diffeomorphism T of II9 onto itself such that there is a differentiably imbedded disk D c J « with T Dc inte­ rior D, n t > 0 T < D = origin of Ei, U « 0 T < D = .E9. Thus using 3.1, the T constructed in 3.2 is a contraction. If all the eigenvalues of a linear transformation L have absolute value greater than one, then L is called a linear expansion. If the derivative at p of a local diffeomorphism T about p is a linear expansion then T is called a local expansion. The inverse of a linear (local) expansion is a linear (local) contraction. In this way 3.1 and 3.2 give information about linear and local expansions. The following theorem was known to Poincare' for dim E = 2. One can find n dimensional versions in Petrovsky [10], D. C. Lewis [6], Coddington and Levinson [2], Sternberg [14] and Hartman [4]. Some of these authors were concerned mainly with the similar theorem for differential equations. 3.3

THEOREM.

Let T: U—±E be a local diffeomorphism about 0 of Euclidean space whose derivative L: E —► E at 0 is a product of Li,Ei—t-El,Li:Et—t-Ei, -1 E = ElxEi where / / L i //, //X- // < 1. Then there is a snbmanifold V of U with the following properties: (a) 0 E V, the tangent space of 7 at 0 is Et, (b) T 7 c V, and (c) there exists a differentiable equivalence R between a local diffeo­ morphism T' about 0 of E whose derivative at 0 is Lt and T restricted to V. (d) V = (X]LiBj where BQ = UflTU and Bj is defined inductively by B / = T - i ( B i _ 1 n S 0 ) . Due to the previous discussion in this section, the hypothesis of 3.3 is mild, merely that no eigenvalue of L has absolute value 1. One may apply 3.2 to the restriction of T to V. Note by applying 3.3 to T _ 1 one can obtain a snbmanifold Vt of U containing 0 whose tangent space at 0 is Et and T restricted to V% is a local expansion. We call V the local sta­ ble manifold, Vt the local unstable manifold of T at 0. Use (*, y) for coordinates of E = El x Et, so that one can write, using Taylor's expansion,

T (
?1

(•, y), Lt y + gt (x, y)).

622

differential equation* and diffeomorphism*

105

The proof of 3.3 is based on the following lemma. 3.4 LEMMA.

There exists a unique C°° map • Et, Ul a neighbourhood of 0 in

$(Lix

Ei,

+ gt {x, $ (x))) = £ 8 # (*) + gt (x, <J> (*)).

Furthermore (x, 0 (x)) £ Ojl* Bj, Bj as in (d) of 3.3. To see how 3.3 follows from 3.4, let V be the graph of
4. Stable manifolds of a periodic orbit. The global stable and unstable manifolds we construct in this section were considered by Poincare" and Birkhoff [1] in dimension 1 for a surface diffeomorphism. The analagous stable manifolds for a dynamical system (see section 9) have been considered by Elsgoltz [3], Thorn [15]. Beeb [11] and in [12]. Suppose T: M —+ M is a diffeomorphism and p £ M is a periodic point of T so that Tm{\>) = p. The derivative £ of T " at p will be a linear au­ tomorphism of the tangent space Mp of M at p. The point p will be called an elementary periodic point of 2' if L has no eigenvalue of absolute value 1, and transversal if no eigenvalue of L is equal to 1. 4.1 THEOREM.

Let p be an elementary fixed point of a diffeomorphism T: M —+ Mf and Ei the subspace of Mp corresponding to the eigenvalues of the deriva­ tive of T at p of absolute value less than 1. Then there is a C°° map R: El-+M which is an immersion (i. e. of rank = dim .E, everywhere), 1-1, and has the property TB = RT' where T': E{ —> Ei is a contraction of Ei. Also R(p)=p and the derivative of R at jp is the inclusion of Ei into M,.

623

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S. SMALE : Stablt manifolds for

PHOOF. One applies 3.3. The map B of 3.3, say B0, is defined in a neighbourhood U of 0 in Et into M. We now extend it to all of Ei to ob­ tain the map R of 4.1. By 3.2 we can assume T' of 3.3 is a (global) con­ traction of Ei. If x £ Ex, let Rx = T~N R0TNx where JT is large enough so that T'Nx£ U. I t may be verified with little effort that R is well-defined and satisfies the conditions of 4.1. The map R : El —y M, or sometimes the image of R, is called the stable manifold ofp or T at p. The unstable manifold of p or T at p is the stable manifold of T _ 1 at p. These objects seem to be fundamental in the study of the topological conjugacy problem for diffeomorphisms. An (elementary) periodic orbit is the finite set \Ji(tTip where p is an (elementary) periodic point. The definition of the stable manifold for an elementary periodic orbit (or sometimes elementary periodic point) is as follows. Let

where 0 < l < » i and Ei is a copy of Ei. Thus the stable manifold of a periodic orbit is the 1 — 1 immersion R of the disjoint union of m copies of a Euclidean space. The stable manifold of a periodic point B is defined to be the com­ ponent of the stable manifold of the associated periodic orbit in which B lies. The unstable manifold of a periodic orbit (periodic point) is the stable manifold of the periodic orbit (periodic point) relative to T~\ 5. Elementary periodic points. Let Q) be the set of all diffeomorphisms of classe Cr of a fixed compact G manifold M onto itself, o o > r > 0 . Endow
5.1

THEOREM.

Let i f be a compact C manifold, r > 0, and Q) the space of C diffeo­ morphisms of M endowed with the C topology. Let B C ^ b e the set of T with the property that every periodic point of T is elementary. Then 5 is a countable union of open dense sets. We prove the following stronger theorem which implies 5.1 (since 5= r\p(z+5p, Z+ denotes the positive integers). 5.2

THEOREM.

Let
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differential equation! and diffeomorphisrnt

107

PROOF.

We first show that S, is open Q>. Let TQ 6 5",, Tt —>• T0 in D, T< € D. It must be shown that T< E £"p for large enough •'. Suppose not. Then there exist Pit = 1, 2,..., | p , | at a?0 has no eigenvalue of absolute value 1. On the other hand the derivative of Tf* at *4 for all t has an eigenvalue of absolute value 1. This is a contradiction since Ti—t-T in the C topology, r > 0 and X{—+X0.

We next show that Sp is dense in Sp-i, p > 1, 5"0 = <7). This will finish the proof of 5.2. Let Sr be the analogue of Sv with elementary replaced by transversal. Then it is sufficient to prove: (1) (Ep fl 5p_i is dense in S^.x, and (2) Ep is dense in Sp. We first do the main step, (1). For p = 1, we use the following easily proved lemma. 5.3 LEMMA.

Let TxM—i-M. be a diffeomorphism. Then a; £ If is a transversal perio­ dic point of T of period p if and only if the graph r of T* and the dia­ gonal A in M x M intersect transveraally at (p,p) (i. e. the tangent space of A and r at (p,p) span the tangent space of M x M at (p,p). Then a general position theorem of differential topology applies to yield that S{ is dense in 93 (see Thorn [16]). Let T E 5p_! and /?,,..., /J* be all the periodic points of T of period

T* x) where | ♦ |

0. By possibly choosing 61 smaller we can assume that any set U of diameter < 2d is contained in a coordinate neighbourhood of M and hence that T(U), has the same property. Next for tCM, let U(x), V{x), W{x) be neighbourhoods of radius d, 1/2', 1/34 respectively. Let (Ua, Va, Wa) for a = l,... ,q be a finite set of these such that U Wa = if, for each a choose a coordinate neighbourhood Ea 3 GLTUa. Then using the linear structure of J8a, T — /S~i: D„ —► E„ is a well defined map where 8 = T*- 1 and D a = (# € Ua \ S-1 x E Ha\. By Sard's theo­ rem, see e. g. [16], choose a map ga: D„ —> Ea, small with its first r deri­ vatives so that T — /8 _ 1 / g „ : (Ta—yXlt, has 0 has a regular value. Starting with a = 1, let Tl = T outside Uit Ti=T + gt on Vt using 3.2a. Then Ti

625

108

S.

SMALK

: Stable manifold* for

restricted to F, has transversel periodic points of period p as can be seen as follows: If x £ Fj and T,p (x) = x, then Tf (x) = T(p_1) Tj x and Ttx = /8 _ 1 x. So (Tj — S - 1 ) * = 0 and since T, — S-1 has a regular value at 0, the de­ rivative of Tjp — I at x is non-singular and a; is a transversal singular point of Ti of period p. One makes the same construction for a = 2 , . . . , q, making sure that ga is so small with respect to the « bump function » that the difieomorphism retains its desirable qualities on N and Wt,..., Wa-i • This proves that Fp is dense in -F p _i. We finally show that Sp is dense in Sp. Let T0 £ Sp . Then by 5.3 the periodic points of T0 of period

M such that T = T0 outside of Nt, T approximates T0 , and T has ft as an elementary periodic point. This can be easily done using 3.2a and the fact that linear transformations with no eigenvalue of absolute value 1 are dense in all linear transformations. This finishes the proof of 5.2. REMARK.

If Ti S, then given an integer N, there exists only a finite number of points of period < y of T. This follows from 5.3. Hence T has only a coun­ table number of periodic points.

6. Normal intersection. Two submanifolds Wt, Wt of a manifold M have normal intersection if for each x e IF, (\ Wt, the tangent space of W, and IF, at x span the tangent space of M at x. A diffeomorphism T: M —>- M has the normal in­ tersection property if when /?,, /?2 are generic periodic points of T, the stable manifold of fit and the unstable manifold of /?2 have normal intersection (this definition is clear even though the stable manifold is not strictly a submanifold). Let Q) and S be as in the previous section and let £ be the subspace of 3 of diifeomorphisms with the normal intersection property. 6.1. THEOREM.

£ is the countable intersection of open dense subsets of Q). (The first theorem of this kind seems to be in [13]).

626

differential equation* and diffeomorphitms

109

Let oar basic manifold M have some fixed metric and let 2T((x), for E > 0, x E M, denote the open £ neighbourhood of x in M. Let R+ be the set of positive real numbers. 6.1a

LEMMA.

For each p£Z+, there exists a continuous function E : Sp —y R+ with the following property. If T E Sp, x 6 M is a periodic point of T of period <j>, then CX[3re(r)(ar)n W (x)) c TF(x), where W(x) is either the stable or unstable manifold, W'(x) or Wn(x) respectively, of x with respect to T. PBOOF.

It is clear that on an open neighbourhood Na of each Ta E Sp that one can find a constant function E„ with the property of E of 6.1a. Let E„ , Na be a countable covering of Sp of this type, a = 1, 2,.... Then let E' = £, on J5T, , E' = min (€ t , E8) on 2?8 — JTj, min (£4, E2, Es) on .tfj — 3F, — N~2 etc.. Then E1 is lower semi continuous on Fp. Finally, for example by Kelly, General Topology, New York 1955 p. 172 one can obtain the E of 6.1a. Now if x is a periodic point of TtFp of period
THEOREM.

E* is open and dense in D. Note that (6.1) follows from (6.2) because ^ = ^pklZ+^k For 6.2 we first remark that E is clearly open in Q). Hence in view of 5.2 it is sufficient to show that E£ is dense in 3P . Let TtEp. Denote by ft,..., ft. the periodic points of T of period <j> with stable and unstable manifolds R V = WT(ft), r = *, w. We will consider only approximations T' of T which agree with T on some neighbourhood V0 of the ft , and so that ft, i = 1,..., fc0 are precisely the periodic points of T' of period .»ro. Let f « = T—»*T•""•*. For * E WY* one takes n < n0 < 0. Then f is a well defined 1-1 immersion.

627 110

S. SMALB : Stable manifold$ for

Now fix i,j, l
IF/ 3 T'-'- (r2))) r 2 )) r _ k (i/j). Here Mt is the least period of ft, ntj that of ft and )) is interpreted as to mean «contains an open set containing». Such Yt clearly exist from 3.1, 3.2 and 4.1. If 2" is an approximation of T agreeing with T on a neighbourhood VQ of the ft let $ Yt = 37, t = 1, 2. Then without loss of generality we can assume (6.3')

Wtm' => T'm (Ii))) Yi)) T' k (£<"')

w/' 3 T'- 1 - (r,))) r2)) r'~* (if). Hence it is sufficient to find such a T' with Y( and Yi having normal intersection. The compact subset GL (T0mi rx— Y{) is so to speak a fundamental domain of T"*» restricted to WV*. Thus one may find without difficulty connected open sets Zi, Zt in W" with compact closures which are each disjoint from their images under Tw» and in addition Zx\JZtZ3 C£(T m i Yi — Yt). Let P = O Z i { 2 , , ( f i n T 2 )|*>0) 0=O2i{T-'(Z1)|I>lj The following is easily checked. 6.4 LEMMA.

pnr-nz.n r,) = Q ynT- i (^ 1 nr 2 )=ts

628

differential equations and diffeonorphismt

111

Let Up, Uq, V be open sets such that Up Z> P, Uq Z> Q, V Z3 T-1 (Zt (\ P,) and U,{\ T— Q, 17, H V= Q. By the Thorn tranaversality theorem [16] and a suitable patching by a C00 function (similar) to 3.2a) one can find an approximation T' of T with the following properties: a) T = T on a neighbourhood VQ of the /?< and the complement of V in M. (b) T'lT-1 (Zj)] and Yt have normal intersection (i. e. W) and 2" [T -, (£|)] have normal intersection on T [T-1 (Zx)\ fl r 2 ). Suppose now that x 6 T/ fl Yi, and jTm 0 and T'mxZ YJ (b) Z\ is T'[T-i(Z,)] and so T"»xlT' [T-*(Zt)]. (c) there exists a neighbourhood of T'mx in Y'2 which is in Y g . It can be shown without difficulty that (a), (&), and (o) are consequences of the choice of V. Now one carries out exactly the same procedure with Zj = q (Zt) re­ placing Zi in the argument. This gives us an approximation T" of T' with the desired properties of 6.2. 7. Elementary singularities of a vector field. We now pass from the diffeomorphism problem to the case of a dyna­ mical system. Let M be a compact & manifold 1 < r < oo and /? the space of all C vector fields on M with the Cr topology. One may put a Banach space stru­ cture on 0 if »• < oo. In any case /? is a complete metric space. A singularity p of X on M is a point at which X vanishes. Let p be a singularity of X on M. Then using some local product structure of the tangent bundle, in a neighbourhood TJ of p, X is a dififerentiable map, X: U —>• Mp, whose derivative A at p is a linear transformation of Mp . We will say that p is an elementary singularity of X on M if the de­ rivative A of X at p has no eigenvalue of real part one, and transversal if A is an automorphism. Let Q. be the subset of (i such that if X 6(2 X has only elementary singularities. 7.2 THEOREM.

6 is an open dense set of /?. To see thas, one first checks the following lemma.

629

112 7.3

S. SHAM : Stable manifold* for LEMMA.

Let X be a vector field on M. Then are i f is a transversal singular point of X if and only if X, as a cross section in the tangent bundle meets the zero cross-section over M transversally. From this and the transversality theorem of Thorn [16] one concludes 7.4

LEMMA.

Let Q' be the subset of /S of vector fields on M which have only tran­ sversal singular points. Then C is an open dense subset of /?. Now 7.2 follows from 7.4 as in the proof of 5.2 where 2r was shown to be dense in S,. Note that if XtG, or even Q', by 7.3, the singular points of X are isolated and hence finite in number.

8. Elementary closed orbits. Let y be a closed orbit of a vector field X on a manifold with asso­ ciated local diffeomorphism T: U—>-2 about p G y fl 2. Then y will be called an elementary (transversal) closed orbit of X if T has p as an ele­ mentary (transversal) fixed point. 8.1

THEOREM.

Let QQ be the subspace of Q (of section 7) of vector fields X on M such that every closed orbit of X is elementary. Then Q$ is the countable intersection of open dense sets of /?. L. Marcus [18] has a theorem in this direction. Also B. Abraham has an independent proof of 8.1 [17]. If y is a closed orbit of X on M, then one can assign a positive real number, the period of y as follows. Let x £ y, q>t, (x) = x where t0 > 0, y>t (x)r*x, 0 < t < t0. Then t0 is an e invariant of y, the period of y. For a positive real number L, let S L C S consist of X on M such that, if y is a closed orbit of length < L, then y is elementary. Since GQ = 0 , QL > with 7.2, 8.1 is a consequence of the following. 8.2

THEOREM.

For every positive X, QL is open and dense in 6. The proof is somewhat similar to the proof of 5.2. First that QL is open in Q follows from a similar argument to that of 5.2 used in showing that Sp is open in S. We leave this for the reader. It remains to show: QL, i s dense in Q. Let XiQ. The first step is to

630

differential equation* and diffeonorphitm*

113

construct a finite number of open cells Ua of M of codimension 1, transversal to X such that Ua 3 Wa where Wa is a closed sub-disk of Ua such that every trajectory of X passes through some Wa. It is a straightforward matter to show that such a set of (Ua, Wa) exists. Fixing a now, tbe next step is to approximate X by X', a vector field on M eqnal to X outside a neighbourhood of Wa so that if y is a closed orbit of X' of length < L, iutersectiDg some fixed neighbourhood of Wa in TJa, then y is elementary. The existence of such an approximation is suffi­ cient for the proof of 8.2. The construction of the approximation X' of the proceeding paragraph is based on tbe methods of Section 2 and 5. We outline bow this is done. Let F a be a compact neighbourhood of Wa in Ua. Then let Da C Ua be the set of points * of Ua such tbat -Fa the associated difleomorphism, say really defined on some neigh­ bourhood of J)a in Ua. Now apply tbe methods of 5.2 to approximate T by T' such that T is defined in a neighbourhood of Da and that T has only generic periodic points. Now using the construction of Section 2 and 3.2a one defines the above X' using 1". 9. Stable manifolds for a differential equation. The following is tbe global stable manifold theorem for singularities of a vector field. 9.1. THBORBM

Let X be a C°° vector field on a 0°° manifold generating a 1-parameter group (pt, with an elementary singularity at x0ZM. Let Ei OM+ be the subspace of tbe tangent space of M at x0 corresponding to the eigenvalues with real part negative. Then there is a 1 — 1 C°° immersion y> : Et —>■ M with tbe following properties: (a) X is everywhere tangent to y(®i) a n a a s * -*■ °°> 9>t (*) —*• X0 for all a>€ y ( I , ) . (b) y> (0) = x0 and tbe derivative of y> at x0 is the inclusion of Ei into Mx,. PROOF

It can be checked that the map E of 4.1 satisfies 9.1 using of 9.1 or its image is called the itabl$ manifold of x0.

631

111

9.

SHALE

: Stablt manifold* for

One has a stable manifold associated to an elementary closed orbit of a differential equation by the following theorem. 9.2.

THEOREM

Let y be an elementary closed orbit of a differential equation X on M generating a 1-parameter group 2 be an associated local diffeomorpbism of y at x with derivative L at X, and Ei the linear sabspace of Mx tangent to 2 corresponding to the eigenvalues of L with absolute value < 1. Then there exists a contraction Tt : El —>- Bi with the following true. The construction preceeding 2.1 applied to Tl defines a ma­ nifold M0 with a vector field X 0 on ;l/ 0 . Then there is a 1 — 1 immersion yt: IU0 —>• M mapping X 0 into X up to a scalar factor and y>(p) = x where p is the point of M0 corresponding to (0, 0) of Ei x R (in the definition of M0). For the proof we only need to note that y is defined first in a neigh­ bourhood of 0 x R and then extended to M0 by the device used in the proof of 4.1. Then y> or its image is called the stable manifold of y. The unttable manifold of a singularity or closed orbit of X on M is the respective stable manifold with respect to — X. In general M0 is either 8' X E, or the twisted product. If X is a dynamical system on a manifold M, we say that X has the normal intersection property if the stable and unstable manifolds of X have normal intersection with each other. Fixing compact M, let G„, /S be as in the previous section and £ 0 be the set of X in 6 0 with tbe normal inter­ section property.

9.3.

THBOEEM

y?0 is the countable intersection of open and dense sets of /?. This theorem and its proof are somewhat analagous to (6.1). For the proof of 9.3, let e : QL —>- R be defined in a completely analo­ gous fashion to the e of (6.1a) where QL is defined in section 8. Let X(.Q.L and x be a singular point of X or a closed orbit of period 0. Then 9.3 is implied by the following.

632

differential equationt and diffeomorphitnu 9.4

115

THSOBEM.

SAIS is open and dense in Q). As in section 6, for the proof of 9.4, it is sufficient to approximate a given X 6 QL by a vector field in stlis • Also just as in section 6, one defines maps f and submanifolds T{ of M. The only difference in the proof from that of 6.1 is in the details of the construction of the approximation itself. One uses here exactly the ap­ proximation in [13] page 202. We will not repeat it here, but only remark that one can do it a little simpler than in [13] by changing l o o a finite sequence of Euclidean cells one at a time. This completes the proof of 9.3. We conclude by remarking that if one takes for M, the 2-sphere, then sd0 is open as well as dense in £ that each X € srf.0 has only a finite num­ ber of closed orbits, and by a theorem first stated essentially by Andronov and Pontrjagin, X is structurally stable. In this caee, i. e., M = 8*, density of s%0 in p was first proved by M. Peixoto [8].

633 116

S. SMALK : Stable

manifold*

for

REFERENCES

1. 2.

O. D. BIHKHOFF, Collected Mathematical Paperi New York 1950. CODDINGTON and LBVJNSON, Theory of Ordinary differential equation, McGraw-Hill, New York 1956. 3. L. E. ELSGOLTZ, An eetimat* for the number of tingular poinit of a dynamical tytttm defined on a manifold, Amer. Math. Soo. Translation No. 68, 1952. 4. P. HARTMAN, On local homeomorphitmt of Euclidean tpaee, Proceeding! of tht Sympoeium on Ordinary Differential Equations, Mexico City, 1959. 5. S. I.BF8CHETZ, Differential Equation; Geometric Theory, New York 1957. 6. D. C. LEWIS, Invariant manifoldt near an invariant point of unttable type, Amer. Journal Math. Vol. 60 (1938) pp. 577-587. 7. R. S. PALAIS, Local Triviality of the retlrietion map for embedding*, Comm. Math Heir. Vol. 34 (1980) pp. 805-312. 8. M. PBIXOTO, On ttructural ttability Ann. of Math. Vol. 69 (1959) pp. 199-222. 9. M. PBIXOTO, Structural liability on 2-dimemioual manifold*, Topology Vol. 2 (1962) pp. 101-121. 10. I. PKTROV8KT, On the bthavior of the integral curve* of a tyitem of differential equation* in the neighbourhood of tingular point. Keo. Math. (Mat. Sbornik) N. 8. Vol. 41(1984) pp. 107-155. 11. G. KBBB, Sur oertain** propriM* topologique* det projeetoire* det lyiteme* dynamiquee, Aoad. Boy. Belg. Cl. 8ci. Mem. Coll. 8° 27 N° 9, (1952). 12. 8. SMALK, More* inequalitiet for a dynamieal tyitem, Boll Amer. Math. Soe. Vol. 48 (1940) pp. 883-890. 13. 8. SMALK, On Gradient Dynamical Syttemt, Ann. of Math. Vol. 74(1961) pp. 199-206. 14. 8. STBRNBKRO, Local contraction! and a theorem of Poincari, Amer. Journ. Math, Vol. 79 (1957) pp. 809-824. 15. R. THOM, Sur «ne partition en eellulet aitociee a une function tur une variitl, C. R. Aoad. Soi. Parie Vol. 228 (1949) pp. 973-976. 16. R. THOM, Quelquet propriM* global** det variiUi differentidtl**, Comm. Math. Heir., 28 (1954), pp. 17-86. 17. R. ABRAHAM, TranivtrialUy of manifold* of mapping! to appear. 18. L. MARCUB, Structurally liable differential tyUeme, Ann, of Math. Vol. 73 (1961) pp. 1-19.

T1POGRAFIA

"ODBRISI,,

XDITBlfll

-

OTJBBIO

1 8 6 3

634

S. S M A L E U.S.A. *

A STRUCTURALLY STABLE DIFFERENTIABLE H0ME0M0RPH1SM WITH AN INFINITE NUMBER OF PERIODIC POINTS

The purpose of this announcement is to define a differentiable homeomorphism of the 2-6phere S* which on one hand is structurally stable in the sense of Andronov-Pontryagin [1] and on the other hand has periodic points of arbitrarily high period, and a non-locally connected minimal set.

; Fig. 1.

n

(fT~

"\ Fig. 2.

According to 12), this answers a question raised by Andronov. (One may take the induced flow on S* x Sl). We recall the definition of structural stability. If T, Tl are differentiable ( O ) homeomorphisms of a closed C*° manifold At, they are equivalent if there is a homeomorphism h:M-*M such that hT"=>T1h. They are eequivalent if h may be chosen pointwise within 8 of the identity. Let d, be a C1 metric on the space x„ of differentiable homeomorphisms of M. Then Tet m is structurally stable if given e > 0 there is a o > 0 such that if dl(T,T1)<6f then T and T1 are e-equivalent. We now describe the differentiable homeomorphism TiSP-^S? with the properties stated above. For convenience we describe T on the plane £• thinking of £ , = S t — p, Tp — p. The differentiable homeomorphism T is to map the regions as indica­ ted in Figure 1 IB-+Blt etc.) and to further satisfy: a) The Jacobian matrix of 7 has eigen-values with absolute value less than one on the exterior of T~' (outer boundary of B) and £,. b) That T-* (SflSj) be two vertical rectangles Ru Rt in S (as in fi­ gure 2), and that T on each of Rx, Rt be «quite» close in the C1 sense to the obvious linear map on Rlt Rt. One can now construct a natural 1-1 correspondence between the points of tf— n 7 * ( S ) (2 the integers) and the collection of all doubly ■MS

infinite sequences of ones and twos with a decimal point,... 1211, 21112... ... etc.

635 366

S. Smale

(U.SJ.)

Here the periodic points in K correspond to periodic sequences. One can extend the numerical notation to give actually a normal form for 7 which i6 used to construct the homeomorphism required in the example. Reference1. A. A. A n d r o n o v and L. S. P o n t r y a g f n , Systemes grossiers, DoM. Akad. Nauk SSSR, vol. 14. 1937. 247-250. 2. L. M a r k u s. On the Behaviour of the Solutions of a Differential System Near a Periodic Solution, with Applications to the Theory of Structurally Stable Systems, Technical Report 8. O O R project 1469, University of Minnesota, 1959.

636

Diffeomorphisms with Many Periodic Points STEPHEN SMALE Introduction Although this paper is motivated by problems in ordinary differential equations, we consider explicitly the topological conjugacy problem for diffeomorphisms. Recall that C° diffeomorphisms T,T': M.-+M. are topologically conjugate if there exists a homeomorphism h: M —*-M so that Th = hT' where M is some differentiable manifold (see [10] and [11] for a background of this work here). We consider here aspects of this problem which are related on one hand to the symbolic dynamics of Hadamard, Morse, and others (see Gottschalk and Hedlund [5]), and on the other hand to the homoclinic points of PoincarS and G. D. Birkoff. The "shift automorphism" (on m symbols) of symbolic dynamics, described in section 1, is a homeomorphism of a Cantor set with a number of interesting properties which include periodic points of arbitrarily high period and a minimal set, itself homeomorphic to a Cantor set. We recall that p 6 X is a periodic point of a homeomorphism T: X —*■ X if Tx(p) = p for some A. The smallest positive A is called the period of p. A minimal set of a homeomorphism is a compact invariant set with no proper compact invariant subsets except the empty set. THEOREM A. On every manifold M of dimension greater than one, there exists an open set U in the space of diffeomorphisms of M with the C topology with the following property. If T e U, then there is a Cantor set Q. <= M, invariant under T such that T restricted to Q is topologically equivalent to the shift automorphism on 2 symbols.

This means that if T: A —*■ A is the shift automorphism there is a homeomorphism h: A —*■ Q with Th = AT. COBOLLABY. On every manifold M of dimension greater than one, there is an open set of diffeomorphisms, C topology, with each element possessing an infinite number of periodic points. 63

«U7

64

STEPHEN SMALE

This shows that the answer to problem A of [12] is negative, for every manifold of dimension greater than 1, in fact. That is, the diffeomorphisms (and dynamical systems also, via [11]) axiomatized there are not dense in all diffeomorphisms. Previous to the research of this paper, N. Levinson had written me that his paper [6] would already yield this answer for problem A for 2-dimensional difiPeomorphisms and 3-dimensional differential equations. His paper and those of Cartwright [3], and Littlewood [7], indicate strong relations between the theory developed in this paper and the theory of a 2nd order differential equation, time dependent, in particular that of van der Pol with a forcing term. I presume that in view of G. D. Birkhoff's theorem on homoclinic points, see below in the Introduction, he would have known the answer to this problem A in dimension 2. One last note on this problem is that by a different example, inde­ pendently, B. Thorn, unpublished, shows the answer to problem A to be negative. His example is an open set of diffeomorphisms of a 2-dimensional torus which have no "contracting" periodic points. A point x e M is called a homoclinic point of a diffeomorphism T:M^ M if lim TJ* = lim T~nm — y, x ^ y «|-»0O

«-»00

(so in particular the limits exist), m a positive integer, for example 1. Homoclinic points were first found by Poincare' in the restricted three body problem and he was aware of the complexity that they contributed to the nature of dynamical systems. Birkhoff has pursued the study of homoclinic points (see [1], [2]). THEOREM B. / / D(M) denotes the space of diffeomorphisms of M with the C topology, r > 0, there is a subset D0 which is the countable intersection of open dense sets of D with the following property. IfxsMisa homoclinic point of Te D0, then there is a Cantor set Cl cz M, xeil, and p such that T'il = Q and T" restricted to Q is equivalent to a shift automorphism of symbolic dynamics.

One should note that the idea of Theorem B is excluding a few diffeo­ morphisms, the mild hypothesis, x is- homoclinic point, leads to a rather strong conclusion. COBOLLABY. If T B D e , then in every neighborhood of a homoclinic point of T there is an infinite number of periodic points of T. This follows from the well-known [5] and easily proved fact that the periodic points of a shift automorphism are dense in the Cantor set. Birkhoff [1] proved the above corollary where the manifold was the

638

DtFFEOMORPHISMS WITH PERIODIC POINTS

65

plane using arguments which do not seem to generalize to higher dimensions. The theorem that there exist dhTeomorphisms of the n-sphere which are structurally stable and have an infinite number of periodic points was announced in [9], [10]. Our work here takes us a large part of the way toward this result, but we postpone the complete proof to another paper. Everything is considered from the C°° point of view, e.g. diffeomorphisms and manifolds are always C°°. Some conversations with Dr. R. Abraham on the generalization of Birkhoff's theorem have been very helpful. We take this opportunity to make some corrections to our paper "Morse Inequalities for dynamical systems" [12]. First, as we see in [11], as was first pointed out to us by Harold Rosenberg, the stable manifolds for a closed orbit may be twisted and hence non-orientable. Thus in the Morse inequalities one must either assume that the stable manifolds are orientable, or that the coefficients are Zt. Also the axioms on page 49 should be augmented by the analogue of Axiom 5 for vector fields page 43. §1

We discuss here some elements of symbolic dynamics. See Gottschalk— Hedlund [5] for more information with historical references. Let 8 be a finite non-empty set with m elements. Let I, j be integers or °o» 0 <, I <; oo, — 1 <; j <; oo, [—I, j] the set of integers between —I and j , inclusive when I, j are finite. Let Alj be the set of functions from [—I, j] to S, A\ a set with one element. Let each of £ and Z (the integers) be provided with the discrete topology and Alj the Compact open topology. Define a topological isomorphism T: Ay—»-^4J^J, 0 <, I — 1, —1 <,jy by r[a] (i) = a(» — l).ThenT: A™ -*-A^, is called the shift automorphism on m symbols. We note the following properties of T: A^ -*■ A™. For the proofs and further analysis of T, see [5]. (1.1) (a) The periodic points form an invariant countable dense subset of (b) There exists a minimal set, homeomorphic to a Cantor set. let If *o ^ l> h^h P — P<J>J> lo>Jo) b 6 t n e m a P P- A\ -*■ A\ by restricting a function from [l,j] to [l0, j0], For the proofs in Section 3, we include the following facts. Let af be the composition Aj

► Aj_1

► Ai_1

defined

639

66

STEPHEN SMALE

and pf: A° —*■ A?_x be a special case of the above defined restriction. The following is easily verified. (1.2) LEMMA. Let /?1( 0t e A?. Then pf+xtft) n of+i(Pt) has exactly one element if and only if Oj(fti) = pj(fit). Define 6: A_^ x Aj -*■ Aj by taking d(
§2

To obtain some geometric picture of what is going on in this section, we refer the reader to the remark at the end of Section 6. Here we assume only that there is given a compact set R of a Hausdorff space M and a map T: R—+M, which is a homeomorphism onto its image. Define B 0 ^ = R and inductively (on j), B? = T(BJ_j) n R, Similarly define B*_, = T-\tf-f

n B°), J ^ 1, and Bj = Bf_t O Bj,

Also let BJ" = O,>0 Bj, #„ = n ^ . ! Bj, BZ = n ^ . ^ o B J , so that Bj is a well defined closed subset of R for all I, j satisfying — 1 <.j <, oo,

0<,l<,

oo.

REMARK.

(2.1)

Note that 5? = T(B?_! n BL^.

LEMMA.

Ifl0^Koo,

j 0 <, j ^ oo, «A«n Bj <= Bj;.

PROOF. We show inductively that (a) BJ* c B j ^ a n d ^ g^ c ^ - ^ From this 2.1 follows immediately. Note B% = T{R) O R <= R = B°_x so that (a) is true for j = 0. If B?_x c B° , then T ^ ) c (B°_2) so B? = TfB"^) n i f c T(B»_2) O fi = B^_x proving (a). The proof of (b) is similar. (2.2) LEMMA. For Q<,l<,, T restricted to B\ is a homeomorphism onto Bj+J. PROOF. First note that it is sufficient to oonsider the case I, j < oo. Since T is a homeomorphism, it is sufficient for this to show that (a) T(BJ) c Bj;} and (b) T-H^Ti) c Bj. For (a) recall Bj = B? n B*_lt Bj;} = Bf+l n &'* and T(Bj) O R = B? +1 . So T(BJ) c B? +1 if T(BLj) <= R. But this is true sinoe I > 0, BLx = T-H#-{ O B°) and T(rf.j) c= T(T-lff^) = BLl1 «= B. 1 1 Since T(B'_1) <= B ^ , T)BJ) c BL^ and (a) is proved.

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DIFFEOMORPHISMS WITH PERIODIC POINTS

67

For (b) T- x (5^ +1 ) c T-\TE§) = &}, so it remains to prove But so it is sufficient to show But this is clear. This finishes the proof of 2.2. Let C\ be the set of components of Bj. From 2.1, let r: Cj-*-C^ be induced by inclusion. From 2.2, let t: C\ —*■
*■ Cj+\

If /3 eCj, we will write asejS.^c J2, etc. where /J is considered to be a subset of R. §3

We refer again the reader to the remark at the end of Section 6. Suppose as in Section 2, T: R —*■ M is a homeomorphism onto its image where R is a compact subspace of a Hausdorff space M. We assume moreover that T satisfies the following axiom. Axiom 1. R is connected, B% has a finite number of components, say m, m>l. Furthermore if fteCLi, P,etf, I, j < oo, then / ^ H / ? , has exactly one component. Let Aj be as in Section 1, where the set 8 used in defining A^ is the set CJ. The goal of Section 3 is to construct, under the assumption of Axiom 1, an isomorphism : Alj—+ Cj with certain naturality properties. More explicitly the goal of this section is to prove 3.1. Recall r: Cj -»- Cjj is induced by inclusion, t:Cj-*- Cj+J by T. (3.1) THBOBBM. There is an isomorphism (1-1 onto map) : Alj-*-Clf, 0 <. I <; oo, — 1 <; j <; oo, with commutativity in the following diagrams: Aj——► L/j

Aj

—■—► G^

1- . 1'

1' . 1'

641

68

STEPHEN SMALE

First observe that by considering .4", as C" are inductive limits of A\, C\, respectively, (see also Section 4) it is sufBcient to prove 3.1 for the case I, j < oo. Define first a map d: Cl_1 x Cj -*-CJ by taking for d{ftif (it) the com­ ponent /?t n /?, of Bj. By Axiom 1, d is a well-defined isomorphism. Furthermore d'1 is the restriction of r, x r, to the diagonal where rx: C\ —>Cl_x, rt:C!j-+-Cj are special cases of the r induced by inclusion. This is all quite analogous to the 6 denned at the end of Section 1. Let 8t be the following composition

and rt:Cfj—*- C%-\ the previously defined inclusion. We have the following analogue of 1.2. (3.2) LXHHA. Suppose 0V /S, e (fj. Then «i"^11(/?1) n r^V/^i) *°» *»*ci«e{y one element if and only if ry(/?,) = «>(/?,). PROOF. Consider the following (non-commutative) diagram.

Here the various f, r0, etc. are all induced by inclusion, tf, tj+1 induced by T. Let a = tM(d{f}v rjf1^,)), where d:C$ x CLX -»-Cj is as above, and Suppose now r ^ ) = *,(&). WewiHshowfirstthatae«,+,i(/?1) n r^iO^i) This amounts to proving (a) «y+1(a) = /?x and (b) fy+i(a) = /?,.

642

DIFFEOMORPHISMS WITH PERIODIC POINTS

69

For (a), using definition of * m , we need to show ^(a t ) = & or fj{i{p\, V/"1)) = Pv ® u t *^"8 "• c ^ ear ^ r o m *** definition of d, f,. For (b) first note that by 2.3 the following commutes /TO

°<+i

J*iL~L/nrl

•"^i

K- I' Hence it amounts to proving ^ _1 /5 t = fa,. Since ft_x X r0: C^_x -»- C°_1 X (7Li is an isomorphism, to prove (b) it is now sufficient to show (bx) f

t-if<*i = fi-4ilPt

and

(b») V*i = rJilfir

For

(bi) w e h*^ f»-iff V» =

«y^t = r ^ by hypothesis. Then (b) is finished using the definition of a, and the various r's. We leave for the reader the task of checking the uniqueness of a. To finish the proof of 3.2 it must be shown that if «j"+i(0i) C\ rf+iifit) consists of a single element a, then rt(fl^ = «/(/Sg). This follows from the commutativity of the previous diagram. We proceed now to the proof of 3.1. First define ^ from A0^ to C°_v A% to CQ, to be the canonical iso­ morphisms. Now <j>: A$ —*■ Cfj is defined inductively. Suppose has been defined up to A°j_x BO that the following diagrams commute: A-i

*"W-i

-^/-l

*"W-i

■Af-2

*"W-2

-^>-2

*"67-2

We wish to complete the induction by defining : A® -+Cj with the corresponding diagrams commuting. If a 6 A], let ^(a) = r^fo,*) O <S/"1(^a). By 3.2, this is well-defined. The hypothesis of 3.2. in this case follows from commutativity in the above diagrams and 1.2. From 1.2, 3.2 and the inductive hypotheses, one can obtain an inverse of 4>: Aj—+ Cj, so is an isomorphism. Next define from^41_1 to Cl_1 in the unique way so that the following commutes: V_i

*> v-_x

i-

!■

A\

>Cl

643

70

STEPHEN SMALE

By considerations similar to those used in the construction of if>: A? —>• Cj, and using analogues of 3.2 and 1.2, one obtains an isomorphism : A_x -*-Cl_lt each I, with commutativity in the following diagrams.

A'-i-UC-t i-i -l

I'

■ + Cl l - l

-1 _ * A - l

Here a, s are the compositions respectively,

A^-^A^-UA'-?, Let ^ : ^ —*• Clj be defined as the isomorphism which makes the follow­ ing diagram commute:

->c<

4-

From previously discussed relations between the maps d and r, and similar relations between d and p, it follows that the following diagram commutes.

A\-UC\

It remains for the proof of 3.1 to prove commutativity in the 2nd diagram in its statement. We first take care of the following special case

A)—U-CJ

or equivalently that

commutes.

A)

+ C)

v

v

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DIFFEOMORPHISMS WITH PERIODIC POINTS

71

Now using (5 and d it is sufficient for this to note that the following two diagrams commute. A0,

*—+(?>

\r+

U t+i

A

tl-

_*r° * °*+i

Next consider the following diagrams which define pit ir{, for » = 1,2.

These commute by previous considerations, so that we have commutativity in

^-A\ Aj+1

* cj

1 *-A_1 x Al+1

^^

►C-i X W+i "*

^t+i

which finishes the proof of 3.1. §4

To the structure T: R—*M of Section 3 with Axiom 1, we add the following Axiom. Axiom 2. Dim fl£ = 0. We will prove the following theorem. (4.1) THEOREM. Suppose R is a compact subspace of Hausdorff space M, T: R-*- M a homeomorphism satisfying Axioms 1 and 2. Then there exists a Cantor set £2 <= R, TO. = Q, and T restricted to Q is topologicaUy equivalent to the shift automorphism on m symbols, m of Axiom 1. Define £2 = B%; this is invariant under T by 2.2. By Axiom 2, 2?£ can be identified with C%, so that we can consider : A " —► £2. Then using 3.1, 4.1 is implied by the following lemma.

645

72

STEPHEN SMALE (4.2)

^ : i ^ - > Q i « continuous. Both A%, C™ = B^ = Q are inverse limits of the systems (A\, p), (C\,r) respectively. Since : A\-*-C\ is map of these inverse systems, by 3.1, and continuous, its limit : A^-t-C™ is continuous (see [4]). LEMMA.

PROOF.

I* To verify Axioms 1 and 2 in our applications, we need some linear theory. Let Ex, Et, EY x Et = E be Banach spaces (finite dimensional in our applications). For 0 < ft < 1, 0 < e, let B(p, e) be the set of bounded linear transformations of E, taking the following form in the product structure of E.

where ||Z|| <, ft + e, || 7|| <. e, and there exist bounded linear transforma­ tions L:E1^-E1, WL-^W^H + E, A:Et-+Elt \\A\\^e, B:E1--E1, ll-B-1!! 0 Bt(fi, e). (5.1) LKMHA. Given 0 < ft < 1, r\ > 0, there exists an e > 0 tcith the following property. If T e B*(/t, e) is the form

then for aUveE^ \\cv\\ <, t]\\av\\. PROOF. It can be assumed without loss of generality that ft -f r\ < 1. We use induction on the following hypothesis: Jf,(e): for T, = T0T,_i, T0e B(/t, e), T^e

T

<=C

B^ft,

c)

J;).for»=o,i-M,

(i) M l ^ JjIMI (ii> lk»i_it»ll ^ (A* + >?)IMI for all t; e ^ Note that if we can find a fixed e > 0 so that Jf^e) is true for all I, then 5.1. is proved. It is easily checked that there is an e > 0 so small that (a) (ft -\- s)tj -\e
646

DIFFEOMORPHISMS WITH PERIODIC POINTS Also /a,

6A _ lafi^i + V s - i . aj>t_i +

b$t_i\

If ^i_i(e) is true then for any v e Elt we have for (ii) a,« = ajOj^r + 60c,_1t> = J5La,_xr + B^Cj^v So IMI ^ ll-B"1!! IMI ^ ll-B-'MI ^ ll-£oi_i» + AcuwH ^ WLa^vW - \\ACl_iv\\ Also l|£»i-i»ll ^ (A* + eJ-'Ik-ifll So IMI 2> ((A* + «) _1 - i?«)lk»i-i»ll ^ (A* + »y)-1l|a,-if|| (i)

MI^IWIIa l -i«ll + Wllc«-i«ll <. «lk»i_i«ll + (A* + e)»?llol_1r|| ^ ((A* + «)»? + e)lk»«-i«ll ^ »?ll<»i-i«li ^ »?IMI

by (ii). This prove 5.1. For any t > 0 define the sector St of El x 0 to be the set {x e E\x = 0 or d(xl\\x\\, EiXO)^ t}. Then 5.1 has the following consequence. (5.2) LEMMA. Given 0 < p<\, rj > 0, there exists e > 0 awcA thai ifTe B*(p, e) then T(E x O ) c 8n. Actually one has the following generalization of 5.2 which we do not use. (5.3) LEMMA. Given 0 < ;* < I, tj> 0 there exists ex > 0 unrt *Ae property that if T e B*(/i, ex) then T(8ti) c S„. PKOOF. Let e > 0 be given by 5.2 so that if t; e Et X 0, T e £*(A*, e) then TveSr}. Let J7 be the set in E of all vectors of the form Tv, Te B*(p, e), t e ^ x O . Then it can be seen that U contains a sector 8^ of E1 X 0 for some 0 < et < e. Now if v e 8H, Te B*{p, e^, the following proves that Tv e #„. First v = TtVi, ViSEi x 0, Tie B*(p, et). Then Tv = TT& and since B*(/i, Ei) is closed under products, this shows Tv e ST], proving 5.3.

73

647

74

STEPHEN SMALE §6

Here we reduce the proof of Theorem A of the Introduction to 6.2 which is proved in Sections 7 and 8. Let D2 be the solid ball of radius 1 in Euclidean space E', D{ that of radius 1/10. Let R = D* x D"-* <= Ek x E"-* = E". Let T, a be some translations of £*, .ffn_* respectively which move distances by J unit. Then let J^ = (T X e)(DJ; x D»-*)( Rt = (T- 1 X e)(2)J x D—*) so that i21, Rt are disjoint subsets of R. Here e denotes the identity map. Let Rx = (e x ? _i ), be so that J^i, R'2 also are disjoint subsets of R. Let c: D* -»• D{, d: D"-* -»• D^~k be the natural contractions, shrinking the radius by -&, and define f^: ^ -*■ R[, ry = (e x ff)^-1 X d)(r X e)- 1 r2: R2 -*■ R'2> rt = (e x ■ E" is a C™ injection denned on a neighborhood U of R in .E" such that (1) T0RCiR = R'iKJ R'z (2) T(t1T0(R)nR) = R1vRt (3) T 0 maps 2 \ onto R[ by rx T 0 maps RL onto i?8 by rt (4) T0((7) O tf is contained in a ball of radius 2 in En. We leave to the reader the task of verifying that there exists such diffeomorphisms. (For Theorem A, the case jfe = 1 is sufficient.) The following will be proved in Sections 7 and 8. (6.2) THEOREM. There exists an e > 0 with the following property. If T: U —*■ En is a diffeomorphism which satisfies (a) (b)

\\T(x)-T0(z)\\<e,xeU, \\T.x\\-T0^<e,zeU,

then T restricted to R satisfies Axioms 1 and 2 of sections 3 and 4. (Here T.x is the derivative of T at x.) The preceding theorem leads to Theorem A as follows. Extend T0:R^>-En to a diffeomorphism T^E^^E* which is the identity outside of the ball in En of radius 3. From techniques of differential topology, it follows that such an extension exists (see, for example, [8]). Now let M be any manifold of dimension greater than one and En <= M

648

DIFFEOMORPHISMS WITH PERIODIC POINTS

75

a coordinate system. Let T0: En-*-En be as above (with Jfc = I, for example) and extend T0 to a diffeomorphism of all of M by making it the identity outside of E*. Then 6.2 and 4.1 apply to yield Theorem A of the Introduction. Remark 1. We have constructed only the simplest examples in n dimensions which could be used for Theorem A. In these examples the corresponding m of Section 1 is 2. The reader will be able to construct further similar examples such that this m is an arbitrary positive integer. Also by changing the orientation in the definition of r, preceding 6.1, one obtains an example of this type where T: R—*-M, R c M, and R U TR is not imbeddable in En. The subsequent analysis covers all such cases. Remark 2. To obtain some geometric feeling for what is going on here and the earlier sections, it may be helpful for the reader to verify that T0 itself of 6.1 satisfies Axioms 1 and 2, and perhaps even to go through Sections 2 and 3 with this example in mind, for small k, n. §7

We prove here that for e small enough, the T of 6.2 restricted to R satisfies Axiom 1. Let n x : R -*■ Dk, II,: R -*■ D1*-* be the respective projections with R as in 6.1. The tangent space of R at any point is a direct sum Ek X E*-*. Let Sn be the sector of Ek X 0 as defined in section 5. (7.1) LEMMA. Given r\ > 0, there is an e > 0, so that if T satisfies 6.1 with respect to e, then the following is true: Let ft e Cj, x e /?, y e R with r'+ty = x. If v e E* X 0 is a tangent vector to y, then Tj+V) 6 S„. Here C°f refers to that of Section 2 defined by the above T. k PBOOF. This is a consequence of 5.2. Take p = ^5, El = E , Et =E*-*, B = identity. The derivative of T0 on Rt u Rt is

(t ;)• Thus in a neighborhood of i ^ u Rt, the derivative of T will be in B{p, e) and that of TM will be in B*(fi, e) (since T*(y) e JBX u Rt, 0 ^ i < j + 1). Thus 7.1 follows. (7.2) LBMMA. Let 0 = & O /9t, ft e Cj, /?, e Cl_v j , I < 00 (all with respect to T of 6.2). / / e of 6.2 is small enough, there exists a diffeomorphism ipt-.p-i-D* x D"-* which is given by (I^T 1 x I^T-o+^A where A:/?-»-/5x/S is the diagonal map and differentiability of y>„ is with respect to the induced structure on /J as a subset of R. Before we give the proof we note a consequence.

649

76

STEPHEN SMALE

(7.3) COBOLLABY. For e > 0 small enough, T of 6.2 restricted to R satisfies Axiom 1. PROOF OF 7.2. We take e so small that in 7.1 for v 6 <S„, the angle between r\ and E* x 0 is less than ir/8. It must be shown that the composition y>g satisfies: (a) (b) (c) (d) (e)

well-defined differentiable of rank n onto 1 - 1.

(a) It is sufficient to show that if x e fa T'(x), T~u+U(z) are in R. In fact since fa <= Bfj, fa<= B*_lt x e /? if and only if T'(x) e R for -<j +\)<. (b) This follows from the definition of ipg. (c) Let xe fa Then the kernels (n^ 1 )." 1 ^) and (II.T-"*").- 1 ^), say, L„ Lt respectively are linear subspaces of the tangent space Rm. If it can be shown that Lx O Lt is the zero vector, then ipe must have rank n. Consider Lt. A vector v e Rx will belong to Lt if and only if (T~lM,),J,v) liesintf* X 0, or equivalently if v e {T'+^E* X 0) where y = P-w+«(a:). Thus L t <= iS, by 7.1 where £„ is the »; sector of E* X 0 and any non-zero vector of S„ has angle with Ek x 0 less than ir/8. A similar argument shows that ifveLlt then v has angle with 0 X .ff"~* less than n-/8. Thus only the zero vector is in Lx O Lt and (c) is proved. For (d) and (e) we show there exists an inverse to tpe. This is a consequence of (7.4) LBHMA.

faeCl_v

Let —Kj

< co,0^l<

oo, x e D", y eD —*, fa e Cf,

Then

[ft,n (*tT>)-Hx)]r\[fi,n (»,r-«+«-l(y)] is a single point. PBOOF.

Let Bm = Ilf^*) = a; x D"-*, * , = I l f *(y) = D* X y, Wt =

T-l(E,) O fa, Wt = T"+»(EV) r\ fa. Then (7.3) says TFX HIT, is a single point. Now inductivity define &}(y) = Tt-B^xfo) n # _ i ) where B°_i(y) is ^ , ; thus Bj = V^j,^B^(y). Then IP, is a component of £j(y). Then by induction on j it is not difficult to see that II x restricted to FP, is a diffeomorphism Wt —*■ D* and the tangent plane of Wt is within i/ of Ek X 0 with »7 as before.

650

DIFFEOMORPHISMS WITH PERIODIC POINTS

77

Similar considerations apply to Wx. Putting this information together yields that Wt n t is a single point. This finishes the proof of 7.4 and hence 7.2. §8 The goal of this section is to verify that T of 6.2 satisfies Axiom 2 for s small enough. This is done through complementing 7.2 with the case where I and j are allowed to become infinite. We first prove (but not use here): (8.1) LEMMA. Let /? e C0^. Then Fix restricted to P is a homeomorphism P -*• D*. Similarly if px e C*,, ITt restricted to piis a homeomorphism from Pi to D"-*. Here C\ refers of course to the T of 6.2 for sufficiently small e. PBOOF. We claim that if p e Cf, x e D*, then the diameter of 0 n (x x D*~k) is less than (1/2)'. On one hand this statement clearly implies 8.1, and on the other hand it is consequence of the following lemma. (8.2) LEMMA. Let a. be a compact subset of R with the property that for each x e D*, (x x D"~*) n a has diameter -*) O o^ has diameter a<_1and/?1 = n£ =1 a , l I ' e C . , , a' =3 a*~ so that & H pt — n i 1 1 (a < O a'). Then a a <-i <"* a ' - 1 a n ( i from 7.2 we get that /3X O ftt is non-empty. Then from 8.2 and the analogue of 8.2 for T - 1 , the diameter of /?x O /?, must be zero, proving 8.3. §9 The main goal of this section is to prove Theorem B of the Introduction. For this proof, we depend on the theory of stable manifolds [11]. We recall in fact from this paper, the following theorem (Theorem 6.1 of [11]). Let D(M) be the space of C diffeomorphism of M, C topology and D 0 the space of T e D satisfying the following condition: The periodic points of T are elementary and the stable manifolds have normal intersection with the unstable manifolds of T. (9.1) THEOEEM. D 0 is the countable intersection of open dense subsets

ofD.

651

78

STEPHEN SMALE

Now note that we can assume that the periodic points of T e D 0 can satisfy a slightly stronger condition without loss of generality, namely Sternberg's non-degeneracy condition of [13]. Thus by Steinberg's theorem [13] in the neighborhood of a point of M of period p, there will exist local coordinates in which T" is linear. (I suspect that with a little effort, the use of this deep theorem could be bypassed.) We can clearly assume without loss of generality in the proof of theorem B that the m in the definition of homoclinic point is 1. Let now p be a homoclinic point of T e D0, and let y = lim T"p = lim T"*p. Then y is a fixed point of T with, say, stable manifold W, unstable manifold Wu and p e W n Wu. Now by the previous remarks, we can suppose that V is a neighborhood of y of the form V = {||*J| < r0, ||*2|| <£ r j , (xv xt) eE1xEt

= E

and that T restricted to V is linear and of the form

I'-M*

I)

where At: i?, -*-Elt At: Et-*-Et are linear transformations, {{A^W < « < 1, \\At\\ < u < 1. We can assume without loss of generality in the proof of Theorem B that peV. (9.2) LEMMA. There exist compact domains with smooth boundary, K in W,J inW*,peKnJ,dKr\J = Q,KndJ = 0,K,J diffeomorphic to disks, and K <= interior T~lK, J c interior TJ. PEOOF. Following Section 3 of [11], there exist K, J with all the above properties, except for perhaps the boundary properties. Then using transversality theorems (see e.g. Thorn [14]) one can find suitable approxi­ mations of K, J satisfying 9.2. The K and J of 9.2 will intersect in a finite number of points, plt. . ., pt, including the original homoclinic point p and y. Of course, the pt are homoclinic points, except for y. We assume as before that the pt and J lie in V. Let m be large enough so that Tmi(pt) e V for each i and w^ ;> m, and let Z»< be the derivative of T™ at piy so that L(: El x Et -*■ Ex x Et. If we write

652

DIFFEOMORPHISMS WITH PERIODIC POINTS

79

according to this produot decomposition then by the transversality of the homoclinic point pit A: E1-*-El ia invertible. Choose e > 0 so that for each », if \\8 — Lt\\ < e, and

then S1 is invertible. Let Vi be a neighborhood of p{ such that T^iy) e V and \\T?{y) — Lt\\ < e if y e U(. (Here TT(y) is the derivative of 2*" at y.) (0.3) LBHHA. There exists X > 0, t 0 such thai for all k ;> k0 the deriva­ tive T^T^T* satisfies the hypotheses of the linear transformation of 5.2 on the subset of T~x(Vi) consisting of points z such that T*(z) e V for m + X<^j

<.m + X + k. PBOOF. This derivative is of the form L„8LQ where L0 is as before and 8 satisfies the condition of the 8 previously denned, i.e., Sr below iB invertible. Then

U i \ \\AS\\ < 1. Now 9.3 is clear. Choose X and k0 of 9.3 valid for all the Ut at once and let U = u<_i T-xUt and DT = {xt e Et: \\xt\\ <>r<. r j . The following lemma can be proved without great difficulty. (9.4) LEMMA. There exists r > 0 andfc;> i 0 such that

T-l(KnJ)

<= T-<*+*+*>( r - v x x>r)n ( T - v x Z)r) <= u.

Now consider Tk+m+* restricted to T-<*+"»+*>(T-V x Dr) By 9.3 and 9.4, and by the following sections 7 and 8, we obtain as before that this pair satisfies axioms one and two. Then by 4.1, Theorem B is proved. REMARK Of course the conclusion of Theorem B applies to homoclinic points of diffeomorphisms T satisfying weaker assumptions than T t e D 0 . For example, all of this is valid for the homoclinic points Poincare originally found in the restricted 3-body problem, and thus one obtains information about the "motions" in this problem. COLUMBIA UNIVEBSITY

REFERENCES

[1] G. D. BIBKHOFF, Nouvelles recherches sur lea systemes dynamiques, "Pontifical Memoir," Collected Works, Vol. 2, Amer. Math. Soc., New York, 1960. [2] , On the periodic motions of dynamical systems, Collected Works, Vol. 2, Amer. Math. Soc., New York, 1950.

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[3] M. CABTRIOHT, Forced oscillations in nonlinear systems, Contribution* to the Theory of Non-linear OieiUatione, I, Princeton University Press, 1960. [4] EDLBKBEBO and STKKNROD, Foundations of Algebraic Topology, Princeton University Press, 1952. [5] OOTTSCHALK and HKDUJND, Topological Dynamic*, Amer. Math. Soc., Colloquium Publications, Vol. 3d, Providence, 1956. [6] N. LBVINSON, A second order differential equation with singular solutions. Ann. of Math., SO (1949). [7] J. E. LTTTLKWOOD, On non-linear differential equations of the 2nd order, IV. The general equation y" + kf[y)y' -f g{y) = bkp{+), + = t + a, Ada Math., 98 (1967), p. 1-110. [8] R. PALAIS, Extending diffeomorphisms, Proc. Amer. Math. Soc., 11 (1960), p. 274-277. [9] 8. SKALE, Report on the symposium on non-linear oscillations, Kiev Math. Institute, 1961. [10] , Dynamical systems and the topological conjugacy problem for diffeomorphisms, International Congress of Mathematicians at Stockholm, 1962. [11] , Stable manifolds for differential equations and diffeomorphisms to appear. [12] , Morse inequalities for a dynamical system, Bull. Amer. Math. Soc., 66 (1960), p. 43-49. [13] S. STKRNBKRG, On the structure of local homeomorphisms of Euclidean space, II, Amer. Jour, of Math., 80 (1968), p. 623-631. [14] R. THOX, Quelques proprietes globales des varietes differentiables, Comm, Math. HeU>., 28 (1954), p. 17-86.

654 STRUCTURALLY STABLE SYSTEMS ARE NOT DENSE. By S. SHALE.

In this note, we give a negative answer to "the problem of structural stability"; are the structurally stable differential equations dense in the C topology in all (first order, ordinary, autonomous) differential equations? MAIN THEOKBM. There exists a compact 4 dimensional manifold Id, an open set U in the space of Cr vector fields, C topology, r > 0, on M such that no X € 17 is structurally stable. This problem has been stated explicitly in the above form by the author on several occasions, see e.g. [10] and [11 ] . However the problem is older, considered for example by Soviet mathematicians, and by Peixoto who gave an affirmative answer [7] for the 2-disk and [8] for compact 2-manifolds. Evidence for a positive answer in higher dimensions was given by the author [12], [13] where it was shown that certain examples of differential equations having an infinite number of periodic solutions were structurally stable, and Anosov [2] gave further examples with the same properties. The notion of structural stability was introduced by Andronov and Pontriagin in 1937 [1] (see also Lefschetz [ 5 ] ) . We recall the definitions now. A differential equation (1st order, ordinary) X, will be a Cr tangent vector field on a C" manifold M, which we will assume compact for purposes of this paper. Of course X, through its solution curves, generates a 1 para­ meter group of diffeomorphisms <j>t of M. An equivalence between differential equations X and X' on if is a homeomorphism h: M-* M which sends a sensed solution curve of X onto one of X'. Fixing a metric on M, it is an (-equiva­ lence if it is pointwise within < of the identity. The Cr differential equations on 31 form a normed space with the C norm. Then X is structurally stable if given < > 0, there exists S > 0 such that if || X—X' \c- < 8, then X' is (-equivalent to X. We will first construct a diffeomorphism g of a 3-dimensional manifold (a 3-toru8) and use this to construct a differential equation X, on a 4-dimensional manifold. This X„ has the property that there is a neighborhood N of X, in the C topology such that no X in N is structurally stable. Received June 1, 1065.

491

655 8. SHALE.

492

The starting point is the linear transformation of R' given by the matrix /a A —le \0 where B — f

b d 0

0\ Oj r/

, ) has determinant 1, o, 6, c, d are integers and 0 < r < 1,

r small. We assume further that the eigenvalues of B are real and not ± 1. Then S as a transformation of the x-y plane induces a diffeomorphism B„ of the 2-torus T. Also A as a transformation on R* induces a diffeomorphism A„: TXR - » T X f i with TX 0 invariant and all other points oi TXR contracting towards T X 0 under 4 0 . Of course A„ restricted to T X 0 is essentially the same as B„ on T. We shall now recall some of the structure of this discrete flow (see [2], [3], [10], R. Thorn [unpublished]). Let II: R*->T be the covering map, and p —11(0,0) so that p is a fixed point of So- Then the two l-dimenaional eigenspaces of B project under II into a stable and unstable manifold through p (see [11] for these definitions of these terms) which we will denote respectively by W'(p) and W"(p). Both W'(p) and W«(p) are dense in T. The periodic points of B„ are dense in T (this fact may be proved algebraically, one may use the Generalized Birkhoff Theorem of [13] or Anosov [2]), and each of these also has a 1-dimensional stable (and unstable) manifold associated to it. These will be disjoint, parallel (using the Euclidean structure on R*) to W'(p) and each is dense in T. Now we pass to the extension A„: TX B-+ TX R of B„. It is easy to see that the periodic points of A„ coincide with those of B„ as do also the 1-dimensional unstable manifolds which we will denote from now on by ^x"(?«)> *—1>V ' ,?* the periodic points of A* The stable manifolds now however are two dimensional, being in T X R of the form W»(gj) X R where W»(j») C T is the 1-dimensional stable mani­ fold of B0: T-*T for the periodic point qt. From now on we denote this 2-dimensional stable manifold W*(qt) X 5 by Wt*(qt). Thus the stable manifolds of the periodic points of A„ are a countable, dense family of parallel "planes" in TXRLet J , denote the restriction of A, to TX [—1,1] C TXRIn a neighborhood 8 of radius | about the point 6— (0,0,2) of R', let C be a linear transformation which contracts in the x-y plane through b, leaving b

656 STRUCTURALLY STABLE SYSTEH8.

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fixed and leaving tbe z-axis invariant, expanding on it away from &.. Denote by (7„ the induced transformation defined on the corresponding neighborhood So —IMS) of 11,(6) in TXR, n,: R'->TXR our covering map, and let

j — {(o,o,*)e:rx-ff | o ^ * g 2 } . Now it is easy to define a diffeomorphism / on a neighborhood of T X [— 1,1] U S0 U J in T X R into T X R which is I„ on T X [—1> 1]> Co on So, and leaves J invariant with no fixed points on J, except 0 and

nt(&). In fact we may assume / is extended not just to a neighborhood of T X [—1> 1] U So U J, but to a diffeomorphism of the whole compact mani­ fold TX8' *TXS', where 8/ is [—3,3] with end points identified. Our desired diffeomorphism g is obtained by perturbing / in a neigh­ borhood 0 of a point a in TX8' as follows. We assume r — j . Let «, —^(0,0,}) and Go — n t (.#,/, (0,0,$), N,(x) being the neighborhood of * of radius «. Then let a—f-1^) and G —/-X(G0) so since r — J, /(G) n f f - f Using the obvious linear structure on 0, let ? —/ outside (? and on G let g — f + rfh where & is the vector (1,0,0), <£ is a non-negative C" function with compact support in G and non-degenerate maximum value 1 at et andfinally17 > 0 is small enough so that g is a diffeomorphism. Note h is transversal to the "planes" JrV(?i). We now show that if g": M-+M is any diffeomorphism sufficiently close to g in the C topology, then there exists a diffeomorphism g": M~*M arbitrarily C" close to g' with the property that g" and g" are not topologically conjugate, at least by a homeomorphism pointwise close to the identity. (Thus g1 is not a structurally stdblt diffeomorphism; see [10], [11] for elaboration of this). One can best study a small C" perturbation g* of g by using theorems of Perron [9] put into a general setting in the spirit of Anosov [2]. We expect to expand on this point in a future paper. In the meantime we argue as follows. Pint, for g" C1-close enough to g, there is an invariant torus T CM — TX8' corresponding to f X O and C'-close to it. For this see Kupka [4] or Moser [6]. Next, g' restricted to 2" is topologically conju­ gate by a homeomorphism close to the identity to g: r x O ^ f X O ; we [2] or [3]. Let the stable manifolds of g* of the countable dense set of periodic points qi on I" be denoted by Wt* (},')'. The tangent 2-plane to each point of Wt»(qi')' is close (arbitrarily, depending on the C distance of g* to g) to the plane Tr",*(?i) (which is independent of i, of course) on T X [—1,1].

657 494

S. SHALE.

This is a consequence of Section 5 of [13]. Similarly, the W t ' ( g / ) ' filled up r x [—1,1] densely. The intersections of WV(?t')' n «w the z-axis with the i-axis will be a countable dense set 2'. Denote by Wf the 1-dimensional unstable manifold of the fixed point (0,0,2) of ^ and by Wt*' that corresponding to the pertur­ bation g'. Now for all g' (and g") sufficiently close to g, we can have precisely one of two cases as follows. Case 1: The first point on Tf1«', near the bump f (a), ordering by 2 ' is on some WV (}«')'. Case 2: It is not. The following two facts finish our picture of g". 1. By letting g" — j ' + i^A (see the definition of g), arbitrarily small ij, we can suppose g" is in Case 2 if g' is of Case 1 and rice versa. 2. If g' and g" are in the opposite cases there is no homeomorphism h of M close to the identity such that g'k — hg". This can be seen by iterating g' and g". From any diffeomorphism g of a compact n-manifold M onto itself, one can construct in a canonical way, an ordinary differential equation X on an (n-j-1) manifold with a cross-section g: M—*M (see [11]). In this way the various relevant properties of g correspond to those of X. Therefore, from what we have shown, the main theorem follows. The example in this paper surely reduces the importance of the notion of structural stability. One might be further discouraged from studying the global qualitative theory of higher dimensional, ordinary differential equations. We believe, however, that this study can be constructive and are preparing a paper in this direction based on an axiom, which we consider to be of central importance, axiom A, described as follows. Let X be a smooth vector field on a compact manifold M which generates a l-parameter flow t: M-*M is a smooth l-parameter flow on a differentiable manifold, the derivative ^»: Tu-*Tu defines a 1-

658 STRUCTTJBALLT STABLE SYSTEMS.

495

parameter group of the V-B maps of the tangent bundle, called the derived

flow. A 1-parameter group of V-B maps V»: E-*E over a compact space M is called contracting if for some Riemannian structure on Tl\ E-*M, the following two estimates hold:

1) | * ( r ) | £ a | r | r « 2) IMOII^M'I*"'"

,

t^Q.rtE, t^0,v€E.

Here a, b, c are positive constants, and J v Q is the norm of v in the metric on the fiber to which v belongs. Finally ,:M—*M, a U-structure on O (generalizing Anosov [2]) is defined as follows. The tangent bundle T(M) of M restricted to O, Ta(M), is a Whitney sum Ta(M) —Da» ®Da'®X (topologically) such that

Ea'-DQ' + X,

Ea'-Da' + X,

are invariant under the derived flow $»: T(M) -*T{M) of t, it- Ea'-*Ea* is contracting (as a 1-parameter group of V-B maps) and eV,: Ea*-*Ea" is expanding. Here X is the line bundle denned by X, X the tangent vector field generating «V For a smooth flow 4>,: M -» M of a compact manifold then our axiom is: Axiom A. structure.

The set of non-wandering points CIC1I of 4>t has a U-

REFERENCES. [1] A. A. Andronov and L. S. Pontriagin, "Svstimes Oroaaiers," Doklady AJeademU Ifauk, TOL 14 (1937), pp. 247-261. [2] D. 7. Anosov, " Roughness of geodesic flows on compact Riemannian manifolds of negative currature," Soviet Mathematics, vol. 3 (1982), pp. 1098-1070. [3] 7 . 1 . Arnold and Ja. <J. Sinai, " Small perturbations of the automorphisms of the torus," Soviet Hathemaiict, vol. 3 (1962), pp. 783-787. See also correction of same, Boviet Mathematics, vol. 4 (1983) page preceding 561. [4] I. Kupka, "Stabilite dee varietes invariantes d'un champ de vecteurs pour lee petitee perturbatione," Oomptee Rendut de I'AoademU dee Science* de Parie, TOI. 258 (1964), pp. 4197-4200. [5] S. Lefschets, Differential Xquaiiont: Geometric Theory, New York, 1957.

659 496

8. 8XALB.

[6] J. Hoser, On invariant manifold* of vector field* and symmetric partial differential equation*, in Differential Analysis, Bombay, 1984. [7] M. Feixoto, "On structural stability," AnnaU of Mathematie*, TOL 69 (1969), pp. 199-122. [8] ," Structural stability on 2-dimensional manifolds," Topology, ToL 2 (1962), pp. 101-121. [9] Perron, "Die Stabilitatsfrage bei Differentialgleichungen," Uathematitche Zrittohrift, TOI. 32 (1930), pp. 703-T28. [10] S. Smale, Dynamical tyatem* and the topologioal oonjugacy problem for diffeomorphitmt, Proceedings of the International Congress of Mathematicians, Stockholm, 1982. [11] , " Stable manifolds for differential equations and diffeomorphisms," AnnaU delta Souola Normal* Superior* di Pita, Series III, vol. XVII (1963), pp. 97-116. [12] , A ttructurally ttable differential* homeomorphitm leitk an infinite number of periodic point*, Report on the Symposium on non-linear oscula­ tions, Kier Mathematical Institute, 1961. [13] , Diffeomorphitm* loith many periodic point*, M. Morse Symposium, Dif­ ferential and Combinatorial Topology, Princeton, 1966.

660

Dynamical Systems on n-Dimensional Manifolds s. SMALE Department of Mathematics University of California Berkeley, California

In this survey paper, we study the problem of the global behavior of a first order, autonomous, differential equation on a compact manifold, or what is the same thing, a 1-parameter group of diffeomorphisms of this manifold. In the spirit of the papers [6], [8], we worry only about the general case or almost all differential equations in some sense. Furthermore, to simplify the presentation, we assume that there is a cross section (see [9]), so that we will be looking at diffeomorphisms
661

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S. SMALE

Recall that x e M i s a wandering point if there is a neighborhood U of x such that U | „ i > B a i ^ 7 * ( ( / ) n I / ^ 0 . Then xe Q if it is not a wan­ dering point [2]. Theorem I [7]. If - M satisfies these axioms, then the stable man­ ifolds give a cellular decomposition of the manifold M, i.e., M is the finite union of cells W{ which are partially ordered: W{ < Wf, if W{ is con­ tained in the closure of Wf. The unstable manifolds give rise to a "dual" decomposition. In a fairly satisfactory sense, this theorem gives a "phase-portrait" for

662 DYNAMICAL SYSTEMS ON rt-DIMENSIONAL MANIFOLDS

485

so as well. Then the homoclinic points are dense in T and in fact Q — T. Furthermore C approximations/' of/possess all of these properties [/]. In [77] a 3-dimensional example g: M* -*■ AP is given with the above torus as an invariant submanifold which has the property that no C-approximation is structurally stable. Thus the axioms stated earlier are indeed substantially restrictive, and the problem poses itself to relax these axioms so that: (1) The previous examples may be incorporated into the new framework; (2) if a diffeomorphism satisfies the relaxed axioms, then a finite decomposition of the manifold generalizing the previous one can be obtained in a natural way. This problem has a solution which we announce here. If 99: M-*■ Mis a diffeomorphism, and Q c Mis compact and invariant under q>, tp-1, define a {/-structure on Q to be a continuous splitting of the tangent space TX{M), each x e Q, TX(M) = Vxu + Vx' such that: (a) If D■ Ty(M) is the derivative, y =
v e vx; || Dqr(xm || < t>' || v ||, w e Vx\

|| D

(x)(w) \\>CX'\\w\l

p < l, r > 0 X > 1, s > 0.

For example, for the diffeomorphism of the torus described previously, there is a (/-structure on Q = T, and in general, Anosov [7] studies the case of a (/-structure on the whole manifold. In the case of diffeomorphisms satisfying the previous Axioms 1-3, there is a (/-structure on the finite set of periodic points given by eigenspaces of Dq?"{x), when . The periodic points are dense in Q. If A c M is a compact invariant set of y: M —*■ M, then A is indecompo­ sable if it is not the disjoint union of two invariant compact subsets (A "in­ variant" always means
663 S. SMALE

486

M as the disjoint union ui^rV^Q,), n

where rV'(Qi) is defined as the set

>0

of x e M such that (p (x) *" °> i2<. If satisfies Axiom B if: Axiom B. If Wu(Qi) n W'(Qj) ^
664

DIFFERENTIABLE DYNAMICAL SYSTEMS*'• PART I. DIFFEOMORPHISMS

1.1. Introduction to conjugacy problems for dlffeomorphisms. This is a survey article on the area of global analysis defined by differentiable dynamical systems or equivalently the action (differentiable) of a Lie group G on a manifold Af. An action is a homomorphism G—»Diff(Af) such that the induced map GX Af—»Af is differentiable. Here Diff(Af) is the group of all diffeomorphisms of M and a diffeomorphism is a differentiable map with a differentiable inverse. Every­ thing will be discussed here from the C" or C point of view. All manifolds maps, etc. will be differentiable (O, l ^ r ^ w ) unless stated otherwise. In the beginning we will be restricted to the discrete case, G=Z. Here Z denotes the integers, Z+ the positive integers. By taking a generator/£ Diff (Af) i this amounts to studying diffeomorphisms on a manifold from the point of view of orbit structure. The orbit of x £ Af, relative t o / , is the subset {/"(*)| m£Z\ of M or else the map Z—*M which sends m into/•(*). The finite orbits are called periodic orbits and their points, periodic points. Thus x £ Af is a periodic point if f*(x) = x for some m £ Z + . Here m is called a period of x and if «* — l, x is a fixed point. Our problem is to study the global orbit structure, i.e., all of the orbits on Af. The main motivation for this problem comes from ordinary dif­ ferential equations, which essentially corresponds to G~R, R the reals acting on Af. There are two reasons for this leading to the diffeomorphism problem. One is that certain differential equations have cross-sections (see, e.g., [114]) and in this case the qualitative study of the differential equation reduces to the study of an associated diffeomorphism of the cross-section. This is the reason why Poincare' [90] and Birkhoff [19] studied diffeomorphisms of surfaces. I believe there is a second and more important reason for studying the diffeomorphism problem (besides i{s great natural beauty). That is, the same phenomena and problems of the qualitative theory of ordinary differential equations are present in their simplest form in the diffeomorphism problem. Having first found theorems in the dif*The preparation of this paper was supported by the National Science Founda­ tion under grant GN-530 to the American Mathematical Society and partially supported by NSF grant GP-S798.

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[November

feomorphism case, it is usually a secondary task to translate the results back into the differential equations framework. Assuming M compact, we put on Diff(AO the topology of uniform O convergence. We will usually keep M compact because for noncompact M, there are different behaviours at infinity that one could consider. See, for example, [86]. These lead to different problems and we don't wish to get into such questions here. One of the first things that one observes is the need to exclude degenerate elements of Diff (Af). For example, given any nonempty closed set FQM, there is/£Diff(Af) such that the fixed point set Fix(/) = F. For a number of reasons, if F is not discrete we would like to exclude such/. The set of /£Diff(Af) such that Fix(/) is discrete (or finite, since we assume M compact) contains an open dense set of Diff (Af). This leads to the notion of generic properties of diffeomorphisms. A Baire set of a complete metrizable space is the inter­ section of a countable number of open dense sets. Then a generic property is a property that is true for diffeomorphisms belonging to some Baire set of Diff(M). We will never speak of generic/£ Diff (Af) (this is usually taken to mean that/ has a lot of generic properties!). Thus "Fix(f) is finite" is a generic property and a little more since open dense is stronger than Baire (see §1.6 for more details and references). It is important in proceeding to consider formal equivalence rela­ tions on Diff (Af) which will preserve the orbit structure in some sense. Furthermore associated to each equivalence relation there is a notion of stability. More precisely if the equivalence relation on Diff (Af) is called E, / G Diff (Af) is called E-stable if there is a neighborhood N(f) of/ in Diff(Af) such that i f / ' £ # ( / ) (or/' approximates / suffi­ ciently), then/and/' are in the same E equivalence class. It would give a reasonable picture (see [ i l l ] , [U2]) to have a dense open set UC.Diff(M) such that our equivalence classes could be distinguished by numerical and algebraic invariants. This is, in fact, our goal.2 If this is to be the case, the desired equivalence E on Diff(Af) should have the property that the .E-stable diffeomorphisms are dense in Diff (Af). With this background we look at some particu­ lar equivalence relations. The notion of conjugacy first comes to mind. S a y / , / G Diff (Af) are differentiably (or topologically) conjugate if there is a diffeomorphism (or homeomorphism) h: M—*M such that hf=f'h. Dif­ ferentiate conjugacy is too fine in view of the above considerations. This is due to the fact that the eigenvalues of the derivative at a fixed point are differentiate conjugacy invariants. The notion of stability

666 1967]

DIFFERENTIABLE DYNAMICAL SYSTEMS

749

associated to topological conjugacy is called structural stabOity, and for some time it was thought that structurally stable diffeomorphisms might be dense in Diff (M). This turned out to be false [116]. Thus by our earlier consideration we should relax our relation on Diff (Af) of topological conjugacy. Before doing this we introduce some basic ideas about G. D. Birkhoff's nonwandering points [15]. If /£Diff(Jlf), x£M is called a wandering point when there is a neighborhood U of x such that U\m\>9f*(U)OU**0. The wandering points clearly form an invariant open subset of M. A point will be called nonwandering if it is not a wandering point These nonwander­ ing points are those with the mildest possible form of recurrence. They form a closed invariant set which we will always refer to as

n=n(/).3

We propose now the equivalence "topological conjugacy on 0." That is /, /GDiiT(A0 are topologically conjugate on ft if there is a homeomorphism h: Q(f)—*Q(f) such that hf—fh. The corresponding stability will be called simply Q-stability. So/EDiff(Jf) will be called Q-stable if sufficiently good approximations f are topologically con­ jugate on fi. In general one can speak of topological conjugacy for homeomorphisms and even two homeomorphisms of different topological spaces, / : X—*X, f: X'-*X'. Then the conjugacy A is a homeo­ morphism h: X-+X'. We end §1.1 by giving some notations and conventions we follow. Anytime the topology on Diff(M) is involved M will be assumed compact. Simply connected X means TLi(X) and H.t(X) are trivial. We sup­ pose that our manifolds are always connected. Dim M means the dimension of M. The tangent bundle of a manifold will be denoted by T(M), the tangent space at x£M by Tt(M). The derivative of / : M—*M will be denoted by Df and considered as a bundle map Df: T(M)—*T(M). At a point x£Af, it becomes Df(x): T,(M)—*Tfw(M). An immersion is a differentiate map such that the derivative at each point is injective. A closed invariant set A of/£Diff(3f) will be called indecompos­ able if A cannot be written A =AiWAj, Ai, At nonempty disjoint closed invariant subsets. Finally if X is an eigenvalue of a linear transformation «: V—*V, we will define its eigenspace £x=» {*G V\ («—X/)"(x)««0, some >»£Z + }. Then X will be counted with multiplicity dim Ex. Two earlier surveys on this subject are [85] and [112].

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S. SMALE

Part I is the heart of the paper, including a number of new ideas, and is devoted to problems spoken of in this section. Part II briefly extends the results to the ordinary differential equation case (G = R) and Part III discusses other aspects of the differential equation prob­ lem. Part IV is devoted to possibilities for more general Lie groups G. I would like to acknowledge here many very helpful discussions with other mathematicians. This includes especially D. Anosov, A. Borel, A. Haefliger, M. Hirsch, N. Kopell, I. Kupka, J. Moser R. Narasimhan, J. Palis, M. Peixoto, C. Pugh, M. Shub and R. Thom 1.2. The simplest examples. This section is devoted to giving a description of a class of Q-stable diffeomorphisms which are the sim­ plest as far as the orbit structure goes. To develop or even define these diffeomorphisms, we will need the basic idea of a stable manifold. A linear automorphism u of a (say real) finite dimensional vector space V, u: V—* V will be called hyperbolic if its eigenvalues X, satisfy |X,| 7^1 all i. We emphasize that complex eigenvalues are permitted. The automorphism « will be called contracting if |X,J < 1 for all t, expanding if |X,| > 1 for all t, and of saddle type otherwise. Thus the inverse of an expanding automorphism is a contracting automor­ phism and vice versa. Observe that for hyperbolic «: V—*V we have a canonical, invari­ ant (under u) splitting of V, V=V'+ Vu (direct sum) where V' is the eigenspace of u corresponding to eigenvalues less than 1 in absolute value and Vu the eigenspace of the remaining eigenvalues. Thus u restricted to V' is contracting and u restricted to Vu is expanding. This gives rise to the following familiar picture for such u. V

v

»

FlGUKB 1

vT

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Note that the hyperbolic elements of the general linear group GL( V) are open and dense. Now suppose / : M—*M is a diffeomorphism with a fixed point PE.M (a local diffeomorphism / : U-*M, U an open subset of M, PG: U, f(p) =*p would be sufficient for some of the following discus­ sion). The derivative of/at p, Df(p), may be considered to be a linear automorphism of the tangent space of M at p, i.e., Df(p): T,(M) -*T,(M). We will say that p is a hyperbolicfixedpoint of/, or simply a hyperbolic fixed point, if Df(p) is hyperbolic in the sense of the previous paragraphs. We will call a periodic point p of period tn£Z+ of / : M—*M hyperbolic if it is a hyperbolic fixed point of /*. Similarly, p is a contracting or expanding periodic point if D^ip) is a contracting (or expanding) linear automorphism. A (global) contraction of a differentiable manifold V is a diffeo­ morphism g: V—*V which is topologically conjugate to a linear con­ traction (i.e., a linear contracting automorphism) u: V'—*V. Of course a contraction will have a unique fixed point. For hyperbolicfixedpoints we have stable manifolds defined accord­ ing to the following theorem. (2.1) Stable Manifold Theorem. Suppose £ £ M is a hyperbolic fixed point of a diffeomorphism / : M—*M with T,(M) = V'+ Vu the cor­ responding decomposition under Df(p). Then there exists a contrac­ tion g: W'{p)—*W'{p) with fixed point p9 and an injective equivariant immersion J: W(p)->Msuch that J(j>o)<=p and DJ(p0): Tv%{W'{p)) -+TP(M) is an isomorphism onto V*C.TP(M). Furthermore the image J(W'(p)) may be characterized as the set of x £ M with the property f*(x)—*p as m—* 00. Equivariance here means simply that Jg =fJ. Note that the deriva­ tive condition implies that the dimensions of V' and W'(p) are the same. The image of / is invariant under/, and frequently we will identify points under J so that W'(p)dM. In general, / will not be a homeomorphism onto its image (see the toral example of §1.3), so that the original W'(p) and W'(p) is a subset of M have different topologies and this is the only way they differ. Both are called the stable mani­ fold of / at p. When it is important to specify the topology, we will say intrinsic for the original topology and the manifold topology for the other topology on W'{p). For analytic diffeomorphisms of two dimensional manifolds, this theorem was known to Poincare' [90] and used by Birkhoff [16].

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The proof of (2.1) starts by showing the existence of a "local stable manifold," W'u»(p). This is due to Perron [88]. He uses iteration methods in a function space to solve a functional equation for / in a neighborhood of p. Further references to versions of this theorem are [2], [24], [39], [120] (most often these papers concern themselves with the differential equations analogue, so one has to make a transla­ tion of the results). The global theorem, (2.1), follows easily from the local theorem by so to speak "topological continuation." One takes for W'(p) the subset U«Gz+ /-"W'^ip) of M. See [114] for more details. For a hyperbolic fixed point p of a diffeomorphism / : M—*M, the unstable manifold W"(p) is defined as the stable manifold of/ -1 at p. Thus Wu(p) passes through p and is tangent to V" in the notation of (2.1). For a periodic point q of /EDiff(Af),/"(g) =qt m£Z+, one defines the stable and unstable manifolds, W'(p), Wu(q) as the stable and unstable manifolds for q as a fixed point of /**. Although each W'(p) is a 1-1 immersion, there is no reason why W'ip) and Wu(q) cannot intersect each other. In fact as the toral example of §1.3 shows, it may happen that W'(p) intersects W"(p) (this is called a homoclinic point; see §1.5). We now are in a position to describe the examples, or the class of examples, we mentioned earlier. As a prototype it is worthwhile to keep in mind the diffeomorphism ga: S*—*S* of the 2-sphere which can be described complex analytically on the Riemann sphere by z—*2z. The two fixed points are 0 which is expanding and <», contracting. Then W«(0) = S « - » , W(0)=0, W*(co)«oo, W(«>) = Sl-Q. It is easily checked that go is structurally stable. Of course one may con­ struct a similar example on 5" with two fixed points. More generally we will consider /EDiff(M), M compact, which satisfies the following three conditions: (2.2) (1) Q, the nonwandering set, is finite. (2) The periodic points of / are hyperbolic. (3) (Transversal intersection condition) For each p, gEQ, W*(p) and Wu(q) have transversal intersection. It follows from (1) that ft consists of periodic points and (2) that W'(p), W"(q) are defined for p, gEfi. The last condition means that whenever xEW(p)r\W»(q), then T,(W(p)) and T.(W*(g)) span T,(M). It is trivial to check that the above g0: S*—*S* satisfies (l)-(3). Futthermore, consider for the moment, diffeomorphisms of the circle S1 satisfying (2.2). In this case (2.2)-(3) is vacuously satisfied,

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and it is easily checked directly that these diffeomorphisms are open. By perturbing an arbitrary/£Diff(5') so that its rotation number [24] becomes rational and a further approximation to obtain (2.2)-(l) we obtain the fact that these diffeomorphisms are open and dense in Diff (S1) (Peixoto's theorem [84]). As one goes around the circle, the expanding and contracting periodic points alternate. The structural stability in the case is easy to check [84]. If A QB, clos A denotes the closure of A in B. (2.3) THEOREM [lW]*Supposef: M-^M satisfies (2.2). Then (a)/or each />£2, W'(p) is imbedded in M and Af=U, e 0 W'{p) {disjoint union of course). (b) clos W'(p) is the union of W'(q), for q in some subset of ft. If we write 7 ^ 7 ' for periodic orbits 7, 7' whenever Uvey W'(p) Cclos U«eT' W'(q), then ^ is a partial ordering. If 7 ^ 7 ' and pE:t, qEy', then dim W(p) £dim W(q). (c) One has the following Morse inequalities: Mi > Bo, Mt-Mo^Bidim If

Bo,

dim V

Here 3< is the *th betti number of M and Mt is the number of periodic points p such that dim W'(p) =j. The essence of the proof of (2.3) is in a more general context in §1.8. Using (2.3) (b), one may "represent" a diffeomorphism satisfying (2.2) by a diagram where the vertices of the diagram correspond to periodic orbits and oriented segments are placed between orbits 7 and 7' when 7 ^ 7 ' but there is no other 7 " such that 7 ^ 7 / / ^ 7 / . A labeled diagram is a diagram with the following additional data attached to each vertex 7. The additional data is the germ of the topoIogical conjugacy class of/* at x£y where m is the least period of 7. This germ is described precisely by the dimensions of W'(x), W"(x) and whether/: Wu(x)-*W"(*),/: W'ix^W'ix) are orientation pre­ serving or reversing (this is a consequence of the theorem of Hartman [39] and Grobman (see [74] which says that locally a diffeomor­ phism at a hyperbolic fixed point is topologically equivalent to its derivative at that point). (2.4) PROBLEM? (a) Exactly what (abstract) labeled diagrams occur as diagrams of diffeomorphisms satisfying (2.2)?

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(b) Given compact M exactly what (abstract) labeled diagrams occur as the labeled diagrams of diffeomorphisms of M satisfying (2.2)? Note that (2.3) (c) may be viewed as a restriction on the kind of diagrams that can occur. Figure 2 below gives the phase portrait or orbit structure of an example of a diffeomorphism of the 2-sphere satisfying (2.2).

FIGURE 2

Here the main disk is to be contracting into itself with one expand­ ing fixed point p outside. Inside the disk are five fixed points a, c, e all contracting and b, d of saddle type. The diagram for this diffeo­ morphism is given by P

FIGURE 3

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Among other interesting results on this subject, Jacob Palis [82] shows that diffeomorphisms satisfying (2.2) form an open set in Diff(Af). He also shows that the diagram of the perturbation off is naturally "isomorphic* to the diagram of/. Even though the above facts give something of a "phase portrait" (in the terminology of [57]), a number of problems on this subject still remain. For example (2.5) PROBLEM6 [109], [ i l l ] . Are these diffeomorphisms (of 2.2) structurally stable? J. Palis has given an affirmative answer in dimen­ sion 2. (2.6)7 What homotopy classes of continuous maps (homotopy equivalences) admit diffeomorphisms of (2.2) type? A necessary con­ dition which follows from the Lefschetz trace formula is that | A^/**) | ) = 1, so that at x, Tx(W*(p)) and T,(W'(q)) intersect in just one point in Ts(M). Clearly a diffeomorphism possessing a heteroclinic point is not gradient-like. The orbit of a heteroclinic point consists of other heteroclinic points. The interested reader will be able to check that the existence of the heteroclinic point x above forces W'(q) to oscillate strongly as it gets close to p and W'(p). The boundary of W*{q) contains W'(j>). The picture looks something like Figure 4. To obtain a global example one may modify the diffeomorphism of the 2-sphere of Figure 2. The result will be something like Figure 5.

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FlGUlsS

Its diagram is given in Figure 6. We discuss relaxing or dropping some of the conditions (1), (2), (3) of (2.2). The rest of Part I is concerned with weakening (1), so we consider now (2) and (3). It seems to us that dropping (2) or even modifying (2) significantly would take one far from the picture described by (2.3). What happens if (3) is relaxed? Consider the following substitute for (3)?

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P

FIGURE 6

(3') If W'(p) and Wu(q) intersect at all, then there is a point of transversal intersection of W'{p) and Wu(q). With the weaker (3') replacing (3) one still is able to prove (2.3). Moreover with either (3) or (3'), the relation ^ and the diagram are invariant under perturbation. However, with a weakened version of (2.2) there is no hope of proving structural stability as the simplest counter-examples show. In fact for structurally stable /£Diff (M), (3) is satisfied. The bulk of this section is taken from [109] with some updating, a few examples and other points added. On the other hand, many of the ideas go back quite a number of years. Certainly the local theory as mentioned in the text is of this character. Also Poincare" [89], Birkhoff [16], and M. Morse [68] all had some parts of this global picture. Since this earlier work, Andronov and Pontrjagin [6], Elsgolts [30], Peixoto [83], Reeb [94], and Thorn [124] among others had made contributions toward the picture given in this section. Besides giving simple examples of fl-stable diffeomorphisms, the material in this section serves as an introduction to the more general theory of §1.6, where a number of these concepts have natural exten­ sions. 1.3. Anosov diffeomorphisms. The examples of this section (at least roughly speaking) are at the opposite extreme from those of the preceding section in that the whole manifold consists of nonwandering points and the periodic points are dense. This is in contrast to ft being finite as in §1.2. We give first the simplest examples of Anosov dif­ feomorphisms, the toral diffeomorphisms. Consider/o, a 2 X2 matrix with integer entries and determinant ± 1, i.e., /eGGL(2, Z). Then / 0 can be thought of as a linear transforma­ tion of the plane R* which preserves the lattice L of points with integer coordinates. There is an induced diffeomorphism/ of the quo-

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tient R2/L = T*, 2-dimensional torus, onto itself. This diffeomorphism / : T2—>T2 has a fixed point p corresponding to the origin of R2. Now suppose/o is hyperbolic, for example

CDThen p will be a hyperbolic fixed point of / and the stable and un­ stable manifolds W'(p) and Wu(p) will be the image of the eigenspaces of/o under the projection II: R2—>7"* (since/ 0 is hyperbolic, the eigenvalues X, n are real and satisfy | x | > l > | j t | > 0 with 1-dimensional eigenspaces). Since W'(p) is a 1-1 immersion, it winds densely around the torus and similarly with Wu(p). The intersection points, in W'(p)r^W(p) (called homoclinic points, see §1.5), are clearly dense in T2, and it can also be shown that the periodic points of / are dense in T*. This follows from an algebraic argument or one can use the generalized Birkhoff theorem (see 1.(5.6)). For any periodic point q£T2 of period m, the derivative of f" at q can be thought of as/J 1 : R2-+R2 after identifying T9(T*) and R1 by translation. The stable manifold W'(q) will then just be the translate of W'(p). From the Lefschetz Trace Formula (see §1.4), the number of periodic points N„ of period tn is 1 — (\m+nm)+degreef. Then any g in the same homotopy class as / must have an infinite number of periodic points and therefore cannot satisfy (2.2). It turns out t h a t / is structurally stable so that any perturbation of / ' will also have periodic points dense in T*. Everything said about /o extends to hyperbolic / 0 (EGL(«, Z), defining what we will call toral diffeomorphisms. The definitive version of the structural stability of/ is contained in the work of Anosov [7], [8] which we will describe now. We recall that a Riemannian vector space bundle E over a space X is a vector space bundle such that each fiber Ex is equipped with an inner product ( , )z in a continuous manner. This allows one to speak of the norm ]|»|| of a vector » £ £ « . A bundle map between vector space bundles is a fiber preserving if>: E—*E of a Riemannian vector space bundle into itself and will be called contracting if there exists C > 0 , 0 < X < 1 such that for all » £ £ , mEZ+

llf-wll < cx»lM|. I t will be called expanding if there exists d>0, (i> 1 such that for all vEE, mEZ+ | | * - « | | > dr-\\v\\.

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Above, we really are just using the norm in each fiber, not the inner product. (3.1) PROPOSITION. If X is compact, then the property of being con­ tracting or expanding for 4>: E—*E, E a Riemannian vector space bundle over X is independent of the Riemannian metric. PROOF. TWO norms || ||, |[ |l' on fibers of E are related locally, and hence globally by a\\ \\Z\\ }\'£b\\ \\ for some a, b>0. Thus if ||*-(«)|| S«X-|M|, then U-(v)\\'*c(b/a)\i\4'.

(3.2) PROPOSITION. The inverse of a contracting bundle automor­ phism is an expanding bundle automorphism and vice-versa. PROOF.

Suppose

||^"*(w)|| gcAw||w||.

Then

writing m{v)—w,

||*-(w)||fc(iA) , invariant under Df: T(M)-+T(M) so that Df: E'+E' is contracting and Df: £"—►£" is expanding. The Riemannian structure of T(M) restricts to give a Riemannian structure on E' and £" so that this condition makes sense; further­ more, in case M is compact, by our previous comment, the Rieman­ nian structure is unnecessary. (3.3) THEOREM (ANOSOV) ([7], [8]). An Anosov diffeomorphism f of a compact manifold M is structurally stable. Furthermore if there is a Lebesgue invariant measure for f on M, then the periodic points are dense and f is ergodic. Finally the Anosov diffeomorphisms are an open set in Diff(Af)For the proof of the first statement of (3.3) see the exposition of J. Moser's proof by J. Mather in the appendix. For the last sentence, see §1.8.

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It is apparent that the toral diffeomorphisms are Anosov diffeomorphisms; the splitting by / at p translates to each point of P" to give the desired global splitting. From an invariant measure for a diffeomorphism / of a compact manifold M, one can see easily that every point is nonwandering, i.e., il = M. It is from this fact that Anosov concludes the density of the periodic points of/. (3.4) PROBLEM. IS it true that for every Anosov diffeomorphism of a compact manifold M,Q = M, or equivalently, the periodic points are dense in Af?9 A second question is: does every Anosov diffeo­ morphism have a fixed point? Motivation for this work of Anosov comes not only from the toral diffeomorphisms, but more importantly from geodesic flows on mani­ folds with negative curvature, where Anosov's ergodicity solves an old problem. This is the 1-parameter analogue of (3.3) and will be discussed later in our survey. For (3.3), the basic idea of Anosov's proof is to construct through each point p of M, a generalized stable manifold W'(p). This will be a 1-1 immersed cell with the property that for each x£W'(p), the tangent space T,(W'(p)) coincides with E?aCT.(M). Furthermore f(W'(p)) - W'(J(p)), and x, y are in the same W'(p) if and only if d(f*x, /*"y)—*0 as tn—»«. Although each W'(p) is smooth, W'(p) only depends continuously on p (recall that the splitting of T(M) was only required to be con­ tinuous). One may think of the W'(p) giving a continuous foliation of Af. The existence and basic properties of W'(p) are based on old work by Perron [88]. Theorem (3.3) states that the Anosov diffeomorphisms are an open set in Diff(Af). On the other hand Anosov has examples to show that this would be false if one imposed a smooth splitting of T(M) rather than a continuous one in the definitions. The following is a basic and beautiful unsolved problem. (3.5) PROBLEM. Find all examples of Anosov diffeomorphisms of compact manifolds (up to topological conjugacy of course) such that Af=C10 What compact Af admit Anosov diffeomorphisms? Must M be covered by Euclidean space? There do exist nontoral Anosov diffeomorphisms. We will show this now and in fact give the most general known way of constructing Anosov diffeomorphisms. Suppose that G is a connected simply connected Lie group with Lie algebra © and a uniform discrete subgroup T (uniform means that the coset space G/T is compact). Suppose also that/t: G—*G is a continuous automorphism such that/«(T)-r and the derivative at

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the identity/o: T,(G)-*T,(G) is hyperbolic (throughout this discus­ sion it will be helpful to keep the toral case, with G».R", in mind). If T,{G) is identified with ® then fi becomes the Lie algebra auto­ morphism induced from /<>. From this data we will construct an Ano­ sov diffeomorphism / : G/Y—*G/Y. At this writing, this is the most general known construction of an Anosov diffeomorphism. Since the linear automorphism fi : ®—»® is hyperbolic, we get the usual invariant splitting © = ®'-|-®\ Furthermore, (see [114]) there exist constants c, c' such that 0 < c < l < c ' and an inner product on ® so that ||/.'(»)|| C '||«||

allrG®', all«e®".

Next by right translations, identifying ® with T,(G), the splitting and inner product are imposed on the tangent space of every point of G. For this Riemannian metric on G, it is easily checked that fo'. G—*G is given a hyperbolic structure or that/o: G-+G is an Anosov diffeomorphism. Furthermore, this splitting of 1"(G) and the Riemannian metric are both invariant under the action of G on G given by right transla­ tion. In particular they are right invariant under T and SO/Q induces an Anosov diffeomorphism / on the compact coset space G/T. For the existence of the /<> in the previous construction, the next proposition shows that G must be nilpotent. (3.6) PROPOSITION. Suppose that : ®—*® is a Lie algebra auto­ morphism which is hyperbolic as a linear map. Then ® must be nilpotent. For a proof, A. Borel has given me the following reference: let ® be a finite dimensional Lie algebra over a field having an auto­ morphism no eigenvalue of which is a root of unity; then ® is nilpotent. Exercise in Bourbaki with hints: Algebras de Lie, Ex. 21b, p. 124. Now that we know that this construction forces G to be nilpotent, and that T is a uniform discrete subgroup, the results of Malcev [61 ], summarized in [12a], become important. (3.7) THEOREM (MALCEV). (a) A necessary and sufficient condition for a discrete group T to occur as a uniform subgroup of a simply con­ nected nilpotent Lie group is that T be a finitely generated nilpotent group containing no elements offiniteorder. (b) A necessary and sufficient condition on a nilpotent simply con­ nected Lie group G that there exist a uniform discrete subgroup F is that

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the Lie algebra of G has rational constants of structure in some basis. (c) / / r,- is a uniform discrete subgroup of a simply connected nilpotent group d, i= 1, 2, then any isomorphism T\-^Tt can be uniquely extended to an isomorphism Gi—*Gt. The coset space G/T, G, T as above is called a nilmanifold. While (3.6) and (3.7) give some general perspective on our class of homogeneous space Anosov diffeomorphisms, this situation cannot be said to be completely understood. There certainly do exist, how­ ever, many nontoral examples of Anosov diffeomorphisms on nilmanifolds as special cases of the above construction. We give two of them now with dim G = 6. Let G\, Gt be copies of the three dimensional simply connected, nonabelian nilpotent Lie group. We take a basis Xit F „ Z< of ©,-, »'= 1, 2 with the bracket relations [Xit F,] =Zit i— 1, 2 and all other brackets zero. The main group G of our basic construction above will be GiXGj. For each real number X > 1 we define a hyperbolic auto­ morphism /o of G by specifying/o' (fi, ®, ©', etc. as in the above con­ struction) on ® in terms of the basis as follows. EXAMPLE 1

EXAMPLE 2

X\ —► XX\

X\ —y XXi

i

Y1-*\ Y1

r1->x-»r1

Zi ->• X'Zx

Zx -»X-'Z,

l

Xt -* \- Xt

Xt -► X-'AT,

F^X-'F,

Yt-+\*Yt

Zi -> X-»Zj

Z s -> XlZ2

Note that in both examples brackets are preserved. In Example 1, one sees that ©", ®* are both ideals which coincide with nilsubalgebras ®i and ®j respectively. In this case G is the product of the cor­ responding subgroups, G = GUXG: In Example 2, both &u and ®* are seen to be abelian, but they are not ideals and G is not (in the group sense) a product of the cor­ responding subgroups Gu and G'. The next step is to find a uniform discrete subgroup TQG such that /o(r) = T. For this we will use matrices with coefficients in an alge­ braic number field. Let K = Q(3llt), the number field of 3 1 ' 2 adjoined to the rationals, and a: K-+K the nontrivial Galois automorphism (sending 3 1 ' 2 into - ( 3 1 ' 2 ) ) . We may suppose that ®i and ©« are each represented by matrices of the form,

680 19671

DIFFERENTIABLE DYNAMICAL SYSTEMS

X 0 0 0 0

[0 (3.8)

763

z

\ Y 0

X,Y,ZER

and @ =- ®i X ®» becomes the space of matrices

c> A, B each of the form (3.8). We will take To then to be the lattice of ® of matrices of the form

it:) where A is as in (3.8) but X, Y, Z are restricted to be algebraic in­ tegers in K and A* is the image of A under the map induced by
a-7

in G where a > 1. Then if er* : ©—►© is the Lie algebra automorphism, we have the invariant decomposition © = ®,-r-®"+A where ®* is contracting under «•**, ©" expanding and h is invariant pointwise. In fact h is the Lie algebra of the centralizer H of A, of all diagonal matrices. Just as in the previous construction we put a metric on

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® = T,(A) which is right translated around, but contains a degenerate component corresponding to h. On G/H, however, the degeneracy is divided out so that we have an induced Anosov diffeomorphism <(><,: G/H—*G/H. G/H is a 4dimensional manifold which is not contractible, but clearly simply connected. Novikov informed me that he could prove that if / £ Diff (M) is Anosov, M compact, where the dimension of W"(x) is one less than the dimension of M, then xi(Af) is abelian and M is covered by Euclidean space. The two dimensional toral example was first communicated to me by Thom to show that there was an open set in Diff (7^) of diffeomorphisms with no contracting periodic points, therefore implying that diffeomorphisms satisfying (2.2) were not dense. After adding some geometry to the example, I showed it to Anosov when I spoke on the examples of §1.5 in the Soviet Union in 1961. By 1962 Anosov announced his theorem on structural stability in the context of what is called here Anosov diffeomorphisms. Proofs have now appeared [o]. The problem of the existence of (compact) nontoral diffeomor­ phisms was posed by Anosov in his Congress talk, Moscow 1966. Previously, after putting this problem into Lie group perspective, I had consulted many Lie group experts to arrive finally at what is here. In particular, conversations with Boothby, Borel, Hochschild, and Langlands were very helpful. The 6-dimensional Example 2 as well as the explicit algebraic number theory approach were given to me by Borel.12 1.4. The zeta function of a diffeomorphism. Suppose / : M—*M is a diffeomorphism with the property that Nm< » , m = l, 2, • • • where Nm = Nm(f) is the number of fixed points of/". This is a generic prop­ erty (see 61.6). Then following Artin-Mazur [12], one defines the zeta function of / as the formal power series fW = exp2«-i(Vw»)^v'»><". This turns out to be an interesting invariant of/. Of course f/(0 = £(<) is an invariant of the topological conjugacy class of/and even of the conjugacy class "on Q" of/. The zeta function thus contains all the information about the numbers Nm = Nm(f) where Nm counts all the periodic points of period m. But this is different from Km = Km(J) which denotes the number of periodic points of least period m. The number Km is more directly interesting in many respects and it is natural to ask for the relation between Nm and Km. From the definition it follows directly that

682 1967] (4.1)

DIFFERENTTABLE DYNAMICAL SYSTEMS

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PROPOSITION.

£

K, -

I titidm

Nm.

m

Narasimhan pointed out to me that one solves (4.1) for the Kt by the Mobius inversion theorem (see [36]). This gives (4.2)

PROPOSITION.

lH»Umm

Here if l=P\ • • ■ pT where the £< are distinct primes, then u(l) «■ (—1)', u(l) = 1, and if / contains a power of a prime, u(l) = 0. The function /*(/) is called the Mdbius function. Observe that m always divides Km (i.e., 2sTm/m'EZ+). The inspiration for the above zeta function is the Weil zeta function of an algebraic variety over a finite field. Dwork recently proved the rationality of this zeta function, see [lOl] for a general reference. For the differentiable version, there is the following theorem. (4.3) THEOREM (ARTIN-MAZUR [12]). For any compact manifold, there is a dense set of Difi(M) for which the following estimate holds: Nm £ CA-. Here C, k are positive constants which depend only on the diffeomorphism / and Nm = Nm(f). (4.4) COROLLARY. For a dense set of Diff(M), the zeta function has a positive radius of convergence, so it can really be considered a function. Actually Artin and Mazur define Nm to be the number of isolated fixed points of/", while permitting/*" to have an infinite number of fixed points. Thus, for example, they do not know whether the fixed point set is finite for the maps in the dense set they obtain. The proof of (4.3) uses algebraic approximation techniques which go back to John Nash [73]. Actually Artin and Mazur define f(/) for differentiable maps for which Nm < <» and prove their theorem in the more general context of differentiable maps. The following prob­ lem then becomes important. (4.5) PROBLEM. Is f (/) generically rational (i.e., is f> rational for a Baire set o f / ) ? 1 3 This goes beyond their theorem in two ways. First, generically true means true for a Baire set which, of course, is much bigger than sim-

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ply a dense set. Secondly, rationality is stronger than possessing a positive radius of convergence. Rationality is especially important because this means for the diffeomorphism that the poles and zeros of the zeta function, a finite number of invariants, determine the infinite set of Nm. The Nm are of course very important objects to get ones hands on. In the direction of (4.5), Artin and Mazur [12] asked if diffeomorphisms in their dense set have a zeta function which is algebraic. More recently, there has been proved the following (4.6) THEOREM (K. M E Y E R ) . 7//£Diff(Af), M compact, satisfies Axiom A (see §1.6), then the estimate of (4.3) is valid. K. Meyer's proof of this is very simple and if one had the density (see §1.6) for Axioms A and B, this would of course supersede (4.3). We will now examine the zeta function for our examples. If/: M—+M is a diffeomorphism such that Nm<<x> for all m £ Z and A is a closed invariant subset of M, then by definition tiif) = exp ££-i(l/ w *)iV4* m w n e r e N* 1S t n e number of x £ A such that m f (x)=x. (4.7) PROPOSITION. Suppose for f£Diff(M), the periodic points are all contained in the union of two disjoint closed invariant subsets Ai, A» of M. Suppose also that f Aj and f A, are rational (or convergent). Then f/ = f is rational (or convergent) and in fact f (t) =fA,W -£&t(t). PROOF. JV' _L N"

f(/) = exp X (4.8)

JJ"

V'

<"* = exp X,

LEMMA FROM CALCULUS.

tm exp X,

r = fA,(0 • fi,(0.

log (1/(1— y))= 2Z"-i(l/*)y*-

For the diffeomorphisms of (2.2) the following theorem gives the zeta function. (4.9) THEOREM. Suppose for / £ D i f f ( M ) , Q is finite. Then clearly B = UTe/> $2T wfcere P is the set of periodic orbits off, and Qy is the set of points of fi in y. The zeta function of f is the following, where m(y) = period of y,

KO-n-rzTsr By (4.8) 1 / ( 1 - r ( 7 ) ) = exp 2rf-x(l/*)<- (1r) *. Apply (4.7). The Lefschetz number L(p)=L(p, f) of a hyperbolic fixed point p PROOF.

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DIFFERENTIABLE DYNAMICAL SYSTEMS

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of /GDiff(Af) may be defined most simply, perhaps, as ± 1 where the sign is the sign of det(J—Df(p)), I: TP(M)-*TP(M) being the identity. A modern proof of a more general version of the following theorem of Lefschetz may be found in Dold [26]. (4.10) LEFSCHETZ TRACE FORMULA. Suppose/GDiff (Af) has only hyperbolic fixed points and Fix(f) denotes the set of fixed points of /. Then E

L(p) - A(/) where

peFUC/) dim*

A(/) -

E

( - 1 ) ' Trace(/v Hi(M, R) -> ff,(Af, R)).

Here /*, is the induced automorphism of the ith homology group of M with real coefficients. The following proposition follows from the definition of L(p) and the eigenspace decomposition of Df(p). (One may assume Df(p) to be semisimple in the proof.) (4.11) PROPOSITION. For £GFix(/), /GDiff(Af), L(p) = (-l)*A, where « = dim W"(p) and A= + 1 if f preserves orientation on Wu(p) and A = — 1 iff reverses it. The following is well known and will be useful in computing the zeta function for some of the Anosov diffeomorphisms. (4.12) PROPOSITION. Supposef£.Ditt (Af) is such that for every mE.Z+ and every *GFix(/*), L(x, /») = + l . Then f ( 0 - I B S ' ^ W * " 0 ' where Ut) - I I (1 - X«0~l and Xtj,j = 1, • • • , dim Ht(M, R), i

are the eigenvalues (generalized and counted with multiplicity) of /♦,: Ht(M, R)^H{(M, R). PROOF.

By (4.10) dim M

1UJ) = E dim M

(-l)«'Trace(/%, dim Hi

= E (-D E C So we obtain

m = 1,2,3, •• •

685 S. SMALE

768 «o

1 dim it

[November dim H{

iogf(0= E - E (-D E Cf or

r(o = n r<w(-i> dtm Jf <-0

where •o

J dim Hi

logr<(/) = E MM- 1

E (M-

#1

r-l

dim H{

<*> \

E-M-

=E r-l

w - 1 *"

dim Hj

logf<(0 -

E

log(l - IX*)-1 by (4.8)

r-l

and dim Ht

r.(o= n

r-l

i\-t\ir)-\

For any /EDiff(Af). the function f(t) denned in (4.12) is well denned even though L(x, /") is not always 1. It will be called the false zeta function of/and denoted by f(t) or f/(*). It is rational and its expansion counts the periodic points algebraically. In fact, the whole difficulty of the problem of the rationality of the (honest) zeta function is that it counts the periodic geometrically, not algebrai­ cally. Proposition (4.12) shows that under the condition L(x,fm) = l for all x£Fix(f*), and m £ Z + , the false and honest zeta functions coincide. Note that if Nm is the number of points of / of period m counted algebraically, i.e., ffm = E»eFix(/") L(P, /*), then (4.12) shows that {•(<)= exp E»-i(l/ w t )^»l w ano " o n e c311 s e e how the following the­ orem of Fuller [31 ] fits into this context (see also [38]). (4.13) THEOREM. Suppose h: L—*L is a homeomorphism of a poly­ hedron of nonzero Euler characteristic. Then h has a periodic point. Otherwise all the ftm would be zero and f would be one. But the degree of f is minus the Euler characteristic (from (4.12)). (4.14) PROPOSITION. Supposef: M—*Mis an Anosov diffeomorphism such that the corresponding expanding bundle EM is orientable. Then J> is rational and

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(a) if Df: £»—►£" is orientation preserving then f/ = lt

if dim fiber Eu is even,

f/ = Vf/ if dim U

fiber E

" « odd,

U

(b) if Df: E —*E is orientation preserving then f/(0 = f ( - 0 */ dim fiber £" is even, f/(0 = 1/f (—0 if dim fiber £" w odrf. This follows directly from (4.11) and (4.12). It seems likely that looking at a double covering of M, one could furthermore prove that the zeta function of every Anosov diffeo­ morphism was rational. For the toral case of §1.3 defined by hyperbolic /oEGL(«, Z), one finds the zeta function defined explicitly in terms of the eigenvalues of /o. In this case /<> coincides with the automorphism of Hy(Tn, Z) induced by/: Tn—*Tn. By the Kunneth formula the whole cohomology ring of T" is given as a tensor product of H*^1) and so one obtains all of the eigenvalues oif*:H*(Tn)—*H*(Tn) as products of the eigen­ values of/0. One thus obtains easily via (4.14) (4.1S) PROPOSITION. For the toral diffeomorphism f: Tn—*T% defined by hyperbolic /o£GL(n, Z) with the eigenvalues Xi • • • X„ of / 0 , we have:

(a) A(/*)=IL(1-XT). (b) f W - I R i - M * ■■■K»(-1)

,

all (*i, ' • • , i l ) 3 1 i i i < ) ' i < • ■ • < * * ^ »,

(c) f(0 is defined from l(t) according to (4.14) where one checks the appropriate case from the X< with \ X,| > 1. Finally, we remark that through the work of Matsushima [63], Mal'cev [61 ], Nomizu [75], and Kostant [54], one can compute the zeta functions for the nilpotent examples of §1.3 quite explicitly. 1.5. Shift automorphisms and homoclinic points. From the pre­ ceding sections, one might ask whether the set Q of nonwandering points must be a manifold generically (allowing certainly for com­ ponents to have varying dimensions). The examples of §§1.2 and 3 have this property. Here we will see that the answer is no, and in fact give an example of a diffeomorphism of S1, Q-stable, such that ft is the union of a Cantor set and two isolated points. First a description of the shift automorphism of symbolic dynamics will be given (see [14] or [35a] for more details). Let 5 be a finite set,

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discrete topology, consisting of N elements and define Xs to be the set of functions from Z to S provided with the compact open topology (Z has the discrete topology). If a^EXs, the value of o a t w»£Z will be denoted by am and we write a = Ylam. Then a may be thought of as a doubly infinite sequence of elements of S with a decimal point between a 0 and a\, thus a a= ( • • • o_iO0-aio« • • • )• An important special case is where 5 has two elements and here we could assume each a,- is either 0 or 1. For general S, Xs is homeomorphic to a Cantor set. Define a map a: Xs~*Xs by (a(a))m = a m+ i. In terms of the doubly infinite sequences, a shifts the decimal point one place to the right It is easily seen that a is a homeomorphism, called the shift auto­ morphism of Xs. It has been widely studied in ergodic theory and probability [14] as well as topological dynamics [35a]. (5.1) PROPOSITION. The periodic points'of a are dense in Xa and if Ck is the number of periodic points of period k (i.e. fixed points of ak) k>0 then Ck = Nk where N is the cardinality of S. PROOF. The element a= YLam£zXs will be periodic of period k precisely when am = o»+i for all m £ Z . Thus it is determined by ffli, • • • , ak with Oi, • • - , o* arbitrary elements of 5. Given any b= T[bm€zXs and K large, one can choose a periodic approximation a= JJ[am of b with o< = 6< for | t |
(5.2) COROLLARY. The zeta function for a: XS—*XS can be defined as in §1.4 and in fact f (/) = 1/(1 — Nt) where N=cardinality of S. This follows from (5.1) with the aid of (4.8). M. Morse has proved (see [35a]) that there is a subset of Xst homeomorphic to a Cantor set, which is a minimal set for a. To see how symbolic dynamics enters into our diffeomorphism problem, we will first describe an example of a diffeomorphism g mapping a subset Q of the plane into the plane. Here g(Q) is not a subset of Q, but eventually we will use g to define a global diffeo­ morphism / of S2 onto itself. One might think of Q as a neighborhood (not invariant) of an indecomposable piece of the nonwandering points of t h i s / : S*—*S*. Take then Q to be a square in the plane R*, for example, Q= {(x, y)G-R'l | x \ ^ 1 , \y\ ^ 1}. Then g will map Q into the region bounded by dotted lines with g(A) =A' etc. in Figure 1.

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c Pi

I V \ v.

Q FIGURE 1

Pt

B FIGURE 2

We will take any such g which has the following properties. (a) g is a diffeomorphism of Q onto the region in Figure 1 bounded by dotted lines sending A—*A', B-*B' etc. (b) on each component Pi, P» of r ' ( l ( O ) ^ 0 i i w>^ De a linear map (up to a translation). To understand (b) note that as a consequence of it, Pi, Pt will be as in Figure 2 and g(P.) = (?,, i = 1, 2. The reader will be able to verify that the intersections of all the images gm(Q),m = \, 2, • • • or, more accurately, f)Z-i im(Qim)) where Q(») = @r\image g" -1 , is a product of a Cantor set and the interval

1*1 £1-

Define A to be the intersection, f)mez gm(Q'm)), Qo = Q, Q(m} as above for m > 0 and for w < 0 , ()<»>=g»(Qc«+i>), Thus A may be thought of as the set of nonwandering points of g: Q—*R*. The careful reader will be able to check for himself the next proposi­ tion (which is in [115]). (5.3) PROPOSITION. The subset A of Q is compact, invariant under g, indecomposable and onQ, g is topologically conjugate to the shift auto­ morphism a: Xs—>Xs, with the cardinality of 5 = 2 . Furthermore one can prove stability with the less obvious proposi­ tion [115]. (5.4) PROPOSITION. For a perturbation g' of g, A' defined similarly is also compact and invariant under g'. Then g': A'—»A' is also topo­ logically conjugate to the shift a: Xs—tXs. Thus (at least after we globally extend g) we have another example of a stable indecomposable piece of nonwandering points. One may modify the above example in the following way. The image g(Q) may wind half-way around Q before passing through Q the second time, or even wind around Q several times for that matter (Figures 3 and 4). This won't change g: A—»A, but g will be different

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on U(A) where U is any neighborhood of A. The intrinsic picture (with respect to fl) is the same for Figures 1, 3, 4 but they differ extrinsically (in any neighborhood of Q).

FIGURE 3

FIGURE 4

One may further modify the above examples by having g(Q) pass­ ing through Q several times (see Figures 5 and 6).

FIGURE 5

FIGURE 6

In all of these examples it is important to keep the linearity condi­ tion (b) above. Then one may define and analyze A as in the first case. Always g: A—>A will be topologically conjugate to a shift auto­ morphism a: Xa—*Xs and stably so. The cardinality of 5 will equal the number of components in Q(~\g(Q), e.g., three for Figure 5 and four for Figure 6. Thus all of the shift automorphisms occur in the above framework. To really complete this picture, A above must appear as an inde­ composable piece of the nonwandering points of a global diffeomorphism. We construct such an / : S*—*S* now which extends the map g: Q—*R* of Figure 1. Consider Figure 7. We have put the square Q into a disk D*C.R7 and we extend g to go: D*—*D* by mapping G diffeomorphically onto G' and F onto F*. The map go: F—*F* is defined so that it is a contraction about some fixed point Po in F*. This g0: Di—*D* will be a diffeomorphism of D* onto a subset of D* so that the nonwandering set is the disjoint union of A and Pt. Finally one easily extends go t o / : S*-*S* so that the non-

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DIFFERENTIABLE DYNAMICAL SYSTEMS

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FIGURE 7

wandering points 8=ft(/) —AKJpa^Jqt where g0 is an expanding fixed point of / outside of D1. T h i s / is our desired global diffeomorphism. At this point it seems appropriate to give a general way of con­ structing fi-stable diffeomorphisms of S*. Take any diffeomorphism / : 5'—*S* satisfying (2.2) with a con­ tracting fixed point p. Let V be a contracting disk neighborhood of p and redefine/ on V to be gt as described above (the "surgery" of §1.10). More generally l e t / : S1—>S* satisfy (2.2) with a contracting periodic orbit pi, • • • , Pk and let V be a disk neighborhood of pi such that/* contracts V into its interior. Then one modifies / (via surgery again) on UJrJ/*'(F) to obtain/': S2->.SS so that on F , / * is conjugate to g0 above. Finally a straightforward modification of the previous construction allows one to introduce into any diffeomorphism / : S*-+S* satisfying (2.2), indecomposable pieces A topologically conjugate to shift auto­ morphisms on N symbols (cardinality S=N) where N can be any­ thing we like. In all of these examples the indecomposable pieces of ft are shift automorphisms, finite periodic orbits or products of the two. We see easily from previous remarks that the zeta functions of these/: S1—*St are finite products of factors of the form 1/(1— Ntq) where N and q are positive integers. It should be noted that the Lefschetz Trace Formula imposes con­ ditions on what products of the above form can occur in these zeta functions. It restricts the Nit pt that can occur in f(s) 1

-IB-itt-w)- .

One can see an analogy between the shift automorphism and the nilmanifold examples of §1.3 by considering the shift automorphism in the following light.

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Let Zn be the cyclic group with n elements and for each m £ Z let Gm be the abelian group of formal power series (starting a t m), / ( * ) = UlT-m «•** with a{£Zn. Put the structure of a compact group on Gm with the product topology. Define the locally compact group of all power series by G = \JmezGm. The map ': G—*G defined by f—*xf is a contraction while u: G—*G defined b y / - ^ x - 1 / i s an expan­ sion. I t is easily checked that the subgroup T of GXG defined by f = {(A / ) |/(*) a polynomial in G, f(x) = £ ? _ m «<*'. /(*) = 2>-<**'} is uniform (compact quotient) and discrete. The "hyperbolic" auto­ morphism <j>'Xu: GXG—*GXG preserves T and the induced homeomorphism : G/T—*G/T is precisely the shift automorphism on n symbols. One may identity ': G^>G above with g: W'ij^-^W'ip) where PEA-, A as in the example of Figure 1, n = 2, W'ip) = W'ij>)r\K. There is a very close relation between the shift automorphisms discussed above and what are called homoclinic points, first discov­ ered (in the restricted 3-body problem) and named by Poincar6 [90 ]. A homoclinic point of / £ Diff (M) is a point of intersection x £ W'ip) r\Wu(q). If W'ip) and Wu(q) are transversal at x, then x will be called a transversal homoclinic point. As realized by Poincar6 [90 ], homoclinic points complicate the orbit structure of a diffeomorphism considerably. The orbit of a homoclinic point consists (clearly) of homoclinic points. Taking the case p=q, one sees that the existence of homoclinic x forces Wuip) to double back on itself oscillating faster and faster as it does so. For example, for the plane, we will obtain behavior something like that described in Figure 8.


X

0 ^ *t 4
FIGUW8

692 i»67]

DIFFERENTIABLE DYNAMICAL SYSTEMS

775

This is the same phenomena that is occurring in Figure 1, but looked at in a different way. In fact, the best way to understand what is going on in Figure 8 is to imbed it (in some sense) in Figure 1. A great advantage of the horseshoe approach of Figure 1 is that one gets a satisfactory picture of the orbit structure and stability while a given homoclinic point at first glance seems to defy analysis. That is the idea behind the following theorem [U5].14 (5.5) THEOREM. Suppose x is a transversal homoclinic point of /GDiff(ii). Then there is a Cantor set AQM, *EA, and mEZ+ such that /"(A) =A and f* restricted to A is topologically a shift automor­ phism. By (5.1) this implies: (5.6) COROLLARY. In every neighborhood of a transversal homoclinic point of/EDiff (Af), there is a periodic point. We interpolate a little curiosity. Note that for the shift on X sym­ bols, Nm"\m so that by (4.2) and the fact that Km/mEZ+, Km ™ £»/» M(0A"*"—0 mod m for every X, m€-Z+. This number theoretic identity for m a prime becomes X»a»X mod p, or Fermat's Theorem. The material in this section is mainly taken from [115] with a number of examples andfiguresadded. The shift automorphism goes back to Hadamard (but it is even sometimes called the Bernoulli automorphism!) who used it to study geodesic flows on 2 manifolds of constant negative curvature [40]. M. Morse [69] obtained further results in the same context. G. D. Birkhoff [l7], [18] in his works on surface diffeomorphisms, studied homoclinic points. In [17], Birkhoff proved (5.6) in dimen­ sion 2, and in his [18, p. 184] he noted a resemblance between his homoclinic points and Hadamard's shift automorphism. I came across the "horseshoe" of Figure 1 when I was trying to get a geometric picture of a variant of van der Pol's equation in N. Levinson's paper [58]. He had written me earlier that this equation had stably an infinite number of periodic solutions. The "horseshoe" was the first example of a structurally stable (or Q-stable) diffeomorphism with an infinite number of periodic points [113]. Putting the shift automorphism into the group theoretic frame­ work was done with the aid of Cat Moore. 1.6. Unification.16 The work of the present section is motivated by the search for unity in the examples and phenomena of the preceding part of the paper. Anosov's work on hyperbolic structures on mani­ folds gives a clue on how to proceed. The Anosov diffeomorphisms,

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however, are rare among all diffeomorphisms; only certain manifolds even permit them. One is looking for a class of diffeomorphisms which include all of the previous examples in a transparent way and will at least have the possibility of including an open dense subset of Diff(Af) for each compact manifold M. This is provided by diffeomorphisms described now, i.e., those satisfying Axioms A and B below. Suppose then/: M—*M is a diffeomorphism of a Riemannian mani­ fold and AQM is a closed invariant subset. We will say that A is hyperbolic (or has a hyperbolic structure) if the tangent bundle of M restricted to A, Tt.(M) has an invariant (continuous) splitting under Df: Tk(M)-*TA(M), Ti(M)=E"+E' such that Df: £•-►£• is con­ tracting and Df: £"—►£" expanding (see §1.3 for these definitions). The dimension of the fiber of £ ' need not be constant but only locally so. Since A is invariant, Df is an automorphism of the bundle Ti(M) and this with the Riemannian metric gives sense to the above defini­ tion. Note that if A (or M) is compact, one may dispense with the Riemannian metric by (3.1). The simplest examples of hyperbolic sets for diffeomorphisms are first of all the hyperbolic fixed points (§1.2) and the hyperbolic periodic points. The finite union of these cover the case of A finite. Next, of course, the Anosov diffeomorphisms of §1.3 are examples where the whole compact manifold is hyperbolic. Also for the ex­ amples of §1.5, the A homeomorphic to a Cantor set is easily checked to have a hyperbolic structure. In all of the above examples the hyper­ bolic sets consist of nonwandering points and the periodic points are dense in each of them (up to the unsolved problem (3.4)). The fol­ lowing is an example to show that hyperbolic sets need not satisfy either of these properties. Take a diffeomorphism satisfying (2.2) which has a heteroclinic point x£zW,(p)r\Wu(q). The 2-dimensional example of Figure 5, §1.2, will do. Thus fi is hyperbolic and one may extend this hyperbolic structure to the orbit of x. In fact the tangent spaces of W'(p) and Wu(q) at x give the desired splitting at x and similarly for each point in the orbit of x. The orbit of x together with 0 is a closed invariant set and this gives the example. The closure of the orbit of x is an indecomposable hyperbolic set. Recall that a homeomorphism h: X—*X is said to be topologically transitive if there is a dense orbit. Then the dense orbits form a Baire set of X (assuming that X is a compact metric space). We will now consider a diffeomorphism / : M—*M of a compact manifold which satisfies the following two properties [116].

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(6.1) AXIOM A: (a) the nonwandering set S is hyperbolic, (b) the periodic points of f are dense in ft. Of these two properties (a) is the most important in what follows. In fact (b) may even be a consequence of (a).17 The above example however shows that a proof of (a) =>(b) must use the fact that ft consists of nonwandering points. (6.2)

THEOREM (SPECTRAL DECOMPOSITION OF DIFFEOMORPHISMS)

[117]. Suppose/: M—*M satisfies (6.1). Then there is a unique way of writing ft as the finite union of disjoint, closed, invariant indecomposable subsets (or "pieces") on each of which f is topologically transitive: Q = Q» U . . . U 0*.

(6.3) COROLLARY. / / / : M-+M is as above one can write M canonically as a finite disjoint union of invariant subsets M=Uf_j W*(Q<) where W*(ft,) = [xeM\r(x)^ait « - > » }. As the remarks at the beginning of this section indicate, the ex­ amples of §{1.2, 3 and 5 satisfy (6.1). The spectral decomposition theorem gives a little perspective on the question of rationality of the zeta function. The zeta function of such an / will be a product of zeta functions, one for each ft<. It seems plausible to me that each of these zeta functions is rational.18 The results of §§1.4 and 1.5 are consistent with this. We explain why we use the words "Spectral Decomposition for Diffeomorphisms" in (6.2). The decomposition of the manifold into invariant sets of the diffeomorphism is quite analogous to the de­ composition of a finite dimensional vector space into eigenspaces of a linear map. In one case we are considering automorphisms in the category of differential topology, in the other, finite dimensional vector spaces. As Jacques Tits pointed out, one may make this more precise by actually putting a linear transformation (generically) into the frame­ work of (6.2). Suppose then u: V—>V is a linear transformation of a complex n-dimensional vector space. By multiplying by a constant, we may suppose u has determinant 1, i.e., wESL(n, £). We will furthermore suppose that the eigenvalues Xi, • • • , X„ of « have dis­ tinct absolute values which are not one. Consider the induced diffeo­ morphism of projective space u9: Pm~1(Q—*Pn~l(C) defined on co­ ordinates by Zt—*\{Zi. Then u„ will satisfy (2.2) with ft consisting of n fixed points (0 • • • 01 0 • • • 0). The two ways of looking at the spectral decomposition coincide.

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Now we state the second of our two main axioms (introduced in [117]). (6.4) AXIOM B." Suppose that fEDiff(M) satisfies Axiom A and that fl„ W'(Oi), etc. are as in (6.1), (6.2), (6.3). Then if W'(Qi) r\Wu(Qj)^0, there exist periodic points pEQi, qEQj such that W(j>) and W"(q) have a point of transversal intersection. The following generalizes the theorem of Palis, see [82] and §1.2. (6.5) THEOREM.20 The set o//£Diff (M) which satisfy Axioms A and B are open and such f are Sl-stable. Assuming /, 8„ etc. as above, we say that Q ^ Q / if W'(Qt) n W " ( f i ; ) ^ 0 . Then we also have (generalizing theorems of §2) (6.6) THEOREM. ///£Diff(AQ satisfies Axioms A and B, then ^ is a partial ordering which is preserved under perturbation. These theorems (6.2), (6.3), (6.5), (6.6) have no proofs in the litera­ ture, but we will try to give a good sketch of their proofs in §§1.7 and 8, §1.7 for (6.2) and (6.3), §1.8 for (6.5) and (6.6). We say that 0,-, ^fti.^fl,-,^ • • • ^ ft*, is a maximal chain if the Q,-, are distinct, and n is maximal. For every /£Diff(Af) satisfying Axioms A and B, we define the diagram A(/) as follows. A(/) is a linear graph whose vertices cor­ respond to the $2,-, labeled by conjugacy class, and directed 1-simplices join consecutive vertices of maximal chains. The diagram A(f) is invariant under perturbations of/. Generalizing problem (2.4) is (6.6)a PROBLEM?1 What diagrams can occur for diffeomorphisms satisfying Axioms A and B? Given first the manifold M, what dia­ grams can occur for/£Diff(Jlf) satisfying Axioms A and B? Finally one can label the diagrams with conjugacy classes of germs of dif­ feomorphisms on neighborhoods of the ii< as in §1.2 and ask the above two questions for these labeled diagrams. One can see that a prototype of diffeomorphisms satisfying Axioms A and B are those of §1.2 with Axiom 3 replaced by Axiom 3' there. The above results may be construed as saying that we have succeeded in relaxing the hypothesis that ft is finite. The diagram here gives sort of a very generalized gradient structure to these diffeomorphisms. The main point of Axioms A and B and subsequent theorems is that the hypotheses include and unify all known 0-stable diffeo­ morphisms, while describing an open set of Diff (M) which is ame­ nable to study. In fact the above theorems as well as those in the future sections give the beginnings of a structure theory for diffeo­ morphisms satisfying Axioms A and B.

696 x*7]

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Thus the question, "are these diffeomorphisms dense in Diff (M),n becomes particularly sharp. This is not yet settled. In this direction, the first theorem is (6.7) THEOREM. For compact M, the following properties of /EDiff(Jlf) are generic: (a) Every periodic point is hyperbolic. (b) For each pair of periodic points p, qEM, W'(p) and Wu(q) have transversal intersection. This is proved in [55] and [114]. In [86] there is a polished version, which also proves the noncompact case. In [2] there is an account done in the general framework of transversality theory. Note that if/satisfies (6.7)(a) then for each m£Z+, the number of periodic points of period m is finite, i.e., Nm<». The other main approximation theorem is related to Pugh's C1 solution of the problem of the "closing lemma" [91 ]. This can be stated as follows. (6.8) THEOREM (PUGH). Suppose /EDiff(Af), and x£M is recur­ rent in the sense that : Z—+M defined by #(m) =/"(») is not a homeomorphistn onto its image. Then there is a C1 approximation f off such that x becomes a periodic point off. Pugh uses the methods of (6.8) to prove the following [92]. (6.9) THEOREM. For compact M, the property (6.1)(b) is generic in the C1 sense. In other words suppose we put the C1 topology on Diff (if) and let G be the set of fE Diff (M) with the property that the periodic points are dense in Q(/). Then G is a Baire set. Unfortunately the C' analogues for r > 1 of (6.8) and (6.9) are yet unproved?2 Furthermore, in my opinion, it is important tofind"con­ ceptual" proofs of Pugh's important results. We end by stating the three basic problems raised here. (6.10) PROBLEMS, (a) Approximation problem:23 For compact M, approximate (C, large r preferably) any / E Diff (M) by f satisfying Axiom A and Axiom B. In this perhaps the most important property is Axiom A(a). (b) Find all (in some sense) possible indecomposable hyperbolic sets of nonwandering points up to topological conjugacy.24 This in­ cludes, as a special case, find all Anosov diffeomorphisms (such that Q=Af). (c) Find the possible ways of fitting the W'(fi,) together to define / : M—*M as in (6.3) and (6.5). This is sort of a generalized Morse theory type problem and essentially problem (6.6)a.2s

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It could happen that (6.10) (a) has a negative answer. This would add difficulty to the conjugacy problem! One would proceed by adding the corresponding counterexamples to those of this paper and enlarge the unifying framework. On the other hand an affirmative answer to (6.10) (a) would imply that this survey gives the basic framework to the conjugacy problem and that answering (6.10)(b), (c) would be filling in the body. 1.7. Our goal in this section is to give at least a full sketch of proofs of the spectral decomposition theorem (6.2) and its corollary. In doing so we state a general stable manifold theorem and use it to show the existence of canonical coordinates on our hyperbolic sets. We first give some preliminary lemmas. (7.1) LEMMA, (a) / / p is a periodic point of/EDiff (M), and U is an open set in M such that UC\ W'ip) 9*0, then the closure of Um>o/"(C/) (b) Furthermore, if q is a second periodic point and Wu{p)r\W*{q) contains a point of transversal intersection, then lL>of"( U) r^W'(q)^0. PROOF. Note first that by replacing / by a power of / , we may as well assume p and q are fixed points to begin with. Since / : W*(p) —*W"(p) is an expansion, it follows that if a neighborhood of p in W"(p) is in the closure of U„>o fn(U), then so is all of Wu(p). Thus we see that (a) is transformed into a local problem about a neighborhood N of p by replacing U by fk(U)r\N for some large n. In case / is linear in some chart about p the conclusion of (a) is easily checked directly, and finally the general case can be reduced to this one by an appeal to Hartman's (and Grobman's [74]) theorem [39], which gives a local topological equivalence to the linear case. The second part of (7.1) can be proved with little trouble by using the linear Lemma 5.2 of [115] or one can use again Hartman's the­ orem and a topological intersection argument. The reduction to the local case is again clear. A stronger lemma than this, the "X-lemma" is in [82].

(7.2) LEMMA. Let f: M-*M be a diffeomorphism with hyperbolic periodic points pit t = 0, • • • , n such that pa=p». Suppose for each * = 0, • • • , n — 1, XiE.W"(pt)r\W(Pi+i) is a point of transversal inter­ section. Then each xf is nonwandering. PROOF. Let x( for some i be as in (7.2) and U be a neighborhood of x{ in M. Then (U m >o/"(£/))nW"(fr);*0 every j using (7.1) induc­ tively. By (7.1)(a) Closure U»>o/*(C/)DW^"(/>,). This shows x, is nonwandering.

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We next come to the general stable manifold theory which we put into the following form. (7.3) GENERALIZED STABLE MANIFOLD THEOREM.26 Suppose ACAf is a hyperbolic set o//G Diff(Af) (that is, A is compact, invariant with the usual splitting of TL(M), see §1.6) with some metric on M. Then for each xGA, there is an injective immersion J\ = J»: W,(x)—*M with the following properties: (a) * G / . ( W'(x)), andyEMW'(x)) if and only ifd(f*(,x),f*(y))-+0 as m—*«>. (b) f(J.(W-(x))=JKmW(f(x)). Let J.(W'(x)) = W(x) now. (c) lWW(*))={;yGAf|/-(y)-*A, m—► <» j . (d) For x, yGA, W'(x) and W'(y) either coincide or are disjoint. (e) The tangent space of W'(x) at y is £J for each yGA (here £J is part of the data of the hyperbolic splitting). (f) W'(x) and W*(y) are C1 close on compact sets for x, yGA close. For A a point this is the stable manifold theorem for a fixed point, §1.2. In (7.3), W(x) for xGA is called the stable manifold of x. The unstable manifold W*(x) is defined as the stable manifold of/ -1 at x. We will try now to give the history and background of (7.3). Of course it all starts with A a point from Poincare, Perron, etc. as in §1.2. Anosov, using the basic work of Perron, proved (7.3) in the case A is all of M. This is the way he proved the structural stability in §1.3. Seeing the need for a more general version of stable manifold theory, because of Axiom A, I asked I. Kupka if he could give such a proof. In substance at least, he proved the above (7.3). All of the proofs in stable manifold theory, however, have been unsatisfactory from a conceptual point of view. On the other hand, at this writing it appears that the situation has been remedied by M. Hirsch. He seems to have a fully satisfactory proof of the above (7.3). Added in proof. C. Pugh has a good proof of (7.3). From the stable manifold theory we now construct what we call canonical coordinates on Q(f) where /GDiff(Af) satisfies Axiom A. If W*(x) is as in (7.3), then we will denote an e neighborhood of * in W'(x) in the intrinsic (metric) topology by W'(x, «). Then let J^*(x, e) be the set W(x, t)r\Q etc. (7.4) THEOREM (EXISTENCE OF CANONICAL COORDINATES).27 Sup­ pose /GDiff(M) satisfies Axiom A and that xG8=-Q(/). Then there is «>0, independent of x, and a canonical map I,: V—>M where V is a neighborhood of xXx in Wu(x)xV^'(x), which is a homeomorphism of V onto a neighborhood of xintt. On tPM(x, e) Xx, I. is the inclusion Jland on xxW'(x, e), I. is the inclusion J\. The map I. is defined at (P, q)E.VC^'(x, t)xW'(x, e) as the unique intersection of W'(p, e) and W*(q, e) in M.

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PROOF. This follows directly from a systematic application of (7.3) and (7.2). The map is well defined into M by (7.3). The image of I, is in ft by (7.2) and the fact that the periodic points are dense in ft. In a similar way, one checks that a neighborhood of * in ft is in the image of Ix. The injectivity of Ix is a consequence of the stable mani­ fold theorem, that the W'(p) for different p, either coincide or are disjoint. Moving toward the proof of (6.2) we give first the following lemma:

(7.5) LEMMA. Suppose /£Diff(Af) satisfies Axiom A, ft = ft(/) the nonwandering points. Given x&l, suppose N is a neighborhood of x in ft with the local product structure of (7.4). / / U is any nonempty open subset of N, then \Jm*o fmU and Umso fm(U) each contain a dense subset of N. PROOF. It is sufficient to consider just one of the two cases. Let q be a periodic point of U with stable and unstable manifolds W*(q) and W*(q). There exist such q since the periodic points are dense in ft by Axiom A. Now let p be an arbitrary periodic point of N. There are points of transversal intersection x£.Wu(p)r\W(q), JC'GW"(}) (~\W(p), with x, x' in ft. Then xQ\JmiofmU, and so p is in the closure of Vmsof™!/. Since the periodic points are dense in N, this proves (7.5). We now prove (6.2). For x £ft, let N = N(x) be given as in the previous lemma and define ft, = Closure Umezfm(N). From the previous lemma it follows that ft, does not depend on any choices. In fact, it follows equally well from (7.5) that for x, y£ft, either ft, and ft» coincide or are disjoint. Furthermore the union ft = U*ftx is actually a finite union and all of the properties of (6.2) are checked very easily now using the previous lemma. Note that one obtains directly that any open set in ft,- has a dense orbit (i.e., U^"(f/) is dense). From Birkhoff [15] one then ob­ tains topological transitivity. We show how (6.3) follows from (6.2). For each x^M, m—»«> :/"**—»ft from the definition of nonwander­ ing. Given x £ M , we claim there is a unique i such that fmx—»8< as m—► » . For each * = 1, • • • , k, choose open sets F„ I/,- such that VtDUiDQi, Vi disjoint and f-lUJJfUtCVt. Now, given xEM, suppose there exist k, I such that QtAlim,, , „ / " * ^ 0 , ft|^lim»H.„/*,* ?*0, k?*l. Then there exist for each j = l, 2, • • • positive integers f»y, lj such that f*>x£UK, /"*+'/£l/j where my<my+/y<»n, + i for e a c h / Then there exists nit f»y
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Note finally that the following proposition is clear from the previ­ ous material in this section. (7.6) PROPOSITION. Let 12,- be as in (6.2). (a) Then for any *£12,-, W*(x) and Wu(x) each contain a dense set o/I2,-. (b) In particular iff: M—*M is an Anosov diffeomorphism of a com­ pact manifold with Q = M, then every stable manifold is dense in M. (c) W(Qi)nW(Qi)=Qi. One obtains (a) from the topological transitivity and the local product structure on $2,-. (b) follows from (a). One checks (c) by first showing that W"(Qi)r\W(QJCQ using (7.2) and the fact that the periodic points are dense in 12. Then apply (6.2). 1.8. The goal of this section is to sketch the proofs of Theorems (6.5) and (6.6). We begin by introducing the generally useful notion of a filtration of a diffeomorphism. A filtration then of f£Diff(M), M compact, is a sequence of closed submanifolds, M=MoDAfOAfO • • • ~DMk = 0 where each M is an open subset together with its smooth boundary and/(M,)Onterior of M,-. (8.1) PROPOSITION.28 If {Mi] is afiltrationfor /GDiff(Af), then it is also afiltrationfor a C approximation off. Furthermore Sl(f)f^dMi = 0 for each i. This is easily checked. If {M,} is a filtration for/£Diff(Af) then we can decompose J2(/)=QiU • • • KJQk_u $24 compact invariant, by defining $2,=°.r\(.W, —Af,_i). We will call this the Q-decomposition of the filtration. We assume now t h a t / satisfies Axioms A and B, with 12,-, W'(B<)f etc. as in §1.6. (8.2) PROPOSITION. / / W«(Q,)rW(Q,)5*0, then for any pEQif g£fyi W"(p) and W'(q) have a point of transversal intersection. PROOF. This is a consequence of Axiom B and §1.7. Then by (7.1) we obtain (8.3) COROLLARY. If W*($t)r\W($i)*0, then W»(J2y)CClosure W"(Qi).

(8.4) PROPOSITION. / / W"(fli)r\W(Qi+i)^0, t = 0, • • • , m - l and fi0=i2«, then all the 12, coincide. u PROOF. Let periodic points p,ES2< for each *. Then by (8.2), W (pt) and W'(pi+i) have a point of transversal intersection g< for each i. Apply (7.2) to see that each g<EG. But then g,£fl,nQ, + i, so indeed the 12, coincide. From (8.3) and (8.4) follows

701

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(Norcmbar

The relation ^ defined in §1.6 is a partial order­

REMARK. One could define ^ in these alternate ways as long as Axiom B was modified accordingly and the whole theory would be the same. Alternate way 1. fl,gfly if Clos W«(fly)nClos W*(fl,)5*0. Alternate way 2.fl,-^Qyif for any pair of neighborhoods £/< of 0
(8.6) PROPOSITION. Suppose f£Diff(M) satisfies Axioms A and B, withfl«>o/"(W^o)i tn9 some large integer and W» is a small neighbor­ hood offly.One may think of W as a neighborhood of W"(fly). By con­ tinuing in this way one obtains the desired nitrations. Conversations with M. Shub were very useful in the following. To prove fl-stability29 for /£Diff(Af) satisfying Axioms A and B, one generalizes the procedure of Moser in the Appendix written by Mather. Instead of the map A defined there one uses the map

B: BiS(M) X C°(A, M) -> C°(A, M) defined by B(g, A)=gA/-1. Here C°(A, M) is the space of continuous maps of A into M with the uniform topology with A =fl, an indecom­ posable piece of fl(/). C°(A, M) is a manifold and B has its second partial derivative continuous in both variables. A version of the implicit function theorem yields a continuous map h: A A

* M such that h

>M

if. h it A

► M commutes

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(providing g is close enough t o / in Cl). There remains to complete the proof of S-stability of /, two things. First, h is 1-1. Here the Moser argument does not work. The proof goes by (8.7) PROPOSITION. / : A—*A is expansive. This means there is «>0 such that for any x, y £A, x^y, there is n£Z with d(Ji'x,fny) >«. The proof of this proposition follows from the fact that A is a hyper­ bolic set for/. Then the fact that h makes the above diagram commute and is close to the identity leads to the injectivity of h. The second point to check is that when h is denned on each $},• as above, A(Q(/))=0(g). Since the periodic points are dense in fl(/) it follows that h(Q(f))C.Q(g). Furthermore, by (8.6) we may assume that fl(g)Csmall neighborhoods of Q(. Thus the proof of Q-stability is reduced to the study of what happens in a small neighborhood of Q<. The stable manifold analysis finally takes care of this last point. To finish our program, we must show that if g has been chosen close enough to/, then g also satisfies Axioms A and B. AH of this is a conse­ quence of the above provided ti(g) = h(Q(f)) has a hyperbolic struc­ ture for g. This proceeds by showing bounded hyperbolic linear maps on Banach spaces are open using the spectral theory at the end of [96], and then using the values of sections of the Banach space split­ ting to reconstruct the vector bundle splitting. 1.9. On basic sets of diffeomorphisms. This section is devoted to the problem of finding all the possible Q, that could occur in the spec­ tral decomposition theorem (6.2) for diffeomorphisms of compact manifolds satisfying Axiom A. In other words we discuss what is known about Problem (6.10)(b). Expanding on this define a basic set of/£Diff(Jlf) to be one of the H, of (6.2) where/satisfies Axiom A. (9.1) PROBLEM. Find all basic sets up to topological conjugacy.30 Do they always have a rational zeta function?3'Are they all locally the product of a Cantor set and a manifold?32 Can they be given some type of algebraic structure? One can consider a possibly more general, but localized picture by considering / : U—*M with U an open set of M, f a diffeomorphism onto its image with AC.U satisfying (9.2) (a) A is compact and /(A) = A, (b) A is a hyperbolic set for/ (see §1.6), (c) the periodic points of / are dense in A, (d) / is topologically transitive on A,

(e) rw/w-A. Since the basic sets have neighborhoods U which satisfy (9.2) we may consider

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(9.3) PROBLEM. Find all A satisfying (9.2). There is a construction which allows one to replace / : U—*M by g: U—*V in (9.2). On UXZ say points (x, m) and (x1, n) are equivalent if x' =gn~m(x). Then the quotient space V is a manifold (not neces­ sarily Hausdorff) and one has a diffeomorphism g: V—*V induced by (x, m)—►(*, m + 1 ) . Define AQff as the image of (A, 0) under the pro­ jection x : UXZ-*U. ■We will say that A in (9.2) is an attractor if U can be chosen so that f)m>of*(U)=A.. Then when A is the basic set of a diffeomorphism satisfying Axioms A and B, an attractor corresponds to a vertex lying at an extreme point of the diagram of/. A special case of (9.1) and (9.3) is to find the attractors. 3 3 Note that no symbolic flow of §1.5 can be an attractor, but that every Anosov diffeomorphism with 12 = M is already an attractor. We will give an outline of all the ways we know of constructing basic sets; then we will go into more detail. First consider these four groups of basic sets: (9.4) (a) Group 0. These are characterized by dimension A = 0 . (b) Group A. This is Anosov case with Q = M. (c) Group DE. These are derived from expanding maps and will be described subsequently. (d) Group DA. These are derived from Anosov diffeomorphisms and will also be described subsequently. Furthermore one may take finite products of any of these to obtain other basic sets (see §1.10). Group 0 is discussed first. This includes the finite A (periodic orbits) and the shift automorphisms AN of §1.5. It seems likely to me that every basic set in group 0 is topologically conjugate to some closed invariant subset of Ay. Call AC.A.N a subshift if A is closed, invariant, and the periodic points are dense in A. One can ask generally to what extent the subshifts occur as basic sets. 34 The following construction may shed some light on the above im­ bedding problem. Suppose A is a basic set of dimension 0 relative to / : U—*M, Ui\J ■ • • \JUN a disjoint union of local product neighbor­ hoods of (7.4) which cover A (such Ui can always be found). Then let £:AAT—»Ajy be the shift automorphism on the JV-symbols Ui, • • • , Uy and define a : A—*AN by a(x)(m) = Ui where x £ A , « £ Z , and f"(x) £ Ui. Then it is easily checked that a is continuous and equivariant. Can the £/,- be chosen so that a is injective? The following is a non trivial example of a subshift as a basic set. We describe it in the following figure as a diffeomorphism of a 2-disk

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into itself which can be extended to a diffeomorphism 5*- »S* by adding an expanding disk to the original.

0

D

E

y

I

p H

The construction follows in the pattern of those in §1.5. The diagram is simply given by Q where the top vertex is an expanding fixed point and the bottom vertex corresponds to the periodic orbit consisting of Pi, pt and pi. The reader can check that the middle vertex of the dia­ gram corresponds to a subshift A of the shift on five symbols. In fact if A| is the shift space on the symbols a, 0, y, 6, p corresponding to the indicated columns in the figure, then A consists of bi-infinite sequences which do not carry any of the following combinations fiy, fib, fip, act, a/9, yi, 77, 7P. &*, ifi, pa, p0. Generally speaking, relative to a shift automorphism a block is a finite sequence of symbols, e.g., /3, etc. in the previous sentence. A subshift is said to be of finite type if it is of the form, all sequences which do not contain a certain finite set of blocks. Thus the above A is of finite type. Related to the previous problems on basic sets of dimension 0 are the following theorems: (9.5) THEOREM (O. LANFORD). Every subshift of finite type has a rational xeta function. (9.6) THEOREM (R. zeta functions.

BOWEN).

There exist subshifts with irrational

In fact Lanford has improved Bowen's theorem to show that most subshifts have irrational zeta functions. We don't go beyond the discussion of §1.3 on the Anosov case except to remark that it seems probable that if a basic set is a submanifold then the restriction of the diffeomorphism is conjugate to an Anosov diffeomorphism.

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J. Moser has shown me an example of a basic set which is a submanifold but not a C1 submanifold. For the DE group we use the examples of Shub [108] of expanding endomorphisms of compact manifolds—see §1.10. For each expand­ ing endomorphism, we will construct a basic set which is an attractor. This goes as follows. Suppose then / : M—*M is an expanding endomorphism of a com­ pact manifold. Let D be the unit disk of dimension one larger than the dimension of M, with M imbedded in DXM as OXM. Let X satisfy 0<X<1 and define &: DXM-*DXM by gx(x, y) = (Xx, y). Next let 4>: 0XM-+DXM be a C1 approximation of the map OXM —>DXM, (0, y)—*(0,f(y)) such that is an embedding. This is pos­ sible by dimensional reasons (the Whitney imbedding theorem). Let T be a tubular neighborhood of (M) with fibers being the various components of T(~\(DXy), yE.M. Now extend 4> to $: DXM—*T in a fiber preserving way so that ^ is even a diffeomorphism. Our de­ sired map DXM—*DXM is then the composition ^g\=h for X small enough. It can be checked that for sufficiently small X, the set A = n„>0 hm(DXM) has a hyperbolic structure and is in fact a basic set. It is locally the product of a Cantor set and a manifold whose dimension is that of M. The following figure gives DXM and its image under h when the starting point is the expanding endomorphism of Si—*Sl defined by

Finally we show how the DA group (9.4d) goes by giving the first case using an extended type of surgery on the Anosov diffeomorphism of the 2-torus.3* One changes the toral diffeomorphism on a small "square" neigh­ borhood Q of the fixed point corresponding to (0, 0) in R*. Initially we have the square Q=ABCD linearly mapped into A'B'CD1 as in the following figure.

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A

B

V

A T M

iy

/

a D

Without changing the diffeomorphism outside a neighborhood of the boundary of Q, we can change / on Q so that we have three fixed points in Q as illustrated in the following figure.

Now T* can be written as the union of a single two dimensional stable manifold of the fixed point x, W'(x) and a one dimensional basic A. We leave the (many) details to the reader. One can apply this construction to any Anosov diffeomorphism. As this was written, we received a very interesting manuscript of R. Williams [127] on 1-dimensional basic sets, which certainly ap­ pears to extend some of the above results. 1.10. Final remarks on conjugacy problems. We cover briefly a number of final miscellaneous points related to the diffeomorphism problem of part I. The first question is: what role do products play?

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(10.1) PROPOSITION. Let fl,- be the set of nonwandering points of /,GDiff(Mj), i = \, 2. Then the set of nonwandering points of the product /iX/jEDiff(ilfiXMi) is contained in QiXflj. Furthermore if the pe­ riodic points are dense in & and i2», then Q = i2iXQ». This is checked easily from the definitions. (10.2) PROPOSITION. 7// < GDiff(Af,), * = 1 , 2 satisfying (2.2), then so does the product fiXft€.Difi(MiXM2). This follows from (10.1) and the fact that W'{p, q) = W'(p) X W'(q). Furthermore one sees from the definitions that (10.3) PROPOSITION. The product of two Anosov diffeomorpkisms an Anosov diffeomorphism.

is

The last two propositions are essentially contained in (equally easily checked) (10.4) PROPOSITION. i / / t £ D i f f ( A f , ) . * = 1, 2, both satisfy Axioms A and B, then so does the product/IX/J. Furthermore so does f™, m(E.Z, Thus if /i£Diff(Afi) is an Anosov diffeomorphism and if /»GDiff(M 2 ) satisfies (2.2), then the product/iX/» is Q-stable (6.5). This product, however, is not structurally stable. Moreover, there is an open set U, in general, in Diff(AfiX-Mj) n e a r / w i t h the property that U contains no structurally stable diffeomorphisms. This is described in [116], It is the example mentioned in §1.1 to show that one had to weaken the concept of structural stability to get a success­ ful theory. There is also an exposition of this fact in [l 1 ] and a further variant in [87]. We now define a modification of diffeomorphisms related to the notion of "surgery" in differential topology. Suppose/GDiff(M) has the property that there is a compact submanifold with boundary Mu dim M=dim Mi, such that f(Mi) is contained in the interior of Mi. Then it follows that J2 = 12(f) is equal to (interior Afi)nfiU(Af—Afi)nfi=QiUflj where each fl, is compact and invariant. Furthermore, a similar decomposition can be done even for any sufficiently good C approximation of/. This is a special case of the filtrations of §1.8. For surgery, in addition to the above / suppose that g: V—> V is a diffeomorphism of a compact manifold with boundary into its inte­ rior. Suppose further that there is a diffeomorphism h: Cl(Mi—/(Af t )) —»C1(F—g{V)) with gh = hf. An isotopy condition o n / , g is sufficient

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to guarantee the existence of h at least in case Closure (V—f(V)) » 1 X V. Then one may replace Mi by V and redefine / on Afi by g on V to define/': M'—*M', M' the modified manifold. This turns out to be a useful construction. One checks immediately that f>» =* tt o f, o ff l where J is / restricted to Jlf<. A topological version of the ergodic theoretic concept, entropy, has been denned in [3]. In this paper, the authors showed that this topo­ logical entropy is positive for some of the examples we described in §§1.3 and 5. The following problem seems natural. (10.5) PROBLEM. If fi, is a basic set of a diffeomorphism satisfying Axiom A (as described in the spectral decomposition theorem (6.2)), is the topological entropy of Q,- positive?37 As J. Palis pointed out to me, any Anosov diffeomorphism will satisfy Axioms A and B. It is possible that applying some of the sub­ sequent theorems, one could obtain an attack on the problem: For an Anosov diffeomorphism f£Ditt(M), M compact, must Q(f) = Ml (See 3.4). Up to now we have been investigating the dynamical system generated by a single /£Diff(Jlf). One can generalize this situation to a differentiable map (or endomorphism) / : M—*M (without neces­ sarily having an inverse). This/ is not the generator of a group acting on M, but a semigroup Z+ acting on M. M. Shub [108] has studied this problem and found that some of the previous results extend to cover this case and some new features are found here. We state some of these now. The simplest new problem coming up in this context is the endo­ morphism of the complex numbers of absolute value one, f'.^—^S1 defined by f(z) =«", n£Z, n> 1. Is/structurally stable? As for diffeomorphisms, an endomorphism f:M-*M is structurally stable if C1 perturbations are conjugate to/by a homeomorphism. Shub gives an affirmative answer in the following more general proposition [l08]. (10.6) PROPOSITION. Suppose f'.S1—*^ is C1 with derivative every­ where > 1 . Then f is conjugate by a homeomorphism to z—*tn where n=degree f. The general questions on endomorphisms of S1 are not yet very well understood. On the other hand Shub has found a satisfactory generalization of (10.6) as follows. Say that an endomorphism / : M—*M of a complete Riemannian manifold is expanding if for each vETm(M), ||D/"(x)(r)|| £eX"||»||, m£Z+, c>0, X> 1 independent of v, x. Examples of expanding endo­ morphisms are, of course, the circle map *—**" as well as various products of these on tori.

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(10.7) THEOREM (SHUB [108]). Any two homotopic expanding endomorphisms of a compact Riemannian manifold are topologicaUy conjugate. (10.8) COROLLARY. Any expanding endomorphism of a compact Riemannian manifold is structurally stable. In view of (10.7), the following becomes a reasonable problem. (10.9) PROBLEM. Find all expanding endomorphisms of manifolds3* (up to conjugacy). Also, is (10.7) true for Anosov diffeomorphisms of compact manifolds? Shub proves for expanding endomorphisms that the manifold is covered by Euclidean space, and has produced, besides those on tori, examples on the Klein bottle and nilmanifolds. Presumably, eventually a systematic approach will include the Anosov diffeomorphisms and Shub expanding endomorphisms.39 The unifying definition is, in fact, obvious. For hyperbolic fixed points of an endomorphism / : M—+M, Shub defines stable and unstable manifolds, generalizing those for a diffeomorphism. In this case, however, W' is no longer the image of a cell, but can be any manifold (i.e., map M into a point Po€zM and take any small perturbation. Then the stable manifold of the fixed point will be M). On the other hand W" is the image of a cell, but not under an immersion or a 1-1 map. Shub generalizes the approximation theorem (6.7) to endo­ morphisms. Previously, holomorphic endomorphisms of the Riemann sphere had been studied by G.Julia ([50], his "prize memoir").40 Stein and Ulam [119] have made a study of certain polynomial endomorphisms of the plane using computing machines. An extremely interesting problem is the study of maps of finite dimensional manifolds into Dift(M). What are generic properties of such maps? This is called bifurcation theory.41 The most important work on this subject is that of J. Sotomayor [118]. He considers maps of an interval into the space of flows on 2-manifolds, and obtains a pretty complete picture in this case.

A P P E N D I X TO PART I: ANOSOV DIFFEOMORPHISMS BY JOHN MATHER

In this Appendix, I give an exposition of Moser's proof that Anosov diffeomorphisms are structurally stable. (See Theorem 3.3 of §1.3.)

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The main novelty in my presentation is the use of the language of manifolds of mappings, which seems to result in conceptual simpli­ fication. I would like to thank R. Abraham for suggesting that Moser's proof might be simply expressed in the language of mani­ folds of mappings. We let M be a compact C manifold and / : M—*M a C1 diffeomorphism. We let D denote the topological space of diffeomorphisms of M into itself with the C1 topology, H the topological space of homeomorphisms of M into itself with the C topology (compact open topology), and C the C" Banach manifold of continuous mappings of M into itself, where the topology is the C° topology and the manifold structure is denned in a manner similar to that by which the mani­ fold structures on sets of mappings are denned in [l] (of Appendix). THEOREM 1 (ANOSOV). / / / is an Anosov diffeomorphism, then f is structurally stable. More precisely, there exists a neighborhood U of the identity of M, idj*, in H, a neighborhood V of f in D and a continuous mapping g—*h(g) of V into U such that for all g £ F , h=h(g) is the unique solution in U of the equation

hg - fh. If £ is a vector bundle over M, we let T(E) denote the Banachable 2?-vector space of continuous sections of E over M, with the C* topology. If, further, x£Af, we let E„ denote the fiber of E over x. We let f+:T(TM)—*T(TM) be the continuous linear mapping given by/*(«) =Dfozof~1. We will consider T(TM) as a Banach space, with any norm which induces its topology. LEMMA

1. Iff is an Anosov diffeomorphism, then f*—id is an iso­

morphism. REMARK. The converse is also true, but PROOF. By the hypothesis, there exists

will not be proved here. a splitting of T(M) into a continuous Whitney sum T(M)=E'-\-Eu, invariant under Df, such that Df: E'—*E* is contracting and Df: £"—»£« is expanding. Let fo=/*|r(-E') and/„=/»|r(JE-). Then there exists C>0, 0<X<1 such that for all »»£Z + ,

||£|| < ex", ||/ni < a". It follows that/o—id and/; 1 —id are automorphisms of r ( £ ' ) and T(E'), resp. In fact,

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i-9

(fj - id)-1 = - z/; y . Hence,/.—id = —/.(/i 1 —id) is an automorphism. The lemma follows immediately. Let A: DXCXD-^CXC be given by A(gu h, gt) = (hogu gtoh). LEMMA 2. A is once differentiable in its second variable and its "partial derivative,'' DtA:

DXTCX

D-*TCXTC

is continuous in all variables. Moreover poL is an isomorphism, where L - DtA | (/ X (TC)id Xf):

(TQu -> (TC), X (TC),

and p denotes the projection of (TC),X (TC)/ on (TC)/ X (TC)//'diagonal. PROOF. The first sentence follows from the methods of [l ]. Also by the methods of [ l ] , we may make the identifications ((TC)u = T(TM) and (TQ/ = T(f*TM). With respect to these identifications L is given by

z—> (zof,

Dfoz).

Let 5: (r(f»rAf)Xr(/*7\&0)/diagonal-»r(rAf) be the isomorphism induced by (s, <)—»to/_1—5o/-1. Then S o p o L = /» — id. Hence, the second sentence follows from Lemma 1, completing the proof of Lemma 2. By Lemma 2 and a suitable version of the implicit function theo­ rem, there exists a neighborhood V% of / in D and a neighborhood Ui of id* in C such that for all gi, gtEV there exists a unique h"u(gi, gi)E.Ui such that A(lu *, i*) G diagonal i.e., (1)

h o gi = gt o h

and such that (gi, gj)—*u(gi, gi) is continuous. Let Ui be a neighborhood of id*i in Ui such that for all hi, AjG U\, hiohtE. Ui and let V be a neighborhood of / in Vi such that for all gi, giE V, u(gi, gt) E Ut. For all gE V, set h(g) =u(g,f) and h~(g) = « ( / , g).

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Setting h=h(g), k~=h~(g), we have hg=fh and h~f=gh~. Hence h~hg = h~fh=ghrh and hh~f=hgh~=fhhr. Since (1) has a unique solution A£t/i for g\, g i £ 7 i , it follows that hh-=h~h = idtt- Hence h is a homeomorphism, so the theorem follows with Z7=* U%f\H. REFERENCE 1. R. Abraham, Lectures of Smale on differential topology, Lecture notes, Columbia University, New York, 1962.

PART II. FLOWS

II.l. Introduction to flows. We shift now our survey to the case of the group G = R, the real numbers, acting on a manifold M, which for simplicity we will assume compact most of the time. Thus we are studying a 1-parameter group of diffeomorphisms, , simply a flow. A flow t: M—*M defines (or generates) a tangent vector field on M; i.e., for each ac£Mdefine X(x)£TB(M) by (1.1)

<**,(*)/*]_. = X(x).

Thus X(x) is a tangent vector at x on M and i(x) is the solution of the ordinary differential equation (1.1) with initial condition 0o(x) =*• Then the orbit (of 0) through x, t—*t>t(x) coincides with the solution of the first order, autonomous (i.e., X{x) doesn't depend on t), ordinary differential equation (1.1). Conversely, given an ordinary differential equation, simple meth­ ods reduce it to the first order autonomous case and thus one obtains the situation in (1.1) with X(x) given. The fundamental existence theorems of ordinary differential equations (see [25], [56]) yield a solution 0«(x) such that 0 o (x)=*, at least locally, i.e., for M <e. Furthermore these local solutions may be pieced together (see [56]) and frequently this leads to a flow on M. Certainly if M is compact, every (smooth of course) tangent vector field defines a unique flow in this way. In the noncompact case one may change the vector field by a scalar factor to obtain one which defines a global flow. We will consider here only the case where 4>t: M—*M is defined for all t, or an action of R on M, i.e., a flow. Most of Part II is the carrying over of Part I to this slightly more complicated case. We will emphasize some of the special features and new interesting problems encountered in this 1-parameter case. There are three possible types of orbits of a flow t: M-+M. First

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* is fixed point of the flow if <j>t(x)=x all J£2?. A fixed point * can also be characterized as a zero of the vector field defined by . Secondly a closed orbit of ,: M—*M is the orbit through some x with <j>t(x) — x some t?*0. Usually a closed orbit is taken to mean exclusion of the fixed point case so there is a minimum period / 0 >0 such that #«.(*)=*• Finally if t—Hf>t(x) is injective, then the orbit through x is not one of the above types and could be called an ordinary orbit. In topologizing actions of R we assume M is compact. Then the flows as we saw above correspond precisely to tangent vector fields on M. The Cr,r>0, vector fields on M form a linear space and with a O norm, r<«>, a Banach space which we denote by x(M). A generic property of flows will be a property true for a Baire set in x(Af). The most obvious generic property is that the set of zeros of XEx(M) is finite [114]. Proceeding as in §1.1 we look for a suitable equivalence between two flows , and $t on M. A conjugacy between t and ^« is a homeomorphism h: M—+M such that h4>t{x) =4>t(hx). Such an equivalence relation preserves the minimum period of a periodic orbit and thus a conjugacy class will not in general be invariant under perturbation. This implies the need of a weaker notion of equivalence. We say that flows i, \f/t are topologicaUy equivalent if there is a homeomorphism of M sending orbits of i into orbits of yf/t- If perturbations in x(M) do not change the topological equivalence class of X£x(-M). then X is called structurally stable. This concept was introduced in 1937 [6] by Andronov and Pontrjagin for ordinary differential equations on the 2-dimensional disk. On compact 2-dimensional manifolds, the struc­ turally stable flows form a dense open set, simply characterized (Peixoto [84], see also §11.2). However in every dimension higher than three there exist compact manifolds on which the structurally stable flows are not dense [116] (see also [87]). Parallel to Part I this leads to a weakening of topological equivalence as follows. For a flow t on M, G. D. Birkhoff [15] has defined x£M to be a wandering point if there is some neighborhood U of x in M with (U|«i>«,<£#(£/))rW=.0 for some * 0 >0. The nonwandering points (those which are not wandering) form a closed invariant subset of M denoted by ft=2(^,). We will say that flows 4>i, tf>t are topologicaUy equivalent on 8 if there is an orbit preserv­ ing homeomorphism h: Q(i)-*Q(if/t). Then , is Q-stable if sufficiently small perturbations (measured in terms of the corresponding X£x(M) of course) are topologicaUy equivalent on Q to 4>t. It is an important problem to discover whether Q-stable flows are dense in X(A0-

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We end this section by giving a direct relation between the flows discussed here and the diffeomorphism questions of Part I, [114]. A compact submanifold 2 of codimension one of a compact mani­ fold M is called a cross-section for a flow fa on Af if 2 intersects every orbit, has transversal intersection with the flow and whenever x £ 2 , <(*)£2 for some f>0. Then fa induces a diffeomorphism / : 2—»2 byf(x)=fat(x) where tQ is the first *>0 with 0«(x)£2 (see [114] for more details). The topological equivalence class of fa is determined by the topological conjugacy class of/. Orbits of/are in a natural 1-1 correspondence with those of fa by {/*(x)|i»£Z} —»{^i(x)|/£i?}, each x £ 2 . Compact orbits are preserved under this correspondence; thus periodic points of / correspond to closed orbits of fa. There can be no fixed points of fa when there is a cross-section. Cross-sections were used by Poincare' and Birkhoff (see, e.g., [19]).42 There is a converse construction of some importance. Given a dif­ feomorphism / of a (compact) manifold 2 we will construct a flow, canonically, on a manifold Mo of one dimension higher, called the suspension off. This goes as follows. Let a: 2XR—*2XR be defined by a(x, «) = (/(*), K + 1 ) . Then [a"} = Z operates freely on 2XR and the orbit space is a manifold M«. Furthermore the flow fa: 2X-R —*2XR defined by ^t(x, u) = (x, u+t) induces a flow fa on Mt which is our suspension of /. Clearly M<, will have a cross-section 2 0 =T(2X0)CAf" where x: 2XJ?—*Afo is the quotient map. It is easy to check that the associated diffeomorphism of (fa, 2 0 ), /<>: 2#—>2o is differentiably conjugate to our initial/: 2—*2. Furthermore if an arbi­ trary flow fa: M—*L has a cross-section / : 2—»2 whose suspension is !: M9—*Mo, then t and l are equivalent by an orbit preserving homeomorphism.43 This notion of suspension is useful because it allows one immedi­ ately to transfer all the examples in Part I, i.e., the diffeomorphisms of §1.2, Anosov diffeomorphisms as well as those of §§1.5 and 1.6, to examples of flows. From the above remarks, all the stability prop­ erties of the diffeomorphism examples are kept by the suspended flows. 11.2. The simplest examples of O-stable flows. We will say that a fixed point x of the flow <j>t- M—*M is hyperbolic if x is a hyperbolic fixed point of the diffeomorphism fa: M-+M. An alternate way of saying this is as follows: If x is a fixed point of the flow fa: M-*M, then the derivative Dfa(x): Tm(M)-*TM(M) defines a linear repre­ sentation of the real line and so can be written in the form D$t(x) =e,A where A is a linear endomorphism of T,(M). Then x is hyper-

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bolic if and only if none of the eigenvalues of A have real part equal to zero. (2.1) PROPOSITION. If x is a hyperbolic fixed point for theflowt, the stable manifold W'(x) of x relative to t for every t and is contracting for every t>0. Then W$(x) will be called the stable manifold of x for theflowt. One may apply 1.(2.1) to obtain properties of W'(x). Now suppose that x £ J f is in a closed orbit y of the flow <j>t: M—*M. There is a submanifold V of codimension one passing through x and transversal to y. Then V serves as a local version of the cross-section of §11.1, defining a local diffeomorphism/: U—*V, f(x)=x, where U is a neighborhood of x in V. We say that y is a hyperbolic closed orbit of t whenever a: is a hyperbolic fixed point of/. It is easily checked that this definition is independent of the choices x £ 7 and V (see [114]). The local stable manifold WLs(*>/) of x f o r / in U defines the stable manifold W'(y) of y by W'{y) = U teK <£<(Wfoc(*. /))• Then W'(y) is a 1-1 immersed cell bundle over S1 (either a cylinder i?*X5 1 , or a generalized Mobius band). For more details see [114]. The unstable manifolds of hyperbolic fixed points and closed orbits of 4>t are defined as the stable manifolds of \pt=~tFor the suspension of a toral diffeomorphism (§1.3), the closed orbits are hyperbolic and dense in M; but hyperbolic fixed points of any flow are necessarily isolated fixed points. We now describe the analogue of the diffeomorphisms of §1.2 as flows t • M—>M, M compact, which satisfy (2.2) (1) il(t) is the union of a finite number of fixed points Xi, ■ • • , xm and a finite number of closed orbits
n

o f ,.

(2) The xf, 7/ are all hyperbolic. (3) The stable manifolds and unstable manifolds of the xt, fa intersect each other only transversally. (2.3) THEOREM [109].** Suppose the flow 4>t: M-*M satisfies (2.2). Then (a) Each stable manifold Wt of the x, and y, is imbedded and M=UT_+r W't (disjoint union). (b) The closure of one Wt is the union of certain W,. Let W, ^ Wt ifW,is in the closure of Wt. Then g is a partial ordering. If W,£ Wt then dim TFJ^dim Wt. (c) One has the following Morse inequalities:

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i f 0 ^ Bt,

M, - Mt £ Bt - So,

Here B< is the *th betti number coefficients Z», and Jlf<=a<+6< +&,+i where o< is the number of Xj with dimension W'(XJ) =■«' and 6< is the number of 7/ with dimension W'(y/)=i+l. One can see that 11.(2.3) is quite analogous to 1.(2.3). There are a couple of special features in the present situation however. For ex­ ample (2.4) THEOREM (PEIXOTO [84]).45 If dim M=2, then theflow, satis­ fies (2.2) if and only if it is structurally stable. In this case the corresponding X£.x(M) form an open and dense set. ■This theorem gives a rough but quite good picture of flows on compact 2-manifolds. It solves the first basic problem for 2-dimensional flows. A gradient flow t: M—*M on a compact Riemannian manifold is defined by a O function / : M—*R in the following way. The deriva­ tive Df(x) of / at x is a cotangent vector at * and the Riemannian metric converts this into a tangent vector -X"(x) = (grad /)(x) at x. By the familiar procedure (JH-1) from X(x) we obtain our gradient flow t.

(2.5) THEOREM [109]. The flows on any given M satisfying (2.2) contain an open and dense subset of all gradient flows. Since every manifold possesses Riemannian metrics, we see that from (2.5) every manifold exhibits flows satisfying (2.2). Recall the existence of the diffeomorphisms of §1.2 was obtained in this way. Theorem (2.5) gives the bridge between the usual Morse theory for functions on manifolds and the work in this section. This even brings the subject here close to handlebody theory in differential topology and Poincar6 duality on a manifold (this is the duality between the stable and unstable manifolds of a gradient flow). See [95] for one definitely nongradient type example satisfying (2.2). See also [63], [97] for related papers. I have just received a manuscript of K. Meyer [15] in which

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"energy functions* are constructed for the flows described in this section. II.3. Anosov flows. Consider first l-parameter groups of vector space bundle automorphisms 4>t: £—»£. Here £ is a vector space bundle and 4>t is a flow on E such that for each /, 4>t: E—*E is a bundle auto­ morphism (i.e., linear on fibers). For example if if/t: M—*M is a flow on a manifold, the derivatives at each t, <j>t=D^tm- T(M)—*T(M), define a l-parameter group of vector space bundle automorphisms. Assuming £ is a Riemannian vector space bundle, say that such a flow t: E-*E is contracting if there are constants c, X > 0 such that \\t(v)\\£ce-u,allveE, t>0. Then t is expanding if -t is contracting and this is equivalent to the existence of ci>0, / i > 0 such that | | # , ( P ) | | ^ C I « " all <>0, » £ £ (compare §1.3). An Anosov flow on a complete Riemannian manifold M (or just a manifold in case M is compact) is a flow t whose induced flow D,: T(M)—*T{M) on the tangent bundle is hyperbolic in the follow­ ing sense: The tangent bundle T(M) can be written as the Whitney sum of 3 invariant subbundles, T(M) =Ei+Ei+Et where on £ " = £ l t 4>t is expanding, on £*=£»,<£» is contracting and £> is the 1-dimensional bundle defined by differentiating t with respect to /. Examples of Anosov flows are obtained readily from §1.3 and the following easily proved proposition. (3.1) PROPOSITION. / / / : Af—»Af is an Anosov diffeomorphism of a compact manifold, then the suspension off is an Anosov flow. Another important class of examples of Anosov flows are the geodesic flows on the tangent bundles of Riemannian manifolds of negative (possibly varying) curvature (see [8], [13]). (3.2) THEOREM (ANOSOV [9]). If t: M—>M is an Anosov flow of a compact manifold it is structurally stable. Also ifQ=M, the periodic orbits will be dense. Finally if there is an (Lebesgue) invariant measure, then 4>t is ergodic. Applied to the geodesic flows on the tangent bundles of manifolds M with negative curvature, (3.2) yields ergodicity, thus solving an old problem. The constant negative curvature case as well as the case of two dimensional M had been done earlier by G. Hedlund [42] and E. Hopf [45], [46]. See also [34] and [64]. Again as in §1.3. there is the very important problem of finding all Anosov flows on compact M (especially when fi= M). Progress on this problem might contribute to the problem of what manifolds can

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have Riemannian metrics of negative curvature. Is this class bigger than the class of manifolds which possess Riemannian metrics of constant negative curvature? On this point see the problem of Calabi in [31J.

11.4. On counting closed orbits. For counting the fixed points (at least algebraically) of a diffeomorphism, the Lefschetz Trace Formula provides a satisfactory method (see §1.4). This also applies to periodic points, and for suspended flows, these methods will give us some answers as to the nature of closed orbits. For flows in general, it is an outstanding problem to find methods which will tell if there are closed orbits and how many. Seifert's problem [lOS] is the best known question exemplifying this lack of knowledge. That is, does a flow on S* (continuous or dif­ ferentiate) have a closed orbit or a fixed point?46 A related question is: does X, a smooth vector field on D2XS1, the 2-disk cross the circle, transversal to the boundary, have a closed orbit or a singular point? Related to these questions are papers of Fuller [32], and A. Schwartz [104]. Thus an analogue of the zeta function for diffeomorphisms of §1.4 seems quite remote for flows. However we will mention a wild idea in this direction. Let r=T(t) be the set of closed orbits of theflow4>t: M—*2,I where we will assume M to be compact and that there are no fixed points. For Y E I 1 , define /(?) to be the period (minimal period, that is) of y, i.e., l(y) is the first *>0 such that <£<(*) =x for some * £ ? . We will as­ sume then that theflowsatisfies the generic property, { v £ r | l(y) gc} is finite for each positive c (that this is generic follows from 11.(5.6)). Then define formally (another zeta function!) Z(s) to be the in­ finite product

Z{s) = II ftd ~[exp/(>)]—*). rer t-o

The question is: does Z(s) have nice properties for any general class offlows4>f41 In this direction we consider the case that is the suspen­ sion of a diffeomorphism / : V—*V where the zeta function (of Weil, Artin-Mazur, §1.4) is rational. (4.1) THEOREM. / / the zeta function of f: V-*V is rational, then Z+(s) =Z(s) where is the suspension off converges in a half-plane to an analytic function of s, and has an analytic continuation to a nteromorphic function. Furthermore the zeros and poles of this meromorphic function can be computed explicitly in terms of those of f/.

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PROOF (LARGELY DUE TO NARASIMHAN). If Km = Km(f), m =* 1, 2, 3, • • • (as in §1.4) denotes the number of periodic points of / of minimum period m, we get directly from the definition of Z(s) 00

oo

*(*) = n n (i - «—<•+«)«•'-. m-1 * - l

Let oo

W(s) - I I (1 - r — ) * - ' - . Then - l o g W(«) = £

nil

- if, log f-

W

)

m/n

Assuming at first that the zeta function f (/) of / is of the form fO) = (l-X<)- 1 , we have (1.(4.1), (4.8)) E w » # - = X » . Thus - l o g ! F ( 5 ) « ] £ ( l / n ) ( X / V ) » = . - l o g ( l - X / V ) or W(s) = l-\/e' and

*« = n w(t+k) = n (i - x/«^»). t-0

4-0

Then we can see that Z(s) is entire because it is the uniform limit, in every compact set, of entire functions. Incidentally one sees from the explicit form of Z(s), a functional equation Z(s+l)=Z(s)«*/(e*—A). Finally the zeros are clearly the solutions of e,+" = \, k = 0, 1, 2, • • • or s+fe = log \+2irin, n£Z. In the general case we have f (t) = H,-.y(l —Hjt)/{\ —X,0 and ^2m/nKn= ]£<XJ — X/M?- Thus we obtain —log W(s) - - l o g I I . - . , ( l - X . M ( l - M y A « ) - 1 , so Z ( 5 ) = I I . - . i I I r . o ( l - X . A ' + * ) • (1 ~ni/e,+k)-1. The zeros are of the form 5 = log X<+2irn*—k and the poles s = \og Hj+2irni—k (distinguish thet's!). This proves (4.1). The following question then arises. Suppose <£<: M—*M is the sus­ pension o f / a s in (4.1) with f/ rational and / satisfying Axioms A and B of §1.6. Suppose even that M is the 2-dimensional toral diffeomorphism. Now letyf/t: M—*M be close to tj>t. Does Z^((s) h a v e a m e r o morphic continuation to all of C? An affirmative answer would be roughly necessary and sufficient condition for Z(s) to be useful. I must

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admit a positive answer would be a little shocking! A way of looking at this problem is the following. The canonical cross-section 2 for <j>t is also a cross-section for yj/t and the time of first return for if/t is de­ fined by a smooth function p: 2—*R+ (R+ the positive reals) which will be close to the constant function 1. There is a natural 1-1 cor­ respondence f—*y', Y—>r" from the set of closed orbits of #« to those of ipt using Si-stability. Let \y = l(y'), so X7= 2«,ernzP(*.) and - l o g Wx(s) = ^r,y,r(l/r)e~Xyr' where Wi(s) corresponds to the W(s) of the previ­ ous proof. Is there sufficient regularity in the XT to continue Wi meromorphically? There are two other remarks we wish to make about Z(s). First if 4>t is the geodesic flow for a 2-manifold of constant negative curvature, then Z(s) is meromorphic. In this case it is precisely the Selberg zeta function [106], which Selberg defined directly in terms of SL(2, R) and a certain uniform discrete subgroup T. Selberg proved that it is meromorphic in this case and found its zeros and poles as well. Sinai and Langlands pointed out to me this interpretation of the Selberg zeta function and this motivated my using it here. Finally we pose the question, how generally do flows have the l(y) growing slowly enough so that Z(s)has a half plane of convergence?4* II.5. Spectral decomposition of flows. One can extend Axioms A and B of §1.6 to flows. This goes as follows. For flows <£, on compact manifolds M, we have (5.1) AXIOM A'. Thefixedpoints of, are each hyperbolic. The nonwandering points Q consist of this finite set of fixed points F and the closure A of the closed orbits; A and F are disjoint. Finally the derived flow restricted to the tangent bundle restricted to A, Dj>t: TL(M)—*T±(M) is hyperbolic (defined analogously to the Anosov flow in §11.3). Topologically transitive for a flow again means that there is a dense orbit. (5.2) THEOREM (SPECTRAL DECOMPOSITION). Ift: M—*M satisfies Axiom A', then Q can be written uniquely as the disjoint union ftiWOt VJ . . . VJfi* where each Q,- is closed, invariant and each #«: 8<—»Q< is topologically transitive. (5.3) COROLLARY. M=Uf.x W'(Qi) (disjoint union, canonically) where each W'(Sl,) = {xEM\4»(x)^tt.}. (5.4) AXIOM B'. Conditions and notations as above, if W'(Q<) r\W*(Q,)?<0, then there exist periodic orbits (or fixed points) y in

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Q<, a in 0/ such that W*(y) and W*(ff) have a point of transversal intersec­ tion. The following seems to be a theorem although I haven't written out the details.** (5.5) / / 4>t• M—+M satisfies Axioms A' and B' then t is il-stable. One also obtains the openness,filtration,and partial ordering as in §1.6. The approximation theorems are quite parallel to those referred to in §1.6 with the same references in fact. (5.6) THEOREM [55 ] AND [114]. The property offlowsthat the fixed points Xi and closed orbits y, are all hyperbolic is generic. Furthermore generically, the stable and unstable manifolds of the xit y, intersect each other only transversally. (5.7) THEOREM (PUGH [«l]). In the Banach space of C vector fields (orflows),there is a Baire set with the property that thefixedpoints and closed orbits are dense in Q. If t: M-+M, ft: V—+V there is defined naturally the product flow 4>tX$t: MX V->MX V. Note that the product of two (or more) flows containing closed orbits of positive period will contain an invariant torus which will make this product not ft-stable. For gradient flows (nondegenerate) the situation is different and simpler; the product is in this case ft-stable. Note that one obtains the example showing that structurally stable flows are not dense, by simply suspending the example for diffeomorphisms. All the material in §1.5 about homoclinic points and symbolic flows can be suspended to obtain similar results on flows. As mentioned there, I first ran into this phenomena in that form, i.e., in trying to understand Van der Pol's equation (with forcing term). See also [107] for these questions discussed in the flow framework. PART III. MORE ON FLOWS

III.l. Flows with conditions imposed. In this section, we discuss some of the problems encountered in attempting to carry over Parts I and II to flows which satisfy certain constraints, e.g., of the type occurring in classical mechanics. Essentially nothing has been done in this direction, so we just mention some background material, related recent results, and some problems. The main class of flows, beyond the unrestricted ones we have been

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discussing up to this point, are the Hamiltonian flows. Abstractly speaking, a Hamiltonian flow is defined on a symplectic manifold, and this proceeds as follows. A symplectic structure on a manifold M is a 2-form 0 defined on M such that dd**0, and at each point of M, B is nondegenerate; nondegeneracy of 6 at x£M means that the map Q: r,(Jlf)—►rJ(Jlf) is an isomorphism from the tangent space at x to its dual where Q(X)(Y) ■«0(.y, Y), X, Y£TM(M) (for a complete discussion of this material, see [l], [123]). We then say that Mis a symplectic manifold. It fol­ lows that dim M is even. Thus on a symplectic manifold, there is a 1-1 correspondence between 1-forms and vector fields. Now if H: M—*R is a differentiate function (a "Hamiltonian" function), its derivative DH(x)£T%(M) defines a 1-form, which via Q we may consider as a vector field, say XH. Theflow#< generated by XH (at least locally) is called the Hamiltonian flow defined by H. It can be checked that t leaves 6 invariant. In fact, by reasons con­ verse to the above, it is important to consider directly those flows (which we will again call Hamiltonian) ,, say, defined for all t, on a symplectic manifold preserving the symplectic form. Then the nat­ ural global problem for Hamiltonian flows becomes (1.1) PROBLEM.30 Given a symplectic manifold M, find a Baire set , is in (B, one can describe the global orbit structure of t. If M is compact, one may conveniently consider the Hamiltonian flows X. as a subspace of all vector fields, x(-W) with the O topology. Note that a Hamiltonian flow, 4>t, leaves a volume on M invariant, namely the form obtained by wedging the symplectic form 0 with it­ self n*»§ dim M times. Thus it follows that in case M is compact, that the set of nonwandering points, ft is equal to all of M. One has a similar problem, also directly motivated by classical mechanics, for a single diffeomorphism. (1.2) PROBLEM. What is the orbit structure of some Baire set of diffeomorphisms / of a compact symplectic manifold which preserve the symplectic 2-form? Of course in studying these problems, one is only permitted per­ turbations of / to / ' which also keep 6 invariant. The first (and still unsolved) problem that one encounters here is to understand a local problem, the orbit structure in the neighborhood of a fixed point x of/. The difficulty is that the symplectic condition on / means that for the derivative D(x), hyperbolicity is not a generic property. For example, if dim M=2, f preserves a volume and Df(x): T„(M)

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—+TX(M) has determinant 1. One may classify these linear transfor­ mations into the hyperbolic and elliptic types. The hyperbolic is the one already discussed with eigenvalues X > 1 and 1/X. The elliptic case is a nontrivial rotation of the plane. In the elliptic case in gen­ eral, there exist no coordinates in the neighborhood of * in which / becomes linear, and only recently in this local 2-dimensional problem has one even begun to understand what is going on. Birkhoff, e.g. [18], had believed that volume preserving trans­ formations of compact 2-manifolds were ergodic (as well as Ilamiltonian transformations more generally) "in the general case" and based much of his work on this hypothesis. (Recall ergodic means there are no invariant sets of positive measure with measure less than that of M.) Through the work of Kolmogoroff, Arnold, Moser, [S2], [10], [7l], we know now that this is not the case. If x is an elliptic fixed point of 0° f: Mi—^M2, then generically, there is an invariant circle-in every neighborhood of x and t h u s / cannot be ergodic [70]. In the 2»-dimensional analogous problem there is an invariant «-dimensional torus in any neighborhood of x and the diffeomorphism is not ergodic. However one still has not yet a topological description in the neighborhood of an elliptic fixed point of a Hamiltonian diffeo­ morphism and thus it seems especially difficult to know how to pro­ ceed as in the first parts of the survey. Furthermore, the recent work of Arnold and Moser on the Hamiltonian case is still fairly local; the global Hamiltonian picture seems remote. We remark, though, that the examples of geodesic flows on manifolds of negative curvature are Hamiltonian and in this case, (§11.3), the flow is ergodic and struc­ turally stable (on each level surface of the Hamiltonian). We make three last comments on the Hamiltonian problem. First an elliptic point of a Hamiltonian diffeomorphism, say in 2 dimen­ sions, where the derivative is a rational rotation, is degenerate. This is one reason why one must work with Baire sets of Hamiltonian dif­ feomorphisms, not open dense sets. Similarly one cannot expect these diffeomorphisms to be Q-stable, as in Part I. Secondly, we remark that Pugh has shown that his closing lemma applies to prove the periodic orbits are dense in the compact Hamiltonian case [93]. Lastly it should be said that in practice, or in engineering, the dif­ ferential equations, because of friction, are no longer Hamiltonian and could be closer to those described in Parts I and II. In this con­ nection see [85]. After the Hamiltonian problems, the next most interesting case to consider might well be volume preserving diffeomorphisms. These coincide in dimension two with the Hamiltonian ones.

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Volume preserving diffeomorphisms have not been studied from our point of view (although, see [122]).51 For dim M>2, however, none of the Hamiltonian objections apply and in fact the hyperbolic linear volume preserving maps are dense and open among all volume preserving linear maps; very possibly in the higher dimensional case, volume preserving diffeomorphisms might be amenable to study by the methods of Part I. A first question could be to prove 1.(6.7) for volume preserving diffeomorphisms.52 For every volume preserving diffeomorphism / of a compact mani­ fold, il(f) ™ M. Presumably, Pugh's method would show the periodic points are dense. Is / ergodic, a generic property in this context? Oxtoby and Ulam [78] prove such an ergodicity theorem for homeomorphisms. One can also ask whether the program of Parts I and II could be carried out for ordinary differential equations of higher order, say second order to begin with; see [56], [123] for a coordinate free defi­ nition of 2nd order differential equations. This hasn't been investi­ gated as far as I know. The same applies to diffeomorphisms or flows of infinite dimensional manifolds. Holomorphic diffeomorphisms of a complex manifold are much more rigid, but I think that the orbit structure is not generally under­ stood. G. Julia's prize memoir [50] is related to this subject. It con­ cerns holomorphic endomorphisms of the Riemann sphere.53 III.2. Some other work on flows. Here we mainly remark on a couple of recent results on flows which are not so directly related to the preceding. The question of existence for minimal sets poses interesting prob­ lems to the global analyst. A compact manifold (or even space) M is a minimal set for the flow 4>t: M—*M if there is no proper nonempty closed invariant subspace of M. Gottschalk [35] has given a survey of this subject. A main problem is: what M can be the minimal set for some flow? It is not known if the 3-sphere can be a minimal set.54 A number of new examples of minimal flows are constructed from Lie groups in [12»]. See also [29] for examples on S"XSl. An important recent result is that of A. Schwartz [102] which generalizes both the Poincarl-Bendixson theorem for plane regions and Denjoy's theory of C* flows on the torus. The Schwartz theorem says that for any C* flow on a 2-manifold, any (compact) minimal set is either a point, a closed orbit, or a 2-torus. Among other applica­ tions of Schwartz's methods, R. Sacksteder has shown that if G is a finitely generated, finitely presented, discrete group G acting C*

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freely on the circle, then the action is topologically conjugate to a group of rotations. See also [99]. Here acting freely means no ^ ( : S1—*Sl, gEG, has a fixed point. There has been recently also interesting work on the subject of distal actions which we do not go into. Here t: M—*M, compact M, is distal, if in some metric, for any x, yEM, x^y, there is an e such that d(t(x), t(y)) > e all tEG. See for example [28], [33], [67]. PART IV. OTHER LIE GROUPS

IV. 1. Action of an abelian Lie group. We consider briefly here the question of an abelian Lie group G acting on a manifold when G is more complicated than Z or R. Recall first that an action of a Lie group G on a manifold is a homomorphism : G—»Diff(Af) such that the induced map $ : G X A f —>M defined by #(g, m) =#,(m) is C*. The orbit Ox through xE M of such an action is the image of the map px: G—*M defined by p*(g) =t(x). The isotropy group H* of the action a t x is the set of elements hEG such that <j>h(x) =x. Then Hx is a closed subgroup of G and G/HM is a homogeneous space of G. Induced from p, is a 1-1 immersion g«: G/HX—*M. Finally we remark that the x orbit 0, refers to />«, qt, G/Hx or qz(G/Hz) at various times. If there is danger of confusion we will try to be more explicit. A fixed point of the action is an orbit consisting of a single point. Actions <j>„ t(ft are conjugate if there is a homeomorphism h: M—*M such that *,(**) =*(&(*)) for all gEG, xEM. Returning to the abelian case, suppose G is isomorphic to Z-\-Z. One may choose generators/, g£Diff(il/) of G, so fg=gf, and thus one is equivalently studying a pair of commuting diffeomorphisms. More generally one may study two commuting differentiable maps and actually the most studied of such problems perhaps has been the existence of a common fixed point for two commuting maps of the unit interval / into itself. Very recently a counterexample has been found to this problem by P. Huneke [49] and independently W. Boyce [22]. They each construct continuous m a p s / , g: I—*I, with fg^gf and such that there is no x E ^ w i t h / f r ) =x=g{x). These maps are not C1 and thus the differentiable version of this problem remains open. In this direction, A. Schwartz [103] has the strongest result: If / and g are C1 maps, I—*I, there is a fixed point of one which is periodic for the other. Going back to the case of two commuting diffeomorphisms g, / : M—+M, observe g is in the centralizer Z(f) of / , i.e., Z(f) = { g £ D i f f ( i / ) | j / = / g } . Thus a first question in such a study could well be

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(1.1) PROBLEM.55 What can be said about Z(f) for /eDiff(J#)r Under what conditions on / is dim Z(f) < « ? Is Z ( f ) - {f*\mEZ} a generic property? . Work of N. Kopell suggests that this last question may have an affirmative answer. Since a significant class of Q-stable diffeomorphisms (§1.2) are a finite union of contractions up to a finite power, it is important to know Z(f) when / is a contraction. (1.2) THEOREM (KOPELL [53]). Suppose Cf: W—*W is a contrac­ tion. Thus at the uniquefixedpoint xr derivative Df(x): T,(W)—*Ta(W) is a linear contraction. Then Z{f)= {gEDiff(W)\gf-fg, gEC*} is a finite dimensional Lie group. If f is linear, with a further nondegeneracy condition on the eigenvalues, then g is linear. Finally for a dense open set of diffeomorphisms f satisfying 1.(2.2), Z(/) = {/"jmEZ}. In the proof of the structural stability of an Anosov diffeomorphism (see the Appendix of Part I), one obtains at the same time that its centralizer is discrete, even in the group of homeomorphisms of M. Adler and Palais [5] have actually computed this centralizer for the toral diffeomorphisms. It would seem at least a reasonable conjecture that an open dense set of diffeomorphisms satisfying Axioms A and B (§1.6) have centralizer Z ( / ) = { / * j m G Z } * Kopell [53] has studied commuting diffeomorphisms of the circle in more detail. Here, at least among those with periodic points, she has found a dense set of actions of Z-\-Z for which the orbit structure can be understood. She also gives an example of commuting diffeo­ morphisms/, gol S? with the following property: g is the identity on an open set and for C approximations/', g1 of /, g such that fg' =g'f, g' must also be the identity on some open set. Further results on abelian actions are related to the question of degeneracy of some orbits when Rh acts on a given manifold. By tak­ ing generators, an action of Rh on M corresponds to a set of k tangent vector fields on M which commute, or equivalently their bracket is zero?7 In this direction Lima [<50] showed that if R* acts on a compact 2-manifoId of nonzero Euler characteristic, there must be a fixed point or, equivalently, a common zero of the two generating vector fields. In a further paper [59] he showed that two commuting vector fields on 5* are dependent at some point (see also Novikov-Arnold [76]). Extenpions of this last theorem have been made to actions of R* on certain M**1 by Rosenberg, and Sacksteder [98], [lOOj. While on this subject it seems worthwhile to mention that closely related is the result of Novikov [77] who has shown that every foliation of di­ mension 2 on S* has a compact leaf. An account of the basic results in foliation theory is in Haeniger [41 ].

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Recently Adler and McAndrew [4] have shown that the topological entropy of a Chebyshev polynomial is positive. There has not been much work on actions of solvable or nilpotent Lie groups along the line of this section. IV.2. The semisimple case. Here we make a few comments on the problem of studying the action of a semisimple group G on a (com­ pact) manifold. Discussions with R. Palais have been helpful here. There is a vast literature on the subject of a Lie group G acting on a manifold M when G is compact, acts transitively, or acts linearly. The reader can refer to [20 ], [66] for the case of compact G. We only remark that Palais [80 ] (see also [81 ]) has shown a strong form of structural stability for compact actions. Namely if an action \j/a is close to 4>„ 4>' G—>Diff (M), G compact, these actions are conjugate by a diffeomorphism A£Diff(M). Thus ,Qix) =hl>t(x), all gEG. In this case we say is rigid. One systematic treatment of the transitive case is [27], Another aspect of this case is [126]. If G acting on M is semisimple, but neither compact, nor acting transitively, nor linearly, there seems to be essentially no literature, a t least that I know of.58 On the other hand, it would seem worthwhile to make enorts in this direction. These efforts could produce unifying theorems, shed light on the above three special cases, or be useful in geometry or physics. One possibility might be to extend some of the results of Parts I and II. We limit ourselves to a few remarks. In the first place, the evidence is that the richness of actions of a noncompact semisimple Lie group will lie somewhere between the abelian case (extremely rich, e.g., G = R) and the compact group case (few actions, i.e., G acts rigidly as mentioned above). We will try to make this point clearer. In the linear theory, or representation theory, the semisimple case is close to the case of compact groups in that representations (finite dimensional) are rigid. This contrasts to the abelian case where even one dimensional representations of R (up to equivalence) are param­ eterized by R. This motivated the speculation that if : G—»Diff(M) is an action of semisimple G with fixed point x£M, the representation g—>Da: Tt(M)-+Ts(M) determines the orbit structure of in a neighborhood of x. R. Hermann [44] showed this to be true formally, while Guil­ lemin and Sternberg [37] show that this is actually true in the case of an analytic (real) action. On the other hand Guillemin and Sternberg [37] give a counterexample in the C case for G = SL(2, R). This situation, however, is still not yet well understood.

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One might ask whether any action of a semisimple G on a compact manifold is rigid. This is false as the following simple example shows. Let G=*SL(2, R) act on the unit tangent bundle T of a 2-manifold M* of genus greater than one by dividing out a uniform discrete sub­ group T from G. These actions correspond to different complex struc­ tures on M* and thus one gets a continuous family of such actions. One sees this by considering M as the double coset space T\G/K where G/K is the complex upper half plane. See also for example [126]. This does not exclude the possibility of a number of cases of noncompact semisimple G acting rigidly on compact M. Here is one such case. G = S L ( n + l , R) acts transitively on Pn(R) using homogeneous coordinates and in fact S L ( « + 1 , R) has no other homogeneous spaces of dimension less than n-f 1. Thus there is a t most one action, transi­ tive or otherwise, of SL(n-f-l, R) on connected M if the dimension of M is less than N+\, so of course this action is rigid and M must be P»(R) (or a point!). This suggests that semisimple G acting on M of much lower dimen­ sion might be fairly amenable to study. The situation is akin to the work of Hsiang and Hsiang on compact G [48]. The work of Hermann [43] and others on the (equivariant) compactification of homogeneous spaces may be interpreted as studying the action of a semisimple G in the neighborhood of certain noncompact orbits. I have just received a manuscript [47] of W. Y. Hsiang which is related to the material of this section. IV.3. Final miscellany. We end by making some final remarks on the action of a Lie group G. The notion of induced representation which has proved useful in the linear theory has an analogue in the general case which we describe now. This construction generalizes the suspended action of §11.1. Suppose then if is a closed subgroup of a Lie group G and # : H—»Diff (Af) is an action of H. Define an action ^ : #-»Diff(MXG) by &(m, *) = (<&(*»), gh) and let T : MXG~*E be the projection onto the orbit space. One obtains the following dia­ gram w h e r e / is induced by TQ. MXG 1

E

ra

'

> G

!

-> G/H

Here £ is a manifold ao.df:E—*G/H

is a bundle over G/H, Cartan's

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construction in [23]. The action a of G on MXG denned by (m, g) = (*»» i' g) induces an action T = T ( # ) of G on E which we might call the induced action of . This action T: G—»Diff(£) is fiber preserving with respect to / and commutes with translation on the base, r re­ stricted to H leaves / - , ( e i / ) invariant where it is equivalent to the original action . If G = R, H—Z, then T is the construction of the suspension of a generator of Z as in §11.1. If M is compact so is E and if is linear, M a vector space, then £ is a vector space bundle and the action of G on sections is Mackey's induced representation. We saw in Parts I and II that the concept of nonwandering points played a central role. A most important task would be to generalize this idea to a more general group G and to formulate some of the conditions say of §§1.2, 1.3, 1.6 for the case of a general Lie group, or even abelian, or semisimple G.59 Palais in [79] considers a class of actions on noncompact groups which have many properties of compact transformation groups. These actions are quite restrictive in that the isotropy group is always compact and the manifold must be noncompact. These actions, how­ ever, resemble G = Z acting on M—Q. Suppose now that : G—*Diff(M) is an action with a fixed point x£M. Then the map $: G—+Aut(Tx(M)) is a linear representation of G, where $(g) = Dt(x) is the linear automorphism of TX(M) de­ fined by the derivative of 0 , at x. (3.1) PROBLEM. T O what extent (generically) does this representa­ tion determine the action of G in a neighborhood of x, say up to conjugacy. This is a basic local question. In earlier sections we saw aspects of it, starting with the stable manifold theorems, §1.2, in Stern berg [121 ] and Guillemin-Sternberg [37]. In general, the question is very far from being answered. Very likely, the higher derivatives will play a basic role for some groups. Suppose more generally that 0 = 0X is the compact orbit of some x £ M of the action $: G->Diff(Jlf). Thus G-*0T, g-^>,(x) is the orbit map with isotropy group HXC.G acting on M leaving x fixed. The derivative D,: To(M)-+T<,(M) defines a structure of a homo­ geneous vector space bundle (in the sense of [21 ]) on the restriction of the tangent bundle of M restricted to 0. We may generalize (3.1) with (3.2) PROBLEM. To what extent does the group of bundle auto­ morphisms De: T0(M)—*TQ(M) determine the action of G in a neighborhood of 0? Here of course (3.1) is the case 0 is a single point. For the notion

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of equivalence in (3.2), one might take orbit preserving homeomorphism. An earlier special case of (3.2) was discussed in §11.2, where G=i? and 0 was a closed orbit, i.e., the circle S1. For global actions of G on M, it is probably profitable to consider initially very restricted cases, for example, actions on 2-manifolds. In this case, the possible orbits are well known, see Mostow [72]. Another tractable case might be actions with only two orbits. BIBLIOGRAPHY 1. R. Abraham and J. Marsden, Foundations of mechanics, Benjamin, New York, 1967. 2. R. Abraham and J. Robbin, Transversal mappings and flows, Benjamin, New York, 1967. 3. R. L. Adter, A. G. Konheim and M. H. McAndrew, Topological entropy, Trans. Amer. Math. Soc. 114 (1965), 309-319. 4. R. L. Adler and M. H. McAndrew, The entropy of Chebyshev polynomials, Trans Amer. Math. Soc. 121 (1966), 236-241. 5. R. L. Adler and R. Palais, Homeomorphic-conjugacy of automorphisms on the torus, Proc. Amer. Math. Soc. 16 (1965), 1222-1225. 6. A. Andronov and L. Pontryagin, SysUmes grossiers, Dokl. Akad. Nauk. SSSR 14 (1937), 247-251. 7. D .V. Anosov, Roughness of geodesic flows on compact Riemannian manifolds of negative curvature, Soviet Math. Dokl. 3 (1962), 1068-1070. 8. , Ergvdic properties of geodesicflowson closed Riemannian manifolds of negative curvature, Soviet Math. Dokl. 4 (1963), 1153-1156. 9. , Geodesicflowson compact Riemannian manifolds of negative curvature, Trudy Mat Inst. Steklov. 90 (1967) - P r o a Steklov Math. Inst (to appear). 10. V. I. Arnold, Proof of a theorem of A. N. Kolmogorov on the invariance of quasiperiodic motions under small perturbations of the Hamiltonian, Russian Math. Surveys 18 (1963), 9-36. 11. V. I. Arnold and A. Avez, Problimes ergodiques de la mtcanique classique, Gauthier-Villars, Paris, 1966. 12. E. Artin and B. Mazur, On periodic points, Ann. of Math. (2) 81 (1965), 82-99. 12a. L. Auslander, L. Green, F. Hahn et al., Flows on homogeneous spaces, Prince­ ton Univ. Press, Princeton, N. J., 1963. 13. A. Avez, Ergodic theory of dynamical systems, Vols. I, II, Univ. of Minneapolis, Minnesota, 1966, 1967. 14. P. Billingsley, Ergodic theory and information, Wiley, New York, 1965. 15. G. D. Birkhoff, Dynamical systems, Amer. Math. Soc. Colloq. Publ., vol. 9, Amer. Math Soc., Providence, R. I., 1927. 16. , Surface transformations and their dynamical applications, Acta Math. 43 (1920), 1-119. 17. , On the periodic motions of dynamical systems, Acta Math. So. (1927), 359-379. (See also Birkhoff's Collected Works.) 18. , Nouvettes recherches sur les systimes dynamique, Mem. Pont. Acad. Sci. Novi Lyncaei 1 (1935), 85-216. 19. , Dynamical systems with two degrees of freedom, Trans. Amer. Math. Soc. 18 (1917), 199-300.

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(November

75. K. Nonuzu, On the cohomology of compact homogeneous spaces of nilpotent Lie groups, Ann. of Math. (2) 59 (1954), 531-538. 76. S. P. Novikov, The topology summer Institute, Seattle 1963, Russian Math. Surveys 20 (1965), 145-167. 77. S. P. Novikov, Smooth foliations on three-dimensional manifolds, Russian Math. Surveys 19 (1964) No. 6,79-81. 78. J. C. Oxtoby and S. M. Ulam, Measure-preserving homeomorphisms and metri­ cal transitivity, Ann. of Math. (2) 42 (1941), 874-920. 79. R. S. Palais, On the existence of slices for actions of non-compact Lie groups, Ann. of Math. (2) 73 (1961), 295-323. 80. , Equivalence of nearby differentiable actions of a compact group, Bull. Amer. Math. Soc. 67 (1961), 362-364. 81. R. S. Palais and T. Stewart, Deformations of compact differentiable transforma­ tion groups, Amer. J. Math. 82 (1960), 935-937. 82. J. Palis, Thesis, Univ. of California, Berkeley, 1967. 83. M. Peixoto, On structural stability, Ann. of Math. (2) 69 (1959), 199-222. 84. , Structural stability on two-dimensional manifolds, Topology 1 (1962), 101-120. 85. , Symposium on Differential .liquations and Dynamical Systems (Puerto Rico) Academic Press, New York, 1967. 86. , On an approximation theorem of Kupka and Smale, J. Differential Equations 3 (1967), 214-227. 87. M. Peixoto and C. Pugh, Structurally stable systems on open manifolds are never dense, (to appear). 88. O. Perron, Die stabilitatsfrage bei differentialgleichungen, Math. Z. 32 (1930), 703-728. 89. H. Poincarf, Sur les courbes definies par des equations difftrentielles, J. Math. 4° serie 1885. 90. , Les mithodes nouvelles de la mecanique c&este, Vols. I, II, III, Paris, 1892, 1893, 1899; reprint, Dover, New York, 1957. 91. C. Pugh, The closing lemma, Amer. J. Math, (to appear) 92. , An improved closing lemma and a general density theorem, Amer. J. Math, (to appear). 93. , The closing lemma for Hamiltonian systems, Proc. Intemat. Congress Math, at Moscow, 1966. 94. G. Reeb, Sur certaines proprittts topologiques des trajectoires des systemes dynamiques, Acad. Roy. Belg. Cl. Sci. Mem. Coll. 8° 27 (1952), no. 9. 95. L. ReizinS, On systems of differential equations satisfying Smale's conditions, Latvijas PSR Zinatnu Akad. Vfctis Fiz. Tehn. Zinfttnu Ser. 1964, 57-61. (Russian) 96. F. Riesz, and B. Nagy, Functional analysis, Ungar, New York, 1955. 97. H. Rosenberg, A generalization of Morse-Smale inequalities, Bull. Amer. Math. Soc. 70 (1964), 422-427. 98. , Actions of R* on manifolds, Comment. Math. Helv. 41 (1966-67), 170-178. 99. R. Sacksteder, Foliations and pseudo-groups, Amer. J. Math. 87 (1965), 79102. 100. , Degeneracy of orbits of actions of tt? ona manifold, Comment. Math. Helv. 41 (1966-67), 1-9. 101. O. F. G. Shilling, Arithmetical algebraic geometry, Harper and Row, New York, 1956. 102. A. J. Schwartz, A generalization of a Poincort-Bendixon theorem to closed twodimensional manifolds, Amer. J. Math. 85 (1963), 453-458.

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103. , Common periodic points of commuting functions, Michigan Math. J. 12 (1965), 353-355. 104. A. Schwartz, Vector fields on a solid torus (to appear). 105. H. Seifert, Closed integral curves in 3-space and isotopic two-dimensional de­ formations, Proc. Amer. Math. Soc. 1 (1950), 287-302. 106. A. Selberg, Harmonic-analysis and discontinuous groups in weakly symmetric Riemann spaces with applications to DirichUt series, J. Indian Math. Soc. 20 (1956), 47-87. 107. L. P. Shil'nikov, The existence of an enumerable set of periodic motions in the neighborhood of a homoclinic curve, Dokl. Akad. Nauk SSSR 172 (1967), 2 9 8 - 3 0 1 Sovkt Math. Dokl. 8 (1967), 102-106. 108. M. Shub, Thesis, Univ. of California, Berkeley, 1967. 109. S. Smale, Morse inequalities for a dynamical system, Bull. Amer. Math. Soc. 66 (1960), 43-49. 110. , On gradient dynamical systems, Ann. of Math. (2) 74 (1961), 199-206. 111. , On dynamical systems, Bol. Soc. Mat. Mexicans (2) S (1960), 195198. 112. , Dynamical systems and the topological conjugacy problem for diffeomorphisms, Proc. Internat. Congress Math. (Stockholm, 1962), lost. Mittag-Leffler, Djursholm, 1963, pp. 490-496. 113. , A structurally stable differtntiable homeomorphism with an infinite number of periodic points, Proc. Internat. Sympos. Non-linear Vibrations, vol. II, 1961, Izdat. Akad. Nauk Ukrain SSR, Kiev, 1963. 114. , Stable manifolds for differential equations and diffeomorphisms, Ann. Scuola Norm. Sup. Pisa (3) 17 (1963). 97-116. 115. , Diffeomorphisms with many periodic points. Differential and Com­ binatorial Topology, Princeton Univ. Press, Princeton, N. J., 1965, pp. 63-80. 116. , Structurally stable systems are not dense, Amer. J. Math. 88 (1966), 491-496. 117. , Dynamical systems on n-dimensional manifolds. Symposium on dif­ ferential equations and dynamical systems (Puerto Rico) Academic Press, New York, 1967. 118. J. Sotomayor, Generic one-parameter families of vector fields on two-dimensional manifolds (to appear). 119. P. R. Stein and S. M. Ulam, Non-linear transformations studies on electronic computers, Rozprawy Mat. 39 (1964). 120. S. Sternberg, Local contractions and a theorem of Poincart, Amer. J. Math. 79 (1957), 787-789. 121. , On the structure of local homeomorphisms of Euclidean n-space. II, Amer. J. Math. 80 (1958), 623-631. 122. , The structure of local homeomorphisms. I l l , Amer. J. Math. 81 (1959),

578-604. 123. , Lectures on differential geometry, Prentice-Hall, Englewood Cliffs, N. J., 1964. 124. R. Thorn, Sur une partition en cellules associts d unefunction sur une tarUtt, C. R. Acad. Sci. Paris 228 (1949), 973-975. 125. A. Weil, Adeles and algebraic groups, Notes, Institute for Advanced Study, Princeton, N. J., 1961. 126. , On discrete subgroups of Lie groups. II, Ann. of Math. (2) 75 (1962), 578-602. 127. R. Williams, One dimensional non-wandering sets (to appear).

735

NONGENERICITY OF ft-STABILITY R. ABRAHAM AND S. SMALE

We prove here that in general, ft-stable diffeomorphisms are not dense in DifflAf), the space of C diffeomorphisms on a C° manifold M, with the uniform C topology, 1 < r < oo. Recall from [1] that if/e Diffl[M), then x e M i s a nonwandering point off if and only if for every neighborhood U of x e M there is a nonzero integer me Z such thai fm(U) n l / ^ 0 . The set ft = fl(/) of all nonwandering points of/ is a closed invariant set. If A c M is a closed invariant set, A has a hyperbolic structure if and only if the tangent bundle of M restricted to A, T (M), splits into a sum of C° subbundles £* and £", invariant under the tangent of/ Tf: TA(W) -► TA(M) such that 7 / i s expanding on £" and contracting on E' (see [1] for complete definitions). Then/satisfies Axiom A if and only if: (Aa) ft(/) has a hyperbolic structure, and (Ab) The periodic points of/are dense in ft(/). If/ g e DifflM), they are Q-conjugate if and only if there exists a homeomorphism /r.ft(/) -»ft(g) such that g/i = /»/ and/is Cl-stable if and only if there is a neigh­ borhood A?(/) of/e Diffl[M) such that every ge N(f) is ft-conjugate t o / In this paper we construct an open set N <=■ Diffl(T2 x S2) such that every ge N violates both (Aa) and ft-stability. The basic idea is to construct fe Difl(Af) with disjoint closed invariant sets A, and A2, having hyperbolic structures of different dimensions, such that an orbit goes from A, to A2, and another goes from A2 to A,. This implies that A,, A2, and the two orbits are contained in ft(/), which therefore cannot have a hyperbolic structure. Further, this "pathology" is stable under perturbations of/in the C topology. In §1 we establish a criterion for the behavior described above, and in §2 we construct a diffeomorphism satisfying the criterion. §3 establishes ft-instability for this example. 1. We begin by recalling some aspects of the Stable Manifold Theorem [1, §7.3], or [2], or [3]. If A is a compact invariant set o f / e Diffl[M) with hyperbolic structure, T (M) = £* + £", then there is defined for each x e A, a stable manifold W\x) which is a one-to-one immersed cell in M, and consists of points ye M with the property that d{fm{x),fm(y)) - 0 as m -* oo. Then W(x) is defined as the stable manifold at x e A f o r / - ' . Then define W'(\) = \JX(A W{x). Finally, W{x) varies smoothly on compact sets as x varies in A. The type of A is the pair (A, b) where a = fiber dim £* and b = fiber dim £". DEFINITION. A subbasic set for fe Diff(M) is a compact invariant set A c M with hyperbolic structure such that//A is topologically transitive and the periodic points are dense in A. 5

736 6

R. ABRAHAM AND S. SMAI.K

If A! and A2 are disjoint subbasic sets of/, we write A! < A 2 when W*(A,) n W"(\2) ± 0 , and A, « A 2 when there are points p,e A, such that W"(p,) and W"(p2) have a point of transversal intersection. THEOREM. f/ien

IfA, and A2 art? disjoint subbasic sets of/eDiff(M)and A2 « A, < A2, W*(A 1 )nW"(A 2 )eQ = 0(/)-

COROLLARY. / / A , amf A2 are disjoint subbasic sets offe Difl)[Af), A2 « A, < A 2 , and type (A,) ^ ty pe (A2), then f does not satisfy Axiom A. PROOF. Suppose/satisfies Axiom A. If A2 < A, < A2 and x e W'^J n W(A2), then x 6Q(/) by the theorem above, while/"(x) -» A, as m -► oo, and/"(x) -» A2 as m -» - ao. As the orbit closure 0(x) c ft(/), A, and A 2 must be in the same basic set fi( of/(that is, the same indecomposable piece of fi(/). see [1, 6.2]). As ft, has a hyperbolic structure and is indecomposable, dim E?x is constant for all xefl,,a contradiction. The proof of the theorem requires the following LEMMA. Let / : A -♦ A be a topologically transitive homeomorphism of a compact metric space with periodic points dense in A. Then given nonempty open sets K,, V2 in A, there is a periodic point pe V, such thatfm(p)e V1for some m.

From the topological transitivity/ m (K,)n V2 =/= 0 for some m. Let q be a periodic point in this intersection and p = f~"\q). PROOF.

PROOF OF THE THEOREM. Let x e WfA,) n W%A2) and let U be a neighborhood

ofx. Now by the hypothesis A2 4. A,, lV"(p,) and W*(p2) have a point of transversal intersection for p , e A,, p 2 e A2. Let K, be a neighborhood of px in A,, Ul a neighborhood of p2 in A2 such that for every pe Vuqe Vx, W(p) and W\q) have a point of transversal intersection. Choose an open set V2 in A,, U2 in A 2 such that if q' e V2, p'eU2, then W'(q') and W\p') intersect U. Now apply the previous lemma to obtain periodic orbits y, meeting Vx, V2 and y2 meeting I/,, U2. Thus W"(y,) and W*(y2) have a point of transversal inter­ section while W'iyJ and W"(y2) both meet U. Now apply the argument of [1, 7.2] to obtain that/"(I/) r> U ± 0 for some m. This proves the theorem. 2. We now construct a diffeomorphism / e Diff(T2 x S2) having a subbasic set configuration A, « A2 < A,. Let ge DiflTT2) be the Thorn diffeomorphism, defined by the linear isomorphism of R2 having matrix

Cil 2

(see [I, §1-3]), and peT the fixed point corresponding to the origin, which is of type (1,1), that is, dimEJ = 1, dim££= 1. Let AeDifffS2) be the horseshoe diffeomorphism [I, §1-5], which has two fixed points, qx and q2, of type (1,1), at

737 NON-<;I:NKRK:ITY OF A-STABILITY

7

each of which the map h is linear in a C* coordinate chart. In addition, recall that {,
yeW'(q2)nW\qi). Thus with respect to/. A, < A2 < A„ completing the construction. Finally we claim there exists a neighborhood N of / e DififT2 x S2) such that for all g e /V, there are subbasic sets At(g), homeomorphic to A,, i = 1,2, such that A,(0)« A2(g) < A,(0). First, we observe that the local fi-stability Theorem [3, Proposition 3.1] applies to yield the following: There is a neighborhood N0 o f / e Diff(T2 x S2) and for every geN0, subbasic sets A,{g) and a conjugating homeomorphism h(g): A,( 1). This follows from the continuity conclusion of the Stable Manifold Theorem [2]. Thus A,
738

8

Nongenericity of Q-stabHity now follows from the following facts: (A) A diffeomorphism geN satisfying (a) may be approximated by one satisfying (b). (B) A diffeomorphism geN satisfying (b) may be approximated by one satisfying (a). (C) If g,g'eN satisfy (a), (b) respectively, then g, g" are not fi-conjugate. Now (A) is a consequence of a general approximation theorem (see for example [4, p. 100]). Fact (B) is proved easily since the periodic points are dense in A, and the W\z) for z periodic are dense in rV"(A,). Finally we see the truth of (C) by following the orbit of H-"(A2) n W"(z) and its image under a possible conjugacy.

1. 2. 3. 4.

REFERENCES S. Smale, Difftrentiable dynamical systems. Bull. Amer. Math. Soc. 73 (1967), 747-817. M. Hirsch and C Pugh. Stable manifolds and hyperbolic sets, these Proceedings, vol. 14. S. Smale, The tl-stahilily theorem, these Proceedings, vol. 14. R. Abraham and J. Robbin, Transversal mappings and flows, Benjamin, New York, 1967.

UNIVERSITY OF CALIFORNIA, SANTA CRUZ AND UNIVERSITY OF CALIFORNIA. BERKELEY

739

STRUCTURAL STABILITY THEOREMS J. PALIS AND S. SMALE

1. We prove here that a class of diffeomorphisms, i.e. difTerentiable automor­ phisms, introduced in [3] and studied in [2] are structurally stable. Since every compact manifold has such diffeomorphisms, this implies a positive answer to the well-known question as to whether every manifold has a structurally stable diffeomorphism. These results are proved in [2] for manifolds with dimension <, 3. We show also that the similar theorems for differential equations are true. The precise results are as follows. Let M be a compact C°° manifold without boundary, and for r ^ 1 let Diff (M) be the set of C diffeomorphisms of M with the uniform C topology. For/e Diff(M) we denote by fi(/) the set of nonwandering points of /(see [5] for elementary definitions and a general background). Consider fe Diff (M) which satisfies (a) Q(/) is finite, (1.1) (b) Q(/) is hyperbolic, (c) Transversality Condition. Condition (a) implies that Q = Q(/) consists of periodic points. Then (b) means that if x e Q and fm(x) = x, for some m > 0, then the derivative Dfm{x):Tx{M) — TX(M) has its eigenvalues not equal to one in absolute value. From condition (b) one has defined stable and unstable manifolds for each x e f l denoted by W(x), W(x) respectively. Then condition (c) means that for each x, y 6 fl, W(x), Wy{y) have transversal intersection in M. A diffeomorphism / Diff (M) is said to be structurally stable if there exists a neighborhood N(f)
hf=gh. (1.2)

MAIN THEOREM.

If fe Diff(Af) satisfies (1.1) then f is structurally stable.

Using [4], we have the following corollaries: (1.3) COROLLARY. The gradient dynamical systems on a compact Riemamian manifold M which are structurally stable form a dense open set among all gradient dynamical systems on M. Moreover, a gradient dynamical system is structurally stable iff its induced diffeomorphism at time t = I satisfies (1.1). (1.4) COROLLARY. Any compact manifold admits a structurally stable diffeo­ morphism. 223

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J. PALIS AND S.SMAI.K

This paper is mainly devoted to proving (1.2). §5 gives the corresponding results for differential equations and poses a question related to the main result in this paper. Conversations with M. Hirsch and C. Pugh have been very helpful in preparing this paper. 2. In this section we first review the needed background results from [2] and [3]. We then construct for/satisfying (1.1) tubular families of its stable manifolds, as defined below. Let / satisfy (1.1) and fl =* fl(/). If flj is a periodic orbit of Q, let W{Qk)

= U P«n* H"(P) and W W = 1 1 ^ . w»- T^" from C3] («« also P] and M) : (11) THEOREM. The fl» are partially ordered by: fi, <> ilk iff W"(Q,)n \V\QJ # 0 . For all \ £k £ m, W"(flk) a/uf W"(fi4) are properly imbedded submanifolds ofM. We now choose once and for all a simple ordering consistent with this partial ordering, so that fl, < Q 2 < flj ^ ... ^ Clm., £ H,.. the Slk being all the periodic orbits of/ Thus Q, (and perhaps Q2) is a sink (or attractor) and Q„ is a source. Submanifolds are always supposed to be properly imbedded and without boundary unless so stated. By the general theory for pe Per(/), W(p) is a submanifold of M. (2.2) DEFINITION (TUBULAR FAMILY OK Wl(p)). A tubular family Tot W*(p) = T0 is a collection of disjoint C submanifolds {Ty) of M indexed by y in an open neigh­ borhood N of p in W"(p) with the following properties. (1) V= V(T0) = [j ytNTy is an open set of M containing T0. (2) T0 = Tr

(3) Ty intersects N transversally in the single point y. Finally (4) the map V-* N which sends Ty into y is continuous; the section s which sends xeTy into the tangent space of Ty at x is a continuous map from V into the Grassmann bundle over V. Note that s in (4) is not required to be smooth. If Q4 is a periodic orbit of / satisfying (1.1), then we will speak of a tubular. family for W'Cfy). This is simply a tubular family T= {Tx} for W'(p), some peilk together with the submanifolds {/'(T x )},„ 0 „_, n = period ilk. A tubular family T = \Tt) of W'(p) is called invariant iff~"Tr = Tf „,„ where n is the period of p. In this case the corresponding tubular family of H^Jl^) will also be called invariant. A system of tubular families for/is a set of tubular families Tk = {T*}, one for each fl,,. The system will be called compatible when it meets the condition, if 7"x n T'y ± 0 , then one submanifold contains the other. The main goal of this section is to prove that/e Diff(M) satisfying (1.1) has a compatible system of invariant tubular families.

741 STRUCTURAL STABILITY THEOREMS

225

A standard result in Differential Topology is: (2.3) LEMMA. Let N be a C manifold possibly with boundary and J = [0,1} There exist a neighborhood U(A) of the diagonal A in N x N and a C function 0:l/(A) x / -» N such that 0(x,y,0) = x, 0(x,y, 1) = y, 9(x,x,t) = x for all tel, and ifx, y e dN, then 0(x, y, t) e dN. PROOF. By a theorem of Whitney, choose a C" structure on N compatible with its C structure and define 6 via geodesies in some C°° Riemannian metric, for which dN is totally geodesic Thus 0{x, y, t) is the point on the unique geodesic joining x and y at "time t from x." A C retraction r ^ 0 of an open set U c M into a closed submanifold B of U is a C map p:U -» B which is the identity on B.

(2.4) LEMMA (CRETRACTION LEMMA). Let M be a C manifold, B a closed C submanifold. Let A be a compact set in B,U0a neighborhood of A in M and r0:U0-* B a C retraction onto U0 n B. Then there exists a neighborhood UofB and a retrac­ tion r.U -* B such that r/U'0 =» rJU'o, U'0 a neighbourhood of A. Furthermore if B has boundary, the above holds if A c int B and V is a tubular neighborhood "with boundary'' dU,fibered by r over dB. PROOF. The tubular neighborhood theorem yields a C retraction n:U1 -» B where I/, is a neighborhood of B. Let tp be a C°° "bump function", i.e. d> is 1 on a neighborhood of A, with support of

^ 1. Let 0 be the "geodesic function" on B by the previous lemma. Then define r(x) = 7t(x) if x $ V0 and 0(n(x), r0(x); (p{x)) if x e U^ This proves (2.4). Having seen this basic extension process, we now prove a parameterized version of it in the next lemma, which we explicitly use in constructing tubular families.

(2.5) LEMMA Let T={TX) be a tubular family for W'ffl,) and T0 = ( J - n j x Let W be a C submanifold meeting T0 transversally and A be a compact set in W. Suppose that r0:U0-» W is a given continuous retraction of a neighborhood U0 of (dT0)n A (dT0 the point set boundary) which is (i) C on each T„, (ii) yeTxr\ Wn U implies r0~ *(y) <=. Tx and (iii) the family {r ~ !(y)} satisfies the Grassmannian continuity condition of(2.2)-A. Then there exists a continuous retraction r.U -* W, U a neighborhood of T^n A such that (i), (ii) and (iii) hold for r and r restricted to some neighborhood of{dT0) n A isr0. (Here T0 is the closure of T0 in M.) We indicate how we apply this crucial lemma in the construction of tubular families. We take W= W(p), peil^ period Clk = n. A fundamental domain D for/ "" on W(p) is a subset of the form closure (N — f\N)) where N is a compact ball in W(p) such that f~"{N) c int N. Then dD = SE u S„ a disjoint union of two spheres, an exterior one Sg, an interior one S, with f~*{S^) = S,. In the appli­ cations, A will be either 5 £ or D, and i will be less than k.

742 226

J.PAUS ANDS. SMALK

This will be carried out in (2.6). Now we prove (2.5). PROOF. First assumeft,-is a point p. Since the question is restricted to a neigh­ borhood of A, we may assume Wto be compact and A c int W. Let U, be an open neighborhood of dT0 n W such that 0 , r> A <=. U0n A. Then choose a C re­ traction F: t/(T0) -► 7"0 defined off (/, and such that F(W) c H^n T0. This can be done using a tubular neighborhood of T0 which extends a tubular neighbor­ hood of Wn T0 in W. Let Fx = F/7, so F x :7, -» T0 is C for each xeN, where JV, the index set for T, is a neighborhood of fl, in IV"(fl,). Then N can be chosen small enough so that for each xeN, Fx is invertible on neighborhood T0 n /4. Choose a C retraction rr 0 :T 0 n V-* T0n W with V some neighborhood of T0 n X. Then define for x e N a C retraction Jt,: 7, n P-» 7, n Wby nx = F ; ' ° jt0 o Fx. This makes sense off I/,. Also, off U, define via (2.3) a "geodesic function" 0 O for T0 n W and then let 0X be the geodesic function on Tx n IVoff U t induced by the map Fx: Tx -» T0. Now we follow the proof of (2.4) to obtain our desired retraction r. Thus let be a C" bump function which is 1 on I/,, has support in U0 and 0 g ^ 1. Define r - r0 on t/„ r(z) = *,(*) if 2 6 T.-supp * znd r{z) = djjtj, r0z, 0 (z)) otherwise. Thus r is defined in some neighborhood U of T0 n A and one checks directly that r has the desired properties. In the general case T0 = W{Cl(), proceed as above for each component of T0. This proves (2.5). (16) TUBULAR FAMILY THEOREM. Let /eDiff(M) satisfy (1.1) and let (%f) = (J Qfc, w/iere rte fi^ are the periodic orbits of CH,f) simply ordered as above. There exists a compatible, invariant system of tubular families {T*}, each 7* being a tubular family of W"(Q»). Thus Tj = W'(Clk). Moreover, the map n: (j ^ T , - . W"((lk), defined by T^T,) = x = Tj n W(ak) isC onTJp each x and I £ i < k. PROOF. We sometimes let T* be the union [j XfNkTx, restricting Nk when neces­ sary. Proceeding inductively define Tl as simply {^"(pJlpefl,}. Now assume Tl r* _ l constructed as in the statement of the theorem. Let us construct T*. For p e fij of period m, let D be a fundamental domain with 3D = SE u S;. We will define a continuous retraction JI : U{D) -* W(p) using a second induction on T'n U(D) for j = k - 1 1, such that nf =f~"n on a neighborhood. of S£, conditions (i), (ii) of (2.5) are met for each V, I £ i £ k - \ and {n~ l(y)} satisfy (2.5)-(iii). Once this is done, the desired T* is obtained as follows. Let N be the neighbor­ hood of p e f l , in W(p), with 3N = S c . For each xeN there is a unique point yeDS £ and an integer./ £ 0 so that x =/""%). We then set Tj =f~'J(n~l(y)) and Tj = W{p\ { T j } ^ is a tubular family of W(p). This follows from Lemma 1.1 of [2]. It amounts to seeing that (J ^.^Tj is open (which also follows from Hartman's Theorem) and the continuity condition (2.2)-4 is met. This continuity is also a consequence of basic known linear estimates. We now construct the retraction n:U(D)-> W{p) as required above. To give the idea, the retraction is constructed in the following order. From

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U({dT%-') n SE) it is extended to U(T%~l n SE). This induces via/"" the retraction on C/(Tg_1 n S,), which is now extended from U((T^~J) n S£) u U(T^~1 n S,) to V(1$-lnDl Then we extend it from U({dT$~2)n SE) to t/(TS _ 2 nS £ ). This induces the retraction on U((To~2)r> SE) to U{To~2 nSE). This induces the retraction on 1/(7$ ~2 n S,) and we extend it from 1/(7$ ~' n D ) u U(Tj" 2 n S£) u l/(Tj" 2 r. S,) to l/(7£~ 2 n D). We continue with Tj" 3 in the same manner. More precisely let n:Tk~l n U(SE) -» W(p) given by (2.5). For that we take in (2.5) A to be S^, W = W"(p\ 7to be Tk~ \ using the fact that 7* _ ' n Z> is com­ pact and dT0n A = 0 . Since T*~' is invariant and/'^Sj) = S„ the composition /""w defines a retraction of T*"' n {/(S,) into W"(p), denoted also by it. Here l/(S;) = /~"(l/(S £ )). Now apply again (2.5) with A = D to extend n to the desired retraction of 7*" l n U(D) into r ' " ' n D . We proceed in the same way to extend n to Tk~2 n U(D), using again (2.5) and the fact that (d7$" 2 )n D <= ( T j - 1 u Tj)n D. Continuing this process through T*~2, 7*~ 3 ,..., 7 1 yields the desired JI and hence (2.6). (2.7) REMARK. Via iteration b y / the system {Tk} obtained in (2.6) can be ex­ tended to be also/invariant Such a system will be denoted by {7* J }, white {T*-*} will denote a similar system of tubular families of the W{Qk). Tk-'{x\ Tk,u{x) will stand for the element of the family T*\ 7*" containing the point x. (2.8) REMARK. The set of diflfeomorphisms satisfying (1.1) is open in Diff(M) [2]. Let {T*'} ( {Tk-*} be as above. There exists a neighborhood N of/such that for each ge N we can construct compatible, g and g~l invariant systems of tubular families {!*/), {Tj"} of the WiOJg)), W'(nk(g)) where {T*/} = 7 k j and {7*-"} = {7* ,y }. Moreover we may assume that the maps (f>t(g) = S%J and 2(g) = Sj'", geN, are continuous, at / on compact parts of the 7**, 74>" where Sjj-5, Sj1" are the sections on the Grassman bundle of (2.2)-4 defined for T\*, TJ-". This can be checked at each stage of the proof of the Tubular Family Theorem, using the same uniform estimates on linear maps mentioned there. The tubular families that appear in the following sections are the ones given by the maps <j>x, <j)2 above. 3. Let N be a neighborhood of/ in Diff(M). For each geN, gf~l is called a perturbation of/ The perturbation is said to be localized at an open set U of M if it is 1 on M — U, where 1 stands for the identity map. The main purpose of this section is to show that perturbations of/near 1 can be decomposed into simple ones. That is, into localized perturbations leaving invariant the tubular family structure introduced in the previous section. We denote by supp/the closure of the set w h e r e / ^ l. The following lemma was detailed to us by M. Hirsch. (3.1) LEMMA. Let {Uk\ 1 < k ^ n} be an open cover ofM and N be a neighborhood of I. There exists a neighborhood Ntofl such that tffe N, then f can be factored as / = / . ° ••• °/i. where fie N, and supp/ c V\. PROOF. First choose Nx so that if/e Nx then there is an isotopy F:M x I -* M, F0 = 1 and Fx~f. Let now {A,:M -» /?|supp A, = W,,c [/,} be a partition of unity subordinate to {U,} and define p,:M -» M x I as fi( = (1, A, + ... + X,).

744 228

J.PAUS ANDS. SMAI.K

For JVj small, F is close to the projection map n:M x 1 -* M and the maps g, = F0nt are close to 1. Thus gt e Diff{Ml gt <• gfJ, e N and gfj ,(M - VJ) <= W - Wj. Here K is open, VV, c v; and K, c I/,, Since g, = g,_, on M - Wt and &, = / setting f = gi°g[-\ we have/ = / , ° ...<>/,. The lemma is proved. (3.2) LEMMA. Let fe Diff(M) satisfy (1.1). TTiere exist open neighborhoods N of f, Uk = l/»(fij and J > 0 suc/i fftof (a) {fJ(Vk)}0iJ<JA s t s m is a covering ofM. (b) /or cac/i geN,kas above and 0<j£J. fJ(Uk) c 7^' n 7>" n 7** r\ 7*", kJ k (c) T (x) and T -"(y) intersect transversally in a unique point for each x, y efJ{Uk). Here I** stands either for T)J or 1*/ and TkM for T)u or Tj1". REMARK.

We assume that N is such that 1*/ is defined by (2.8).

PROOF. 7)-\ 7}•" are tubular families of W'((lk\ W{Clk). Thus for each k we can choose Uk = Uk(Clk) so that (b) and (c) aie true for j = 0, Uk and T)', T)". Since Q(/) = \j Qk and M is compact, there exists J > 0 so that {/ ; (l/ k )} 0S j<j.is»s»i is a covering of M. Tj J , T)* being invariant (a) and (b) are true for 0 <,} £ J, fJ(0k) and T}*, T)-u. Now y 0 s j s j / W t ) b e i n 8 compact, (2.8) implies the exist­ ence of a neighborhood N as required by the lemma. This proves (3.2).

(3.3) LEMMA. Let \fJ(Uk)} he the covering of M and let N be the neighborhood off as in (3.2). IfgeN and supp gf'' c /•*( Uk) for fixed j , k then there are homeomorphisms rfyn"\M-*M so that for all \ *k i
HereV=f\Uk)Kjf>+\Uk). PROOF. Since the systems of tubular families considered here are compatible, it is enough to construct n',tf satisfying (a) to (c) above for i = k. In this case (3.2) implies that the map n':U - U, n'(x) = 7}-'(x)n ffTj-V'M), « well defined and is, in fact, a homeomorphism. The same is true for nu:V -* U, defined by n"(x) = g1JV-\x))n Tkt'(x). Since s u p p a T ' = Vff(x) = p(x) on I/. Since this is true on M - U, we have ifn'f = g on M. Thus conditions (a) to (c) of the lemma are met by n\ tf and hence (3.3) is proved.

4. In this section we prove the structural stability of diffeomorphisms satisfying (1.1). (4.1) THEOREM. Letfe Diff (M) satisfy (1.1). Then f is structurally stable. PROOF.

Let {f'(Vk)\

be the covering of M as in (3.2). If g is near/ then from

745 STRUCTURAL STABILITY THEOREMS

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(3.1) gf~l —/„<>.,. o/j, each f{ near 1 and supp/, <=fJ(UJ for some j,k where 0 ^ j < J, 1 ^ k < m. Thus from the openness of the diffeomorphisms satisfying (1.1), we have that/-°... °/, °/also satisfy (1.1). The result will follow by proving the conjugacy between /, °... ° / , ° / and / - i ° ••• °/i ° / Thus we may assume that 0 is of the form g = /°f, ff as above. But for 9 in this form, we have g = if °Tf°f>Tf and if being the homeomorphisms given by (3.2). Since if and if pre­ serves, respectively, each element of {7}**} and {T^"}, it is enough to prove that iff and fan conjugate. Let d = iff. We define a conjugacy h between/and d as follows: (4.2) h(x) = lim^md"f-'(x). This amounts to (a) /i(x) = x for x e M - (J m*0fm(Vk), (b) h(x) = d'f-"(x)f0T xe[J mi0r(Uk)*Tidf-Xx)eM - (J > i 0 / - ( l / t ) , (c) h{x) = 7*-«(x) n W"(nt( = # The continuity of h follows from its continuity on W\Cl(), i ^ k. Let us show that h is continuous on W(tlk). It is enough to do so on H^"(fl4) n t/^. Let x„ e Uk be a sequence so that xm -» x e W"^) n t / j . As we showed above /KT^x,,,)) = T*-*(xJ. Consider now the following two cases. If xM e W^ft*) t h e n / - " ( x j 6 Uk for all n £ 0. If xmt WiQ,) there exist y ^ e / " 1 ^ * ) - ^ « B S : 0 s o that f~""ixm) = ym. Thus n„ -► 00 as m -» oo and h(xm) =
746 230

J. PALIS A N D S.SMAI.K

5. We consider here flows or vector fields on M and extend the previous results to this case. Let x be the set of C vector fields on M, with the uniform C topology. This makes x into a Banach space. For each A'ejjwe denote by X, its induced flow. A fixed (critical) point of X is called hyperbolic if x is a hyperbolic fixed point for Xt„,. In a neighborhood of a closed orbit y, X is the suspension of a difleomorphism /defined in a cross section S of y through x e y. We call y hyperbolic if/has x as a hyperbolic fixed point. We now define the analogue of the diffeomorphisms satisfying (1.1) as the vector fields X such that (a) Q(X) is the union of a finite number of fixed points x,,..., xm and a finite number of closed orbits y, y„ of A". (5.1) (b) the X,-, >'y are all hyperbolic, (c) the stable and unstable manifolds of the x,, y, have transversal inter­ section. X, Ye x are called equivalent if there is a homeomorphism of M sending tra­ jectories of X into trajectories of Y. X is called structurally stable if there exists a neighborhood N of X in x s u c n t n a t each Ye N is equivalent to X. (5.2) THEOREM. If X e / satisfies (5.1) then X is structurally stable. To prove (5.2), we first indicate how to carry over to this case the corresponding concepts and results of §§2 and 3. The only novelty here is the concept of tubular families for the stable and unstable manifolds of the closed orbits of the flow. Let Xex satisfy (5.1). For the fixed points of X. we define tubular families of their stable and unstable manifolds as in (2.2). Let now y be a closed orbit of X with period n (not necessarily 3.14 ..!) and S be a cross-section of ;• at x e •/. S is called invariant if X,WK(U) c S. where U is a neighborhood of x in S. In this case we define a tubular family of W'(y) as follows. Let / : U -* S be the restriction of Xn and let {T}(y)}„v be an invariant (under/"') tubular family of Wfix). The tubular family of W'(y) is now defined by T'(X,(y)) = X,{T}(y% for >■€ U and all t. We show that in proving (5.2) we may assume X, and in fact all vector fields in some neighborhood of X, to have invariant cross-sections for its closed orbits. Let 7 be a closed orbit of X with period a, S be a cross-section of y at x e y and z e y, z ± x. From simple properties of flows near a closed orbit (e.g. [1. p. 251]), for each Vin some neighborhood N of X we get a positive C function /H V): M -» R so that (p(Y)Y)„w(U) c S and f>(Y) = I on M - V, where U is a neighborhood of x in S and V a neighborhood of z in M. Moreover, the correspondence Ye N -» p{ Y) is continuous. We now repeat the same procedure for each closed orbit of X and notice that the set of vector fields satisfying (5.1) is open in x [2], this yielding in particular a natural one-one correspondence between the fixed points and closed orbits of X and those of Y, Y near X. Thus, taking the neigh­ borhood N of X small enough, we get a continuous map fi:N->x s u c n that for each Ye N.tfY) satisfies (5.1), it is equivalent to Y with the same trajectories as Y

747 STRUCTURAL STABILITY THEOREMS

231

and its closed orbits have the same period and invariant cross-sections as the correspondent ones for (4X). The definition of a compatible, invariant system of tubular families is the same as before, the invariance being considered for the maps Xp for all t. As in (2.6) one can prove the existence of compatible, invariant systems of tubular families of the W*iXi\ W'(y} and W"(x(), W"(yj\ Again, it is enough to do so for vector fields equivalent to X and we point out that instead of the neighborhood of the fundamental domain considered in (2.6), we take here a cross-section S to the flow whose trajectories together with W(x^ (or W*(y})) form a neighborhood of

W\x) (or W'(y)). The translation to flows of the results of §3 is immediate, once we consider flows corresponding to the maps n'f and (n")~ lg in Lemma (3.3). The proof of (5.2) becomes now very much the same as that of (4.1). We just remark that when X, has no closed orbits, (5.2) provides for Y, near X, a homeomorphism that not only sends trajectories of X, into those of Y„ but it is in fact a conjugacy between the flows. This is the case for gradient flows satisfying (5.1). Using (4.1) (or (5.2) for the corresponding case for flows) and a converse known for some time, we have: (5.3) COROLLARY. Let fe Diff (M), with CUf) being finite. Then f is structurally stable if and only iff satisfies (1.1). This gives a good characterization of structural stability in the Q-finite case and leads us to the question of finding in the general case a similar characterization. Referring the reader to [5] for the concepts appearing here, we pose as a possible answer to this question the following CoNJECTURE./e Diff(M) is structurally stable if and only if/satisfies (a) AXIOM A. fi(/) is hyperbolic and the set of periodic points of/ is dense in

0(f). (b) STRONG TRANSVERSALITY CONDITION. For all x, yeOt[f\

W(x) and W*{y)

have a transversal intersection. REFERENCES 1. P. Hartman. Ordinary differential equations. Wiley, New York. 1964. 2. J. Palis, On Morse-Smalt dynamical systems. Topology (to appear). 3. S. Smalt. Morse inequalities for a dynamical system. Bull. Amer. Math. Soc. 66 (1960), 43-49. 4. , On gradient dynamical systems. Ann. of Math. (2) 74 (1961), 199-206. 5. , Differemiable dynamical systems. Bull. Amer. Math. Soc. 73 (1967). 747-817. I.M.P.A., Rio DE JANEIRO UNIVERSITY OF CALIFORNIA. BERKELEY

748

NOTES ON DIFFERENTIATE DYNAMICAL SYSTEMS STEPHEN SMALE1

A diflierentiabie dynamical system is a smooth action of a Lie group on a manifold. Specifically, this is a homomorphism of a Lie group G into the group Diff'(Af) of C diffeomorphisms of the mar'old M such that the induced evalua­ tion map G x M - t M defined by (g,x) -»d>J[x) is smooth. We shall assume throughout that Af is a compact connected C° manifold and r ^ 1. There are only two Lie groups with which we shall be concerned. Case 1. G = R. This is the classical case. The homomorphism :R -* Diff(M) defines a flow on Af. There is a natural 1-1 correspondence between flows on Af and smooth vector fields on M defined as follows. To the flow d»,: M -» M we asso­ ciate the vector field X(x) = dd>jl,x)/dt\,m0. To go the other way start with a vector field X on Af. Now a vector field on Af is nothing more than an ordinary differential equation defined on Af. Therefore, the classical existence and uniqueness theorems for differential equations produce a unique flow which "solves" X in the sense that X is the vector field assigned to the flow by the correspondence defined above. The flow is defined for all time because Af is compact. Case 1 G = Z. This is the case of discrete time. In many ways it is conceptually easier to work with than the classical case. The action of G on Af is determined by a generator of #(G) in Difl"(Af\ The problem is reduced to studying the iterates of this one diffeomorphism and its inverse. The question we are interested in dealing with is the orbit structure of . More precisely, what are the sets of the form {J(x)\geG} = 0 „ x e Af ? Ox is the orbit of x. If G = R, the orbits are homeomorphic to (1) R, or (2) S1 = R/Z: a closed orbit, or (3) point: &fixed,singular, or stationary point. If G = Z, the orbits are homeomorphic to (DZ, (2) a finite set of points: periodic points. A very simple kind of dynamical system is the gradient dynamical system. Let Af be a smooth, compact Riemannian manifold and let/: M -* R be a diflerentiable function. Df(x) e T* for each x e Af. The Riemannian structure induces a par­ ticular isomorphism between TJ(M) and TJ(Af). Using this identification, Df(x) becomes a vector field denoted — grad(/). , is the flow obtained, as above, by solving the differential equation -grad(/). 4>, has the property that/(,(*)) is a 1

Notes, by John Guckenheimer. 277

749 27K

STI-:I>III-:NSMAI.I-:

strictly decreasing function of f or a constant. It follows that there are no closed orbits of 4>r The fixed points of , arc the critical points of/ An example of a gradient dynamical system is obtained by embedding M isometrically in R" and taking/to be the projection onto the "vertical" axis. The orbits of (/>, are then the lines of steepest descent along M.

We are interested in the global geometric picture of a flow. The question arises as to how this picture changes under perturbation of the flow. We need to make precise the idea of perturbation. The set of C vector fields on a manifold M forms a Banuch space X{M) with the uniform C norm. The norm is defined as the maximum of the first r derivatives of a vector field with respect to some finite covering of A/ by coordinate systems. The norm is unique up to an equivalent norm. We give X{M) the norm topology. Given u vector field X e X(M), a pertur­ bation of A' is any element of a previously specified neighborhood of X in X{M\. In this setting, we make two definitions. DEFINITION. Suppose (.V, 4>,i (Y, '. Now consider the question as to when a gradient dynamical system is structurally stable. Observe that a topologicul equivalence between flows preserves fixed points. THEOREM (MORSE).

Any/: M -» R can he approximated by one with nondegenerate

critical points. The proof of this theorem uses transversality methods. Recall that a nondegener­ ate critical point is a critical point .v where D2J{x) defines a nondegenerate quadratic form on TX{M). Nondegenerate critical points are isolated. Therefore, we have the following

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279

THEOREM. If X = — grad/ is a structurally stable gradient dynamical system, fhas nondegenerate critical points.

We digress somewhat to consider some more properties of arbitraryflowson a manifold M. DEFINITION. Suppose x is a fixed point for the flow <pr The stable manifold W'(x) = {ye M\,: TJIM) -* TJiM) lie on the unit circle (for anyfixedt > 0). The stable and unstable manifolds will be seen to play an important role in determining whether a dynamical system is structurally stable. For a gradient dynamical system (—grad/,,\ the statement that it is a hyperbolic fixed point of , is equivalent to the statement that x is a nondegenerate critical point of/ If x is a hyperbolicfixedpoint of the flow
W'(x) and W\x) are invariant under
rV{p) = p, W(p) is a 2 cell; W\q) is a 1 cell such that W\q) u {p} is a nontrivial one cycle, W*{q) is a 1 cell such that W\q) u {r} is a nontrivial one cycle; and so on. We obtain two cellular decompositions of the torus Tas

r=

u w\x)= xtif.i.rjl

U w\X). xt{p4.r.f|

These two decompositions are essentially Poincare duals of each other.

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STEPHEN SMALE

This particular dynamical system is not structurally stable. This is plausible because W'(r) n W{q) = S1 - {r, q}. By tilting the torus slightly in the right direction, we can make W\q) pass below r and W\r) come from above q. In the tilted system W\q) n W\r) = 0 . Transversality condition for r For each x, y hyperbolic fixed points of d>„ W\x) and W\y) have only transversal intersections; that is, if p6 W\x) r\ W*(y\ the tangent planes of W'(x) and W(y) at p span Tp(M). THEOREM. (1) A structurally stable gradient dynamical system satisfies the transversal intersection condition. (2) Any gradient dynamical system can be approximated by one with nondegenerate singular points which satisfies the transversality condition. THEOREM (PALIS-SMALE). A gradient dynamical system is structurally stable if and only if the system has hyperbolic fixed points and satisfies the transversal inter­ section condition.

In this case, the topological equivalence between a structurally stable system and a perturbation of the system can be chosen to preserve the group action of R on M. This cannot be done for a structurally stable flow with a closed orbit because the period of the orbit is not invariant under perturbation. At this point we shift attention from flows to discrete dynamical systems. Most of what we say is applicable to flows via a procedure described later. Nonlinearity is essential to the example we now consider. Define a map / of the unit square I2 into the plane which has the following properties: (1) / i s a diffeomorphism onto its image, (2) f(l') n I2 is two rectangles (this is where nonlinearity loVces itseif upon us), (3)/is linear o n / " ' ( / 2 ) n / 2 , (4) / h a s a hyperbolic fixed point p. A picture of such a map might look like a horseshoe

/(/2) If W*ip)'znd W'(p) have a point of intersection other than p, then the orbit of this point of intersection lies in W(p) n W'{q). Consequently, the stable and unstable manifolds of p start to oscillate wildly.

752 281

NOTES ON DIFFERENTIABLE DYNAMICAL SYSTEMS

a

.

r*

force*

W\p)

p

By keeping track of the successive intersections of I2 with itself under iteration, wefindthat f) f-"\l2)

= Cantor set x interval,

f] f\i2)

=x Cantor set x interval,

M>0

P) f"\I2) = Cantor set which we denote A.

mcZ

All of the recurrence phenomena of/ occur on the set A. We can extend/to a diffeomorphism of S2 as follows. "Cap" the top and bottom of the unit square by attaching disks D, and D2. Denote by JD = D, u I2 uD 2 . By choosing D, and D2 large enough to contain f{I2\ we can extend / so that f(D) c D. Then extend/to map the complement of D onto the complement of/(D). The extension of/can be chosen to have one sink in D2 and one source in the complement of D. /can be chosen so that the whole geometric picture remains unchanged under C1 perturbation.

The map/| A has a particularly nice description. Consider the set of bi-infinite sequences of O's and l's; that is, \g:Z -* {0,1}}, with the compact open topology. This forms a Cantor set which we denote A2. We write elements of A2 as a = Wuz- There is a shift map a: A2 -» A2 defined by a(a) = b where a /+1 = b,.

753

282

STKPHKN SMALK

PROPOSITION. A can he given coordinates from A2 so that f\s is the shift on A2. Precisely, there is a homeomorphism h:\ -* A2 such that the diagram

A A [h [h A2-A2 commutes. This example describes a level of geometric complexity which we did not encounter in gradient dynamical systems. We show how to obtain a flow with basically the same geometric complexity as the above example. This is done by a general construction for obtaining a flow from a discrete dynamical system. Given a difleomorphism f. M -* M, we define N to be the compact manifold M x R/~ where ~ is the equivalence relation generated by (x. t) ~ (/(x), t + I). The unit vector field along R induces a vector field on N with flow ,. t has the property that <£, :M x {0} -» M x {0} is naturally equivalent t o / The periodic points of/ lie on closed orbits of/ In the horseshoe example, the periodic points of/ in A are dense in A because the periodic points of the shift in A2 are dense. It follows that there is a flow on a 3-dimensional manifold (which is an S2 bundle over S') which has an infinite number of closed orbits. Under perturbation, the flow continues to have an infinite number of closed orbits. This contrasts markedly with the behavior of a gradient dynamical system. Now we turn to the problem of finding generic properties of Diff(M) (or X(M)). A generic property is one satisfied by a Baire set (countable intersection of open, dense sets) in the space. We give a list of all known generic properties of DifP(M), r > 1. (1) The periodic points of/are hyperbolic; that is, iffm{x) = x, Df has no eigenvalues on the unit circle. (2) W\x\, W(y) have transversal intersections for each pair of periodic points .x. y of/ W'(x), W(x) are defined in an analogous manner to the stable and unstable manifolds of a fixed point for a flow. More generally, if/: M -* M is a C difleo­ morphism, we say x ~, y if/(/"(xJ,/"(.v)) -» 0 as m -» x . This defines an equiv­ alence relation on M which decomposes M into stable sets {W,}. We write W'(x) = the W\ such that x e W\. W"(x) is defined similarly by replacing/ with In the horseshoe example, the W'{x) contain horizontal lines in I2. Two recurrent themes appear in our attempt to understand dynamical systems. These are (1) the smoothness of the W\, (2) the transversality of the W't and W\ if smoothness is satisfied. Question. Are the W[ generically smooth?

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The stable manifold theorem states that W\x) is a smoothly immersed cell if x is a hyperbolic periodic point. The definition of structural stability we gave for flows has a natural analogue for discrete dynamical systems. fe Diffr(M) is structurally stable if there exists a neighborhood N(f) of/in Diffr(M) such that if g e N(f\ g is topologically conjugate t o / ; that is, there is a homeomorphism h:M -* M such that DEFINITION,

MLM i*

lh

MZ+M commutes. THEOREM. Iff is structurally stable, then (1) and (2) above are satisfied for x and y periodic. THEOREM. For gradient dynamical systems, (1) and (2) are necessary and sufficient conditions for structural stability. There exist examples of/eDiff(M) for which the {WJ}, {W$ are families of smooth, injectively immersed manifolds of constant dimension, and the two families meet transversally everywhere. Thus the stable and unstable manifolds in this case give transversal foliations of the manifold M. For example, take f0 e SL(2, Z) with eigenvalues X, fi such that \X\ < 1 < \fi\. f0 induces a map/:!" 2 -♦ T2 of the torus T2 = R2/Z2. The stable manifolds of / wrap around the torus parallel to the eigenvector belonging to X, and the unstable manifolds wrap around parallel to the eigenvector of ft. Each stable and unstable manifold is dense in the torus. The periodic points of/are dense. DEFINITION. f:M -* M is Anosov if T(M) = P + £■ where £* and £" are (continuous) vector bundles invariant under Df such that there exist constants 0 < c, 0 < >l < 1 for which

\\Dfm(xM\\
veE',

meZ,

\\Df—(xX»)fl < cA-|»|,

veP,

meZ.

Little is known about Anosov maps. We mention the following. Question. Suppose f0 e SL(n, Z) induces an Anosov toral diffeomorphism / Does/have a minimal set of dimension 1? Problem. Find all Anosov diffeomorphisms up to topological conjugacy. There are partial results with regard to this problem. Some infra-nil manifolds support Anosov diffeomorphisms and these are the only known examples. For the latest work in this direction, the reader is referred to the thesis of J. Franks. THEOREM (ANOSOV).

Anosov diffeomorphisms are structurally stable.

Using a toral Anosov diffeomorphism, we give an example of a diffeomorphism of T2 which has a neighborhood containing no structurally stable diffeomorphisms.

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STl-I'HKN SMAI.K

This example demonstrates that structurally stable systems are not dense on the torus. It follows easily that structurally stable diffeomorphisms are not dense on any manifold with the exceptions of S1 and S2. Structurally stable systems are dense on S 1 ; it is open as to whether structurally stable diffeomorphisms are dense onS 2 . We start with an Anosov diffeomorphism of the torus having a fixed point p. If we choose coordinates centered at p with the axes along the eigendirections then

(a)

(b)

/maps the rectangle (a) into the rectangle (b). Modify/by averaging with a func­ tion that is the identity outside a small neighborhood of />, leaves p fixed, and ex­ pands strongly at p. The effect of the modification is to make p a source. Two saddle points are created on the stable manifold of p. The diagram of stable and unstable manifolds is the following:

————___^__

This new diffeomorphism g we have obtained is called a DA (derived from Anosov) diffeomorphism. The unstable manifold of p is now an immersed open 2 cell in the torus. T2 = W(p) u A1 where A' is a compact, invariant dimensional set. a n d / | A'is topologically transitive. A1 is locally the product of a Cantor set and an interval. A further modification of g gives a counter example to the denseness of struc­ turally stable systems on T2. g can be chosen so that the stable manifolds of/ are not altered; that is. the stable manifolds of g are horizontal lines in our picture. Choose a small disk D about p and consider its first two images under g. Introduce

756 NOTES ON DIFFERENTIABLE DYNAMICAL SYSTEMS

285

a new saddle point q in g(D) - D so that the unstable manifold of q has a "hook" in g2(D) - g(D\ The hook is a point of tangency of W\q) with some stable mani­ fold. Note that we have not changed g outside g2(D). It follows that the stable manifolds inside g2(D)' — g(D) are still horizontal lines. The hook creates a lack of transvcrsality between a stable and unstable manifold. The stable manifolds of periodic points are dense in the torus. Thus, in some neighborhood of the diffeomorphism we have produced, those diffeomorphisms with a tangential intersection between the stable and unstable manifolds of periodic points are dense. We have exhibited an open set of Din"(T2) with no structurally stable diffeomorphisms because structural stability implies that the stable and unstable manifolds of periodic points intersect transversally. Bob Williams found this example by modifying earlier examples of mine. The question now arises as to whether there is not a weaker equivalence relation which might replace structural stability and have the property that a generic set is stable in the weaker sense. Such an attempt led to the concept of Q-stability. DEFINITION. The nonwandering set Q(f) o f / e Difl{Af) = {xeAf|for all neigh­ borhoods U of x, there is an n > 0 such that/"(I/) nU ^ 0}. An (l-conjugacy o f / t o g is a topological conjugacy o f / ^ t o / ^ ; that is, a homeomorphism h :Q(f) -* 0(g) such that

CHf)CtHf) commutes. / is tl-stable if/has a neighborhood N(f) in Diff(M) such that g e N(f) implies g is H-conjugate t o / 0 ( / ) is where the "action" of/ takes place. In this sense, Q-stability is almost as good as structural stability. The example we gave above is fl-stable but not structurally stable. Our next goal is to describe the Q-stability theorem which gives sufficient conditions for a diffeomorphism to be fi-stable.

757 286

STKPIIKNSMAI.K

DEFINITION Suppose f.M-»M is a map, A c M is closed, and /(A) = A. / i s hyperbolic on A if D/:TA(M) -♦ TA(M) is a bundle homeomorphism which splits so that TA(M) = £ * + £" with E', £" subbundles of locally constant dimen­ sion, invariant under Df. and Df contracts £* and expands £". Contraction and expansion here means that the estimates given in the definition of Anosov diffeomorphism are satisfied. Let/e DifT(M). AXIOM A. (a) fl(/) has a hyperbolic structure. (b) The periodic points off ore dense in Si. Note that for r = 1, condition (b) is generic by the Closing Lemma. Condition (a) is not generic. PROBLEM. Find a replacement for (a) which is generic and allows the construc­ tion of smooth stable manifolds. SPECTRAL DECOMPOSITION THEOREM. Iff satisfies Axiom A, then there isa canonical decomposition of SI = Q t u ... KJ Sim; the Q, are closed disjoint sets and / | Q i.v topologically t ransit ive.

For example, the spectral decomposition of the horseshoe is given by CHf) = sink u source KJ A2. The spectral decomposition of the DA is given by SHg) = sourceu A'. The whole torus is indecomposable for the Anosov diffeomorphisms on T1. Question. If/is an Anosov diffeomorphism of M, is SHf) = Af? Axiom A is not a sufficient condition for Q-stability. A phenomenon known as an fl-explosion can occur. DEFINITION. Assume / satisfies Axiom A and has spectral decomposition CHf) = fl,u ... u Si^ A cycle of JO,} is an ordered set |JJ, flj such that W" 1 ^) n W'iCl^ .,) * 0 and W{ilik) r\ W{ilh) ± 0 . The fi,, are to be distinct and k > I. No Cycle Property. J (assumed to satisfy Axiom A) has no cycles. THEOREM (PALIS).

SI- STABILITY f is Si-stable.

Iff is Si-stable and satisfies Axiom AJhas the no cycle property.

THEOREM.

Iff satisfies Axiom A and the no cycle property, then

The proof of the ft-stability theorem depends heavily on a generalized stable manifold theory. This theory stales that if A has a hyperbolic structure for/ then for all v e A, there exist smooth, injcctivcly immersed subccils W%x) of M. The W'(x) vary continuously with x e A in the C topology. If Sit is a component of the spectral decomposition of/ W'(Si,) is defined to be [j^iW'ix). The proof of the O-stability theorem proceeds by a generalized handlebody theory. A filtration of M is constructed starting at the attractors and building via stable manifolds. Very little is known about the structure of the ft, that can occur in the spectral decomposition of a diffeomorphism satisfying Axiom A.

758 NOTES ON DIFFERENTIABLE DYNAMICAL SYSTEMS

287

We end with a few questions and comments about dynamical systems. PROBLEM. Find necessary and sufficient conditions for/to be Q-stable (structur­ ally stable). An important question here is whether Q-stability implies Axiom A. Strong Transversality Condition. Assuming Axiom A, W(x) and W{y) intersect transversally for all x, y e Cl. THEOREM. Structural stability implies the Strong Transversality Condition. CONJECTURE. Axiom A + Strong Transversality Condition implies structural stability. This is true if ft is a finite set (Smale-Palis). PROBLEM. Find generic properties of DiftTM). In particular, find generic stability properties. Is the following property generic? DEFINITION. / is future stable if for all perturbations/', there exists h:M -*M which sends stable sets of/into stable sets of/'; that is, h(W\x)) = W\h{x)) for allxeM. References. A much more detailed survey of part of this material appears in S. Smale, Differentiable dynamical systems, Bull Amer. Math. Soa, 73 (1967), 747-817. This survey contains an extensive bibliography. The survey is up to date as of about a year ago. Results obtained within the past year have not yet been published, though se%craJ preprints exist. ADDENDUM. M. Shub and R. Williams have since pointed out that future stability is not generic using the example in these notes of R. Williams. UNIVERSITY OF CALIFORNIA, BERKELEY

759

THE Q-STABILITY THEOREM S. SMALE

1. We give a proof here of the ft-stability Theorem, announced with the proof sketched in [4]. The paper [4] also provides a good introduction and background for this work. We recall that if fe DifT(Af), the set of nonwandering points, fi = ft(/X is defined as the closed invariant set of x e M such that for any neigh­ borhood U of x,f"\U) n U ± 0 for some m e Z, m =jt 0. If/ g e Diff (M), they are il-conjugate if there exists an ft-conjugacy h:Q(f) -»ft(0), i.e. a homeomorphism h such that gh = />/ We assume from now on that M is a compact C°° manifold and that Diff (M) has the uniform C-topology, 0 < r < 00. Then / is il-stable if there is a neighborhood N(f) in Diff(M) such that every g e N(f) is il-conjugate to/ Recently it was discovered that ft-stable diffeomorphisms are not dense in Diff(Af) in general [1]. On the other hand a large class of diffeomorphisms are Q-stable and it is the purpose of this paper to give a proof of this fact. (Presumably, there is no difficulty in proving similar theorems for ordinary differential equations by the same method.) The main condition on fe Diff (M) is that Cl - CUf) has a hyperbolic structure. This means that the tangent bundle of M restricted to ft, T^M) splits into a sum of subbundles £* and £" as follows: £* is invariant under the derivative Df: T^M) -* T^M) and contracting in that for some c > 0 , 0 < A < 1 and all v e F , m e Z + , ||D/"(t))| <, c)T\v\ (using some Riemannian metric on M\ Finally £" is invariant under Df, and expanding for Df ot equivalently contracting for Df'1 (see [4] for details). Then/satisfies "Axiom A" if AXIOM A: (i) There is a hyperbolic structure on 0(f). (ii) The periodic points are dense in i l One can find in [4] (using the stable manifold theory appearing in [2]). (1.1) THE SPECTRAL DECOMPOSITION THEOREM. ///eDiff(Af) satisfies Axiom A,

then CUf) can be canonically written as the finite disjoint union il = Cl0 u ... u fl^ of closed invariant sets, on each of which f is topologically transitive. These ft, are called basic sets for/ Define ft,
"Z" ft,.}, W'(Clj) = {y e M|/-(y) m " V ^ft,}.

Then/satisfying Axiom A has the no cycfe property if: No CYCLE PROPERTY.Under the above conditions, ft,, ^ ft,, ^ . . . ^ ft^ ^ ft,,,

with distinct im and k > 1 is impossible. An early use of this condition in a closely related context is H. Rosenberg [3]. The goal of this paper is then to prove 289

760 290

S. SMALE

(1.2) THE Q-STABILITY THEOREM. J / / e Diff(M) satisfies Axiom A and has the no cycle property, then f is il-stable. In [4] there are plenty of examples off satisfying the conditions of the previous theorem. The proof of (1.2) actually yields an fi-conjugacy close to the identity. In his thesis, Jacob Palis proves (1.2) for the case il is finite. In [4], the statement of the Q-stability theorem is a little weaker than here. The no cycle property is replaced by AXIOM B. life Diff(Af) satisfies Axiom A with ft, as in the spectral decomposi­ tion theorem, then when H"(flj) n W^fy) 4- 0 . there are periodic points pefl„ q 6 Clj such that W*{pl W'(q) have a point of transversal intersection. These properties are related by the following proposition, proved in [4, (8.4 and 8.5)]. (1.3) PROPOSITION. Iffe Diff(M) satisfies Axiom A, then Axiom B for f implies the no cycle property. The main theorem above moves us toward the yet unsolved (1.4) PROBLEM. Find necessary and sufficient conditions ("practical condi­ tions"!) for/e Diff(M) to be ft-stable. Discussions with Bowcn, Hirsch, Palis, Pugh and Shub among others have been helpful in the preparation of this paper. We take this opportunity to remark that the example in the last paragraph of p. 763 of [4] is wrong, as has been pointed out to us by A. Borel This was to have been an Anosov diffeomorphism of a noncontractible simply-connected manifold. Thus the possibility of such a diffeomorphism remains open. The main examples in that section are unaffected. 2. We are concerned here with neighborhoods of hyperbolic sets and in par­ ticular with some applications of stable manifold theory [2]. Iff. U -* M is a diffeomorphism of an open set U of M onto its image in M let f\U\ m > 0, be defined inductively byf(f~ 1{U) n 10 and similarly for m < 0. Suppose U is an open set of M, f:U -* M is a diffeomorphism onto its image and A c l/is compact, invariant with Axiom A true for A. Then by restricting U to a smaller neighborhood of M, still denoted by U, there is a stable (and un­ stable) manifold theory for/on V [2]. A (local) stable manifold is given for each point of x e A and is denoted by W&x) and W$ = \J„A Wftx). If / is given globally then a global stable manifold W'{x) is defined as the set of y such thgt f(y) 6 W^f\x)) for sufficiently large m and this is independent of U. The unstable manifolds are defined similarly (as stable manifolds for/ - 1 ) and denoted by W\x) etc. Following [2] let W\(x) = W£(x) intersected with the e-ball about x in an appropriate metric, and W\ = Q x€A W\(x\ Then there is X < 1 such that for c small enough, f(W\(x)) c WxJif(x)\ all xe A. From now on we assume that e is this small andfixed.Clearly W'u c WJ c W$ c U, although it is not clear that the first is a proper containment.

761 THEO-STABILITY THEOREM

291

From the definitions it follows that f]k>0 Wl\ = ®i- T 0 0 8 fOT t b * "dynamical system"/: WJ -» W\, A is an attractor. Now apply Lemma (4.2) of this paper (see the remark following (4.2)). Note that this does not involve any circularity! This yields a compact subset Kof W\ with Wlkt c Vc W\, some k £ 1 and /(JO c int K We have thus constructed a "fundamental domain" V - f(V) f o r / : W ; - H ^ or for / : K-» V (outside A anyway). Let F*(A) = V-f(V) and similarly define f(A) relative to WftV). Of course there is uniqueness of F*(\). (2.1) PROPOSITION. Suppose U is an open set ofM,f: U -* M isa diffeomorphism onto its image and A c U is compact, invariant and has a hyperbolic structure. Suppose also that there exists a neighborhood U, o/f*(A) with (Closure(U. >0 /-Xl/,))) n A = 0 . TTien f/we is c > 0 a/uf a neighborhood Q of A in £/ suc/i t/iaf allm > 0 t/ien x e W^(p)/or sow p 6 A.

iff'm(x)eQfor

(12) COROLLARY. If in (2.1), [/ = M andtfW{\) = {x|/~"Xx) -» A, as m -* oo} (as in §1) (ten W"(A) = y « A JP-fx). Simi/ar/y W(A) - [j^ W(x). (13)

COROLLARY. AS

in (2.1X PU>o/"W <= *$•

The proof of (2.1) follows from (14) LEMMA. (Proof to be supplied by stable manifold theorists Hirsch and Pugh, to appear, see [2]\ For any neighborhood U, of Closure F», U->o/"( t ; i) u ^(A) contains a neighborhood QofA. 3. We now concern ourselves with neighborhoods of the "bask sets" ft, of §1. As we have shown in [4], these bask sets possess a "local product structure" which implies that for suitable choice of U = (/(ft,) each Wfl(x\ W$y) intersect in at most one point and ft, consists precisely of these points of intersection. Thus H£(ft,) n H^(ftj) = ft, and so by (12) there is a neighborhood Q of ft, in M such that iff"\x) e Q all m e Z, then x e ft,. Furthermore from §7 of [4], it follows that W"(ft,) n H^ft,) c ft and thus ^ft,)nW"(ft,) = n,. We have the following proposition: (3.1) PROPOSITION. Let ft, be a basic set offe Diff(M). Then there are neighbor­ hoods U = U(fy of il,, N(f) off in Diff(Af) such that ifgeN(f), and g satiates the condition of(2.1) then there is a conjugacy h :ft, -» A, where A = {x e U\g*(x) e V all me Z}. Thus h is a homeomorphism with gh\x) = hf(x) for all x e ft,. After §1 (3.1) becomes essentially a consequence of the stable manifold theory [2]. From [2], in fact, for g close enough t o / it follows that there is a canonical continuous correspondence between the local stable manifolds of/relative U and those of g relative U. In fact W$(f) moves continuously with / and similarly Q

762 292

S. SMAI.li

of (2.4). The stable, unstable manifolds of g induce a hyperbolic structure on their intersections, which is A by §2. In this way the correspondence between the local stable manifolds of/and those of g induces a homcomorphism h:£l,-+ A. This proves (3.1). One can also use the original argument in [4] in conjunction with §2 to prove (3.1). 4. The goal of this section is to prove the following (4.1) LEMMA. Suppose Q, is a basic set for fe Diff(M) (satisfying Axiom A), U a neighborhood of il, and X is a compact subset of M such that f(X) c int X and

[jm>0f-m(X)ijnlz3w(n,). Then there is m > 0, compact Yc M withf(Y) c int Yand fl, c int Y-f c U.

"(X)

We will need the following (4.2) LEMMA. Suppose Q is a compact neighborhood of a subset P of a compact manifold M with fe Diff (M) satisfying C\mi0fm(Q) = p- lhen tnere is a compact neighborhood VofP in Q such that f(V) c int V. REMARK.

The proof is valid for/a homcomorphism of a locally compact space

M, etc. Let A, = f)ra* 2 of m (Q\ Then Ax => A2 o ... is a decreasing sequence of sets whose intersection is P. Choose r such that Arc Q and lei U = Q r^f(Q) n ... nf'-'iQ). Then/(L') c U and f l - i o / l ^ ) = p- Since/'(U) is a decreasing sequence of sets converging to P, there is an n > 2 with/"(I/) c interior U (U is compact). Now choose W compact such that V c int W and f'Wez int U. Let £ = U v(Wnf'~l W). We claim that/"" '(£) <=■ interior £, in which case we are finished by downward induction on n. For this first note PROOF.

(1)

/ " " l ( t / ) c l / c i n t / " - ' ( W ) c (int W ) n i n t / " - ' ( W ) c i n t £

and noting n > 2, (2) f*~ l(Wnf»-

l

(W)) c f!"- 2W = f - 2fmWc: f-2

\n\ U c int V c int £.

This finishes the proof of (4.2). Now turning directly to the proof of (4.1 \ we may assume the V there compact and that n 4 i o / ' ( t / ) c W«(fi,). For x e fl„ r. > 0, define Wt{x) as the set of g e W(x) of distance less than r. from x measured in the intrinsic metric in W'(x). Let H^(Q,) = [jxefli Wt(x) and for £ > 6 > 0 define

ru:,s) = r=w:(nl)-wucil).

763 THEO-ST ABILITY THEOREM

293

Then for e small enough, F" will compact with \Jk> <,/*(?*)c U> an ^ if <5 is small enough, for every xe W\Ci^ there will be keZ such that f\x)eF". One can think of F" as being an enlarged fundamental domain of/on W\tl,). Furthermore since F* is compact and in U*>o/~*(Mi-i) we can choose m > 0 such that f—{X) => f. Let P = JHIUu ( ( V o / W and Q = U \jf-\X). Then H^oAC) = * Choose compact K by (4.2) so that Q => K=> P, /(V) c int V and let V = V Kjf~m(X). Then clearly Ysatisfies (4.1). 5. We apply (4.1) to the situation at hand in thefi-stabilitytheorem (1.2). Thus we assume ft„ / = 0, ...,k are given by the spectral decomposition (1.1) for feDiB(M) satisfying Axiom A. We are assuming also the no cycle property. Choose a simple ordering < on the ft, using indices such that ft,, < ft, < Qj < ... o/~"(t/<) c WW (5.1) PROPOSITION. With the ft,, Ut as above there are compact sets M0 <= M, c ... c M, = M and* positive integers m„ .... m» wif/i fnese properties: f"*(M,-,)

<= int A#„

ft, c int Af, -/--«(M,_,) c l/„

/(M,) c int Af„ and M, c U, s l W«(ft,).

PROOF. First since W"(ft0) n H^ft,) = 0 for i > 0, W"^) = fto- (We recall that M = |J?. 0 VT{Qi and that W"(ft,)n W"(ft,) = ft,). Apply (4.1) with A#,_, = 0 , to obtain Af0 as needed. Now suppose M,_, has been constructed as described above. To apply (4.1) note that we need only to check (J r > 0 /"Wi-1) u ft, => W"(ft,).

But IHO,) = U?-«(»nni) n w(ft,)) = (J.s/^'W ° ^ T O b y t h e no c y c,e

assumption. We get our desired condition from this and the obvious fact that U->o/""W<-,) => W"(ft,). each i < I. Then (4.1) yields an M, which we claim satisfies the above claimed properties of (5.1). All this is immediate except for M, cz (J(s,W*(ft,). This property follows from an induction on / and the choice of Ut. 6. We give here thefinalconstruction of the conjugacy, proving (1.2). From the definition of nonwandering points it follows using the notation and construction of the previous section that there is an e > 0 with this property. If aeDiff(M) is C° within e off then ftfo) c (J?.0int M, -/~"' , (M i _ 1 ); we write ftfc) = Uf-oA/. A, = ft(gf)n int M, -/~""(M,-i). A, of course being compact and invariant Since A, c [/, (3.1) yields the desired conjugacy, with a possible further shrinking of c That is, the global n:ft(/)-*fl(0) is defined for each l, ft,-A, by (3.1). 7. We add a remark to the main result The proof of (1.2) yields also:

764 294

S. SMAI.K

(7.1) PROPOSITION. Diffeomorphisms satisfying Axiom A and the no cycle property form an open set in Diff(M). Furthermore, sufficiently small perturbations g of such fe Diff(Af) admit the same compatible simple ordering on the Ufa) as on Sl/Lf). We emphasize that g and/will not necessarily have the same relations defined by <, on the 12, as Example 3 in the next section shows. It is easily shown that Axiom B on /would imply that il^f) -* SI jig) preserves £ relation. 8. We give here a heuristic account of 3 examples which shed some light on the main theorem (see [1] for example, or [5] for the type of mathematics needed to make these rigorous). 1. This example shows that Axiom B is actually stronger than the no cycle property. It also shows that among diffeomorphisms satisfying Axiom A, Axiom B is not a generic property. EXAMPLE

Idim

W'(Q2)

WW2)

FIGURE I

In a four-dimensional manifold fi3 is a fixed point of saddle type with a threedimensional stable manifold and one-dimensional unstable manifold. Then Cl2 will be a two-dimensional toral diffeomorphism (see [4, §3]) and a neighborhood

765 THF.O-STABIMTY THF.OREM

FIGURE 2a

FIGURE 2b

293

766 296

S. SMAI.I-:

in MA will be a product with a two-dimensional saddle point. Thus generically W"(n3) will meet 3-dimensional W'{Q2) in a point locally. This picture can be completed so that Axiom A and the no cycle property will be satisfied. On the other hand Axiom B must fail because for any pe Cl2< W*(p) has dimension 2, and so it can't have a point of transversal intersection with W(il3). EXAMPLE 2. This is an example of a two-dimensional difTeomorphism which permits an Q-explosion. It shows why Axiom A itself does not guarantee fi-stability. Here in Figure 2a the circles represent fixed points which are sinks or sources while p, q, r are fixed saddle points with degeneracy along their stable manifolds. That is, W\p) meets Wx(q) on an arc, etc., as indicated. Axiom A is satisfied with ft a set of fixed points. Now we make a small perturbation to obtain homoclinic points as in the following picture (Figure 2b). For the perturbed difTeomorphism, Q is an infinite set, far removed from the original Q. EXAMPLE 3. We show now how under the conditions of (1.2), perturbations may destroy the ^ relation. Take a Riemannian manifold M,
767 THEQ-STABILITY THEOREM

297

REFERENCES

1. R. Abraham and S. Smale, Nongenericity ofd-stability, these Proceedings, vol. 14. 2. M. Hinch and C. Pugh, Stable manifolds and hyperbolic sets, these Proceedings, vol. 14. 3. H. Rosenberg, A generalization of Morse-Smale inequalities, Bull. Amer. Math. Soc. 79 (1964), 422-427. 4. S. Smale, Differentiate dynamical systems. Bull. Amer. Math. Soc. 73 (1967), 747-817. 5. , Structurally stable systems are not dense, Amer. J. Math. M (1966), 491-496. UNIVERSITY OF CALIFORNIA, BERKELEY

768 Seminaire BOURBAKI

374-01

22e annee, 1969/70, n" 374

Fevrier 1970

STABILITY AND GENERICITY IN DYNAMICAL SYSTEMS par Stephen SMALE

A general reference to this subject, with examples, written about the summer of 1967 is [7], (reported in a recent Bourbaki Seminar by C. Godbillon). Here I will try to emphasize developements since. An important source of much of this more recent work should appear in the immediate future [1].

For simplicity we restrict ourselves to a dynamical system defined by a diffeomorphism

f

of a compact manifold

M

into itself. This is the case of a

discrete differentiable dynamical system with time represented by the number of times

f

is iterated or the

n

in

f

Most of the results discussed here are

valid also in the case of a dynamical system defined by a 1 st order ordinary dif­ ferential equation.

The space of all dynamical systems will be denoted by by putting the

C

Dyn(M) , topologized

uniform topology on the corresponding diffeomorphism

f ,

1 £ r £ a> . The study of the dynamical system is the study of the orbits 0(x) = {f"(x) | n 6 Z}

of

f

especially from the global point of view. Thus a

natural equivalence relation is topological conjugacy, i.e., topologically conjugate if there is a homeomorphism fh - hg . Clearly such an

h

sends the orbits of

f , g 6 Dyn(M)

h : M -» M f

such that

onto the orbits of

g .

More than 10 years ago I posed the problem of finding a dense open set (or at least a Baire set) of

Dyn(M)

such that the elements of

are

U

U

could somehow

769 374-02 be described qualitatively by discrete numerical and algebraic invariants. Since this problem has been often quoted since, I would like to take this opportunity to revise the problem in the light of what has been learned in these 10 years.

The problem posed in this way is too simple, too rough and too centralized. I believe now that the main problems of dynamical systems can't be unified so elegantly. The above problem however can be split apart so that it makes good sense and in my opinion gives some perspective to the subject. This goes as follows.

One should search for a sequence of subsets U. C U

C ... c U, c Dyn(M) , k

Baire subset of an open set) with U.

is that as

i

increases, U.

mical system, but as of

U.

i

U.

not too large, U. U

dense in

of

Dyn(M) ,

open (or at least say a

Dyn(M) . One main feature of the

includes a substantially bigger class of dyna­

decreases one has a deeper understanding and the elements

have greater regularity (or stability properties). So

U.

should consist

of the simplest best-behaved non-trivial class of dynamical systems, and U. cannot be expected to have very strong stability properties at all.

To give a better idea of what I am saying, I will give a schema of the

U.

now which to some extent illustrates our state of knowledge of dynamical systems. (There will always be some arbitrariness in the exact choice of the first state briefly the choice of the

U.'s

U. .) We will

and the rest of the talk will give

some justification for our choice, defining the necessary terms as we proceed. In each of the following ding

U

U

there is a large class of examples not in the precee-

. The reader may consult the literature cited for many of these.

770 374-03 U

= [f € Dyn(M) | f

satisfies Axiom A, 0(f)

is finite and

f

satisfies the

transversality condition} U„ = {f € Dyn(M) | f

satisfies Axiom A and the strong transversality condition}

U, = {f € Dyn(M) | f

satisfies Axiom A and the no cycle condition}

U. = (f £ Dyn(M) | f

satisfies the main known generic properties}

We have used the unifying language of Axiom A which can be stated as follows : f € Dyn(M)

satisfies Axiom A if the non-wandering set

structure and the periodic points of the closed invariant set of is some

n > 0

with

x e M

f

tracting on

and

T „ ( M ) of

M

of

is

x

there

D(f

)

is contracting on

restricted to n , Df , such that

E

. Finally

said to be contracting if given a Riemannian metric on 0 < v. < 1

U

0

f (U) f"l U / $ . A hyperbolic structure on fl is a conti­

, invariant under the derivative,

E

has a hyperbolic

are dense in 0 . We recall that

such that for any neighborhood

nuous splitting of the tangent bundle Tn(M) = E U + E

Ci = 0(f)

with HDf^x) (V)|| S ciim||v||

for all

Df

Df : E

M , there is

is con­ -> E

is

c > 0 , p, ,

v e Es .

A generic property is a property that is true for some Baire set in

Dyn(M) .

A basic notion for the study of dynamical systems is the notion of stable ma­ nifold. Given

f € Dyn(M)

m

d(f (x) , f^y)) lence class of

-» 0 x

as

and some fixed metric on

M , we say that

x~

y

if

m -» OD . This is an equivalence relation ; the equiva­

is denoted by

W (x)

and called the stable manifold of

x .

The following theorem is a consequence of the work of a number of mathematicians, see especially [1], [5]. THEOREM 1 .- I£ f e Dyn(M)

satisfies Axiom A, than for each

a smoothly, injectively immersed open cell.

x 6 M , W S (x) i_s

771 374-04 It is an outstanding question as to whether the conclusion of Theorem 1 is a generic property.

The unstable manifolds of ted by

f

are the stable manifolds of

f~

and are deno­

Wu(x) .

With this behind us consider the dynamical systems in implies that

Q

saying that if

consists of the periodic points of x e C3 has period

of absolute value then W (x) and

m , then

Cl is finite

f , and Axiom A amounts to

Df (x) : T

1 . The transversality condition of W (x) meet transversally at

U. . That

-» T U.

has no eigenvalues

means that if

x e M ,

x . (Stated in this manner, this

transversality condition coincides with the strong transversality condition cf

If

f € U. , then

f € Dyn(M)

f

has indeed very strong stability properties. Say that

is structurally stable if it possesses a neighborhood of diffeomor-

phisms, each topologically conjugate to THEOREM 2 (Palis-Smale [1]).- I£

f .

f e Dyn(M) with

structurally stable if and only if

0(f)

finite, then

f

f 6 U. .

It was known for sometime via gradient dynamical systems that for any U.

is not empty and more recently

i£

U.

M ,

was shown to be open [6].

Structural Stability is a very strong regularity property (now known to be not generic) and largely via the preceeding theorem,

U.

can be considered to

consist of very well understood dynamical systems of relatively simple character. On the other hand, the proof of this theorem is not altogether simple because of

772 374-05 the fact that structurally stable is such a strong (and subtle) property. To define the remaining terms used in decribing satisfying Axiom A, W (x) and W u (x)

U

, say that for

f e Dyn(M)

f has the strong transversality property if for any

meet transversally at

x e M ,

x .

It has been conjectured that a necessary and sufficient condition for f e Dyn(M) exists

to be structurally stable is that

f e U„ , f / U.

ted example of intersect

f e U

. It is known that there

which is structurally stable. Among the rather complica­

f ^ U. , some have the property that a neighborhood doesn't even

U. because of lacking the strong transversality property. Also for

f

satisfying Axiom A, it can be seen that the strong transversality property is ne­ cessary for

f

to be structurally stable. Via this route it was first found that

structurally stable was not a generic property. Proving that U„ = structurally stable dynamical systems would cement

U„

into our hierarchy.

We pass to U, . To understand the no cycle property, we recall the spectral decomposition theorem which states that if

f

satisfies Axiom A, then 0(f) can

be written canonically as the finite union of closed invariant disjoint subsets n. ,... ,CL

on each of which

Define Vs(n±)

=

f has a dense orbit.

U W s (x) and W u (0 i ) = U Vfu(x) xefl. xen. i

A cycle is a sequence of distinct W5^)

0 ""(n^,) / jt , i = 2

i

Ci.,... ,CL , k > 1 , with the property k , and Vs(f!1) D W ^ ) /ft

. Then

f

(suppo­

sed to satisfy Axiom A) has the no cycle property if there are no cycles. Dynami­ cal systems in U-

have the important regularity property known as fi-stability

which is defined as follows. First say that

f , g 6 Dyn(M) are O-conjugate or

773 374-06 conjugate on 0 hf(x) = gh(x) N

in

if there is a homeomorphism for all

x e 0(f) . Then

f

h : 0(f) -» 0(g)

is O-stable if it has a neighborhood

Dyn(M) of diffeomorphisms which are O-conjugate to

rally stable implies

THEOREM 3 (The

such that

f . Clearly structu­

0-stable.

O-stability Theorem).- I£ f € U

, then

f

is_ Q-stable.

The converse of Theorem 3 is an open problem. More generally one can ask what stability properties of dynamical systems are valid outside of even some dynamical system not in

U.

U, . Can

be structurally stable ? Another version

of these questions is : Does structural stability imply Axiom A ?

or even does

O-stability imply Axiom A ? My feeling is that the questions of this paragraph are very hard and important to settle.

Some other regularity properties are true of define the "zeta function" points of

f

£(f)

£ N t n= 1 n

f € U. . For example one can

where

. It is an open question whether

N

is the number of fixed

n

{(f) having a positive radius of

convergence is a generic property. But on the other hand, the following theorem was very recently proved by J. Guckenheimer [4]. THEOREM 4.- Lf

f 6 U, , then

C(f)

not only has a positive radius of convergence,

but it is a rational function whose zeros and poles are algebraic numbers.

R. Bowen [1], [2~\ , [3] has studied dynamical systems satisfying Axiom A in the direction of ergodic theory and has obtained the following rather striking results.

774 374-07 THEOREM 5.- Let

f

satisfy Axiom A and

Q.

be one of the subsets given by the

spectral decomposition theorem. Then there exists an invariant ergodic measure uf

on

Q. , positive on open sets, zero on points (unless fi. is finite) which

is the unique invariant normalized Borel measure on

Q.

maximizing entropy. The

(measure theoretic) entropy coincides with the topological entropy and this entropy is equal to the log of the radius of convergence of the zeta function of

f .

Also Bowen gets good information on the distribution of periodic points in Ci.

and shows that

(f|n

• ^c)

*s

a

I-automorphism in the " C-dense " case, a

i mild condition which is met for example in the case fl. is connected. I

Let me emphasize again that indeed these last theorems cover situations which are very rich in examples and that I am not giving them here.

What happens outside of of examples outside of U

U,

U, ? At the present time, there are a large number

whose import is that one cannot obtain any dense open

c: Dyn(M) with very strong regularity or stability properties. Some of them

are as follows : Abraham-Smale [1] show that fl-stability and Axiom A are not generic proper­ ties. Shub has an example of an open set in

Dyn(M)

where

Ci = M , where Axiom A

and fl-stability fail. Newhouse [1] shows that if r > 1 , with the C topology, 2 even on S , Axiom A and O-stability are not generic properties. The earlier examples, with

r

arbitrary in the range

1 £ r £

m

, where on higher dimensional

manifolds. C. Simon has recently shown that the zeta function being rational is

775 374-08 not a generic property.

Under perturbation, some features of these examples seem to be preserved. On the other hand the above examples and others need much study to get some understan­ ding of the area beyond

U, .

As far as I know, there has been not much progress in the finding of new generic properties (or study of

U. ) since [7] was written.

REFERENCES

[1]

"Global Analysis", Proceedings of the 1968 AMS Institute on global Analy­ sis at Berkeley. Vol. 1 (of 3 vols) is devoted to dynamical systems.

£2]

R. BOWEN - Periodic points and measures for Axiom A diffeomorphisms, to appear Trans. AMS.

[3]

R. BOWEN - Markov partitions for Axiom A diffeomorphisms, to appear.

[4]

J. GUCKENHEIMER - Axiom A + No cycles »

[5]

M. HIRSCH, J. PALIS, C. PUGH, M. SHUB - Neighborhoods of hyperbolic sets,

Q£{t)

rational, to appear.

to appear, Inventiones Math. [6]

J. PALIS - On Morse-Smale dynamical systems, Topology, 8 (1969), 385405.

[7]

S. SMALE - Differentiable Dynamical Systems, Bull. AMS, 73 (1967), 747817.

776

Beyond hyperbolicity By M. SHUB and S. SMALE

The goal here is to study properties of (discrete) dynamical systems not possessing the frequently analyzed properties of structural stability or hyper­ bolicity (e.g., Axiom A). In particular, we relate the two a priori disparate notions of filtrations and Q-explosions. The context is a compact C manifold M and Diff (M), Cr diffeomorphisms, C topology, 0 ^ r ^ oo, r fixed throughout. Because of Lemma 4, we need dim M > 2, although presumably more work could remove this assumption, at least from the main theorem. We recall that a filtration for / is a finite, ordered collection {Ma}, a = 1, •••, n, where each Ma is a submanifold with boundary of M, dimension Ma = dimension M, with int Ma z> Ma-u and f(Ma) c int Ma> each a. We also suppose that M,. — $, Mn = M. A filtration has two obvious but important properties: (1) stability under even C perturbations, i.e., if {Ma} is a filtration for / , then there is a neigh­ borhood N(f) of / in Diff (M), C topology, such that if g e N(f), then {Ma} is a filtration for g; (2) gives a decomposition of the nonwandering set Q=Q(f). More precisely, recall that Q is the closed invariant subset of points x of M with property, given any neighborhood U of *, there is m > 0 such that /"(IT) D U # 4>. If {Ma} is a filtration for /, then let Qa = (Ma - Ma^) n Q. Then the Qa give a finite, disjoint decomposition of Q into compact invariant subsets. Filtrations exist for any diffeomorphism, e.g., take Mi = <j>, Mt = M. An additional useful property that a filtration might possess and prevent this sort of triviality is as follows. Let Aa = f\mezfm(Ma — Af0_i). Note that Aa is compact and contained in the interior of Ma — Ma^. A„ is the maximal invariant set for fonMa — Af„_!. Clearly, Aa "D Qa for each a. We say that {Ma} is a fine filtration if A, = Qa for each a. We make a detour by stating some problems related to fine filtrations. Problem (1): Let / : D* —♦ D% be a diffeomorphism of the n-disk into itself (even for n = 2) such that A(/) = fl(/) is true for/as well as C perturbations (A(/) = n«>o/"(-D")). Is Q(/) a point? Clearly, Q(/) has the Cech cohomology of a point.

777 588

M. SHUB AND S. SMALE

Problem (2): For a fine filtration {Ma} of / , Mi = f, Qx = #, and we have (Cech theory) H*(Qt) = limit {H*(Mt) -£-» H*(Mt)}. sion of this statement for the other Q„?

Can one find an exten­

General conditions are known for the existence of fine nitrations. Axiom A and the no-cycle condition (there is no need to recall the definition here) imply the existence of a fine filtration and this is a major step in the proof of the Q-stability theorem. For a general reference for this whole subject, see the survey [1], Fine filtrations exist under more general conditions than Axiom A, and it is a subtle question as to whether their existence is a generic property. The answer turns out to be negative by a remarkable, yet un­ published example of S. Newhouse concerning a set of diffeomorphisms of the 2-sphere. This example motivated us to define a fine sequence of filtrations. This is a sequence of filtrations, indexed by k = 1, 2, • • ♦, {Af£}, a = 1, • • •, nt, such that (1) {Afa} refines {Af,*-1}, each h > 1, i.e., for each a, Af * — Af,*-, is con­ tained in Af *-1 — Aft,1 for some /3; and (2) n*>o A* = Q, where A* = (J« ^ a n d Aj is defined for each Jc as before. Then a fine filtration is a fine sequence, constant in k, and it can be shown that Newhouse's example has a fine sequence. On the other hand, a fine sequence of filtrations gives an approximation of a fine filtration and is the best that can be hoped for in general by Newhouse's example. It isn't known if possession of a fine sequence is a generic property for C" diffeomorphism. We say that / in Diff (M) does not permit C° Q-explosions if, given an open neighborhood U(Q(f)), there is a neighborhood N(f) in Diff (Af), C topology such that Q(g) c U(Q), any g in N(f). If / has a fine sequence of filtrations, then it doesn't permit C Q-explosions. This is seen as follows. We are given a fine sequence for / and also U(Q). Choose A: such that U(Q) z> A* z> Q. Now, fix k. For each a, 1 < a g nk, we can find m, q > 0 such that Qa c A*, c /"(Af*) - /"'(Af£-d c U(Q). In fact, we may suppose the same m, q work for all a. The last inclusion will be true for g in a sufficiently small C° neighborhood of / for all a. This implies the assertion. The main goal of this paper is to prove the converse, so: THEOREM. A diffeomorphism f possesses a fine sequence of filtrations if and only if f does not permit C Q-explosions.

Toward proving this theorem, we define an open decomposition for / to

778 BEYOND HYPERBOLICITY

589

be a finite number of open sets Wtt in M, with disjoint closures and such that U« Wm ^ Q . Define Wa = {xeM\ f~m(x) e Wa for some m ^ 0}. Then UaW'a = M and each Wl is open. Say that W, ^ Wa if TFa n TPJ * *. An r-cycle is a set of W« with W8l ^ Wfl, ^ • • • ^ W„r+1 for r > 1, and a 1-cycZe is a W„ with some * $ Wa, m, q > 0, and /"(*), /"«(*) e W„. Then {Wa} has the no-cycle property if there are no r-cycles, r > 0. Let A, = fl-ez fm(WJ. 1. Let {Wa}a*A be an open decomposition for / e Diff (Af) with the no-cycle property and each A, compact. Then there is a filtration {Ma} such that A„ c Af„ — M„_i for each a. Remarks. (1) The ordering of the filtration will be compatible with the relation ^ on the Wa. (2) One canfindsuch an {M„} (perhaps with a bigger indexing set) which refines any given filtration. For the proof, choose a simple ordering on A compatible with ^ and let Ml, = the closure of \Jfs,a W}. This almost does it since A, c Wa c M'a — Af£_t and f(Ma) c Af«. However, f{M'a) is not necessarily in the interior of Mi, (int Ma), and the proof must be a little more elaborate. We proceed inductively to define Ma as follows. Aft = <j> and let Nt = Wt. Then fUso/"(#,) = A,c Nt. Here # , is Cl JV„ or the closure of Nt, and we have used the no-cycle property. Then, since A, is compact, (cf. Lemma 4.2 of [2]), there is a compact neighborhood P, of A, contained in JV, with /(P2) c int (Pt). Finally, choose a compact manifold neighborhood Af, of P, in iV, with /(Af,) c int Af, to complete the first step of the inductive process. The next step begins by letting N, = Af, U Wf. Then, using the no-cycle property, it follows that H«>o fm(Ns) = W"(A,) U A,, where JF"(A,) = {xeM\fm(x) »A,asm > -«>} LEMMA

(note W*(At) = A,). Furthermore, W"(AS) U A, is compact since dW; c A, (here dWt = Cl W? — IF?). Then Af, is constructed from Nt just as in the previous step. The general induction step proceeds similarly, with Nk = Wl U Afj.,, r\mi*fM(Nk) = \JJikWn(ks), and TF«(At)cU/<» W(A,). This proves Lemma 1. LEMMA 2. Let /eDiff(Af) no£ permit any C ^-explosions and a neigh­ borhood J7(Q) be given. Then there is an open decomposition {Wa}aeA with the no-cycle property, compact A„ and \Ja Wa c U{Q). Lemmas 1 and 2, together with Remark (2) after Lemma 1, yield the theorem.

779 590

M. SHUB AND S. SMALE

For the proof of Lemma 2, suppose M has a metric coming from a Riemannian metric and this metric induces a C° metric on Diff (Af). Choose 8 > 0 so that if the C° distance from g e Diff (Af) to / is less than 8, then fi(flr) c U(Q). Choose a finite covering of Q of open convex balls Bit each Bid U(Q), diameter B{ < 8. Let Uf denote the components of U Bf. By shrinking the Bi a bit, we may suppose the Up to have disjoint closure and be finite in number. Say Ua is equivalent to Uf if there is a common cycle containing Uf and Ua and let Wa be the union of members of an equivalence class. Then {Wa} is an open decomposition with no r-cycles, r > 1. Let W'a = {x e M\fm(x) e Wa, someTOS> 0} and Va = W% n JVJ. Then the finite number of F„ are open, U« F„z>Q, and {V„}a has no cycles. Any xeClA„ — A<, leads to a 1-cycle since /"(a;) e Cl A«, all TO. This implies that A* is compact, each a. Finally, one can shrink the Va a bit if necessary to insure that their closures are disjoint. It remains to prove that each Va is contained in J7(Q). The idea of the proof is to create an Q-explosion by taking x e Va — U(Q) and perturbing / by less than 8 to make x a periodic point. To this end, define a chain of 8 balls between y, z for any fe Diff (M) to be a sequence Uit •••, U% of convex open balls of diameter <8, ye Uv zeUH, Ui D Q ¥= $, and for each i = 1, • • •, n — 1, there exists m ^ 0 such that fm(Ui) r\Ui+l* 4>. 3. Given a chain of 8-balls between y, z, there is a geDiff (Af), with the C distance between f, g less than 8, g"(f~l(y)) = f(z), some N>0 and g = f outside (J"=i U(. LEMMA

Postponing the proof of Lemma 3 for a moment, we see how it finishes the proof of Lemma 2 and hence of the theorem. Suppose xeVa, x& U(Q). Since x e Wl n W'a, there exists TO, q > 0 such that y = f(x) e W„, /"*(«) = z e W a . We suppose m, q minimal with this property. Then there is a chain of £-balls between y and z of the B{ used in constructing W. Application of Lemma 3 yields a g having x as a periodic point, contradicting our choice of 8. Part of the idea of Lemma 3 is in the following. Here is where the hypothesis dim Af > 2 is used. LEMMA 4. Let (qi} p^, i — 1, • • •, I be pairs of points on a compact mani­ fold M, all disjoint such that d(qit p%) < 8. Then there is a diffeomorphism t]\M—*M within C distance 8 of the identity such that 7}(qi) = pf for each i.

780

BEYOND HYPERBOLICITY

591

For the proof, one uses disjoint arcs at joining g, to p{, each i and applies a standard theorem from differential topology obtaining TJ whose support is in the disjoint cell neighborhoods of each arc a(. We proceed to the proof of Lemma 3. Let yt e fni(U{) D U(+l, i = 1, • • •, n - 1, where »« ^ 0 is the smallest possible with nonempty intersection. Let z< = f~ni(Vi) and Wi e Uit U ^ 0 be chosen such that /''(w<) e U(. This is always possible since U{ D Q =£ ^. Now by changing / a little, if necessary, we may assume that the points f~\v), V, *, /(«), Vi, ** and /'"(«><), 0 ^ i ^ Z«, are all distinct. Our goal is to perturb / t o g which maps 3/—*wl~*z^ —►«— ;, ►«— , ► • • •z%..i—*z under positive iterates. For this purpose, we consider the pairs of points {(a, $)} = (ylt wj; (f'(Wi), /'(«><)) for 0 < 3 < l{; (/'<(«;«), z<); (i/«, *>«+,), (wif z), and finally, (/(«), /(«)). By Lemma 4, there is a diffeomorphism TJ with support in U«-i.—.« Ui so that 57(a) = ft, 7) has C° size less than d and the support of V c U<-i,-••.« Ut. 7] ° f is the diffeomorphism we were seeking. V o /(/_1(2/)) = Wx? J? o /(«>,) = f(wd) and 7 ° f(fu~lwi)

= *i?

7 ° /(«i) = «V

etc. UNIVERSITY OF CALIFORNIA AT SANTA CRUZ AND BERKELEY REFERENCES

[ 1 ] M. SHUB, Stability and Genericity for Diffeomorphism*, to appear in the Proceedings of the 1971 Brazil Symposium on Dynamical Systems. [ 2 ] S. SHALE, The Q-stability Theorem, Proceedings of the Symposium in Pure Mathematics XIV, Global Analysis, AMS, Providence, R.I., 1970. (Received February 29, 1972)

781

Reprinted from: DYNAMICAL SYSTEMS

©iffj ACADEMIC KESS. INC. NfW YOK AMD LONDON

Stability and Isotopy in Discrete Dynamical Systems* STEVE SMALE Department of Mathematics University of California at Berkeley Berkeley, California

We want to show here that every diffeomorphism of a compact manifold is differentiably isotopic to one that is ^-stable as a discrete dynamical system. I do not recall where this problem first came up exactly, but in Rio de Janeiro, in the month before the 1971 Salvador conference on dynamical systems, Mike Shub and Bob Williams were both discussing this and related problems very much. Shub, in fact, raised the question in a talk at IMPA as to whether a particular diffeomorphism of a 4-torus was homotopic (or isotopic) to one satisfying Axiom A, to an £?-stable one, or to a structurally stable one. He asked the same questions for diffeomorphisms of 2-manifolds or even arbitrary diffeomorphisms. These questions are answered now. It was on the beaches of Rio de Janeiro on one of those afternoons that I has just left a discussion with Mike Shub on this subject when I began to see how by simply using handlebody theory (from those very same beaches 11 years ago ) one has a good picture of how to do this kind of problem. Needless to say, this note would never have been written without the provocative questions and discussions of and by Mike Shub and Bob Williams. * Dedicated to Mr. & Mrs. Alexandre Magalhaes Da Silveira. 527

782

528

STEVE SMALE

Every diffeomorphism of a compact manifold is differentiably isotopic to one satisfying Axiom A and the no-cycle condition.

THEOREM

COROLLARY

Every diffeomorphism is differentiably isotopic to an Q-

stable one. For a reference to definitions and how these things are put together see "Global Analysis" [1]. For handlebody theory see [3]. Shub and Williams have remarked to me that the proof shows the transversality condition can be satisfied; thus by Robbins theorem [2], one can get structural stability in the corollary. The theorem above is reminiscent of the theorem (Palis-Smale in [1]) that every manifold supports a structurally stable vector field (i.e., con­ tinuous dynamical system). In our case here, however, the dynamical system obtained must be more complicated, since the Lefschetz trace formula implies, even for many isotopy classes on 2-manifolds, that there are an infinite number of periodic orbits. There is a purely differential topological aspect of the proof embodied in the proposition: PROPOSITION L e t / b e a diffeomorphism of a compact manifold Mn and Mn = Hn => //„_! D • • • D i / , s W0 be a handlebody decomposition of M (corresponding to a "nice" function; thus H0 is a disjoint union of ndisks, Hl is H0 with compact tubular neighborhoods of 1-disks attached, etc.). Then / is differentiably isotopic to a diffeomorphism g, where g(Hm) <= interior Hm, each m. The proof uses induction on the following hypothesis: Hyp(m): / i s differentiably isotopic to/ O T , where fm{Hk) k — 0, . . . , m. LEMMA


If m < n/2 (or (« — 1 )/2 if n is odd) then Hyp(m — 1) implies

Hyp(m). For the proof of the lemma, the image of each m-handle is moved into Hm, first by a homotopy, then by a differential isotopy, pushing links and knots "off the edges," and finally by an ambient isotopy of M. The procedures by now have become sufficiently standard that we will not pursue this part any further. T o go from the lemma to the proposition let k = n/2 (or k = (n - 1 )/2 if n is odd) a n d / t be the diffeomorphism obtained by this inductive process.

783

529

STABILITY AND ISOTOPY

Then carry out the same process for fkl on the complements, respectively, of i/ n _,, Hn_t, . . . , Hk+l (Poincare" duality trick as in [3]). This will yield a g of the proposition. Now we assume that g is as in the proposition and seek a demonstration of the theorem. Note that the handlebody decomposition is a filtration of the manifold forg in the sense of [1]. We will make isotopies of g which will preserve this filtration property. Then the nonwandering set Q will be broken up into a disjoint union of pieces, each piece contained in the interior of one handle. On the handles we will obtain a horseshoe phenom­ ena after an isotopy (the ref. [4] can be used for the techniques of this part; in fact the Q we obtain will be zero-dimensional). Take a given A-handle A at some stage in the induction and consider how its image inter­ sects a second A-handle A,. The handles A, A, can be represented as products of disks, thus A = D i x Z)»-*, A, = Df x Z)f-*. Let 9>:Z>* X 0 n g-1^)-+Dlk

x 0

be the composition fl*xOn

g~\h,) -+ A n g-'ih,) - ^ A, -* Z),* x 0,

where 0 is the center of the disk, the first map is inclusion, and the last is projection. Let q e Z),* x 0 be a regular value of tp and N(q) be a cell neighborhood of q over which

Ak_y Z3 • ■ • of a manifold for / e Diff(M) gives rise to a Morse theory with type numbers Af{ = Y.t Hi(At, At_i) related to the Betti numbers of M by the Morse relations. For the special case of a handlebody decomposition, At = Hk, these are the usual Morse relations. But now if/satisfies Axiom A and the no-cycle property, one has an associated filtration {Ak}. The asso­ ciated Morse relations impose conditions on the 42< of the spectral decom­ position of/.

784

STEVE SMALE

530 References

[1] S. Chem and S. Smale, Global Anal. Proc. Symp. Pure Math., 1970, 14, Amer. Math. Soc., Providence, Rhode Island, 1970. [2] J. Robbin, On structural stability, Bull. Amer. Math. Soc. 76 (1970), 723-726. [3] S. Smale, Generalized Poincare's conjecture in dimensions greater than four, Ann. of Math. 74 (1961), 391^06. [4] S. Smale, Diffeomorphisms with many periodic points, "Differential and Combina­ torial Topology," pp. 63-80, Princeton Univ. Press, Princeton, New Jersey, 1965.

785 Dynamical systems oa manifolds A dynamical system is a way of describing the passage in time of all points of a given space S. ® can be thought of, for example, as the space of states of some physical system. Then if r is in ®. one unit in time later x will have moved to a point denoted by x,. At time zero x is at x or x». Two units after time zero x will have moved to x» One unit before time zero x was at x-i. If this procedure is extrapolated to fill up the real numbers R, the trajectory x, is obtained, for all time (, a real num­ ber. Thus for each x, x, is a curve in S and represents the life history of x as t goes from — oo to a . If x. is also assumed to be differentiabie in (/, x), then x —* x. can be regarded as a transformation, or diffeomorphism, from S onto © for each t. In this case (©, xt) define a dynamical system. In applications of dynamical systems to fields outside of mathematics, the first goal is to explicate the mani­ fold of states S. A state of a system is information char­ acterizing the situation at a given moment. The first step in this explication process is to define the state in the most obvious unrestricted way. This will give a space that is possibly too large, and natural constraints on physical laws will cut down the unrestricted states to a subset of attainable or physical states. A physical state is one that has an actual possibility of occurring. These physical states form the final manifold of states. A num­ ber of examples in the following section will explain the process given in the previous sentences. EXAMPLES OF STATE SPACES

Example from economics. In the pore exchange econ­ omy of theoretical economics, a model is used with n different commodities, each measured in quantity by a positive real number (fixing a unit of measurement). Thus commodity space P will be the set of points of real Cartesian space 91" with each coordinate positive. A point of P represents a bundle of commodities possessed, for example, by a certain consumer in the economy. It is assumed that there are a finite number of con­ sumers. say m, with the possessions of the ith consumer denoted by x. in P. An unrestricted state of the economy is a point x = (x„ • • •, x.) in which each x, is in P. The space of all these states is denoted by P', the Cartesian product of P with itself m times. Thus an unrestricted state of the economy simply gives the possessions of the consumers of that economy. Now, it is supposed that the total resources in this economic model arefixed,say, at a point w in P. mThus the space of attainable states is a subset W of P de­ scribed by the condition (x„ • • •, x.) is in W if the sum of the x, is w.

786 The Kirchhoff voltage law (KVL) asserts that o» + o, = —oy, or that the voltages around a loop of components sum to zero. The set of states is 3F satisfying these Kirchhoff laws forms a three-dimensional linear subspace K of St*. Any physical state has to lie in K. To obtain the precise set of physical states, another step is needed, in order to bring in the implications of the resistors on (i„ o,). The most common type of resistor is linear in that it obeys Ohm's law; i.e., o, = R,i„ in which R, a some positive constant. For a physical state, Ohm's law must be satisfied as well as Kirchhoffs laws. This cuts down the space of physical states to a two-dimensional linear subspace © for 91*. More generally, any resistor, linear or nonlinear, is de­ fined by its characteristic A., which is a curve in the twodimensional (i„ o,)-plane. Then a generalized version of Ohm's law stipulates that for a physical state in St", the components (i'„ u,) lie in A,. It can be proved in the gen­ eral case that these constraints force the physical states to lie in a two-dimensional manifold - of St*. Suppose, for example, that the resistor is current con­ trolled so that the characteristic A, is the graph in 91* of some smooth real function /. Thus o. = Hi,). In this case it is convenient to take as a representation of 2, the Cartesian (n, oj-plane ST identifying 9f and Z under the diffeomorphism SI* - * 3 C St", which sends (i\, o>) into a state in 3f that is completely determined by i» and o» (see 352). Then the space of states for this circuit is equivalent to 91* or (i>, o,)-space. Simple electrical circuits. This example just given can be generalized to what could be called simple electrical Particle on any surface in P. The previous example circuits. A simple electrical circuit consists of compo­ extends immediately to the case of a particle constrained nents and nodes. The components are circuit elements to move on any (smooth) surface M in Euclidean space that are either resistors, inductances, or capacitors, each £*. The sphere J* is replaced by M everywhere in the dis­ with two terminals. In the circuit, groups of terminals cussion. Thus, for example, the space S of all states for meet at the different nodes. It is assumed that the compo­ this system is T(M) or the set of (x, v) in hi x P in nents have a given orientation so that the current flowing which v is tangent to M at x. TIM) i» called the tangent in one direction is given by a positive real number, and bundle of Si and may be represented as the onion of all the current flowing is given by a negative number if it the fibres TJM) as x ranges over the points of hi. TA.hi) flows in the opposite direction. The unrestricted states of the current have as com­ in the space of all of the possible velocities for the par­ ponents the currents through each component and the ticle at x. Plane rigid body. A point r. and a directed line n, are voltages across each component. So if there are N com­ fixed in B through x„ Then the position of A in the plane ponents, the space of unrestricted states © is ^-dimen­ P is characterized by the position of x. in P and the sional Cartesian space St". The laws of physics impose constraints on these states. angle between n, and a fixed directed reference line n in £*. (For simplicity it is assumed that B has only one possi­ First of all, Kirchhoff laws assert that a physical state bility for its orientation in £*.) Therefore, the space of must be in a linear subspace K of 9f" with dimension K positions or configurations of £ is a three-dimensional equal N. In formal terms Kirchhoffs current law says space or manifold, the Cartesian product of P and the that for each node, the sum of the currents entering the circle S\ P X S\ In this instance, the fact that there is node equals the sum of the currents issuing from the a correspondence between angles and points on the circle node. Kirchhoffs voltage law then requires that the is used. M = P x f is sometimes called the space of voltages across each component of a closed cycle add to generalized coordinates for the configuration of B. If zero. It follows that the set of states in 9f" satisfying (x, #) is in P x •Pi then (x, #) stands for the configuration Kirchhoffs current law and Kirchhoffs voltage law of B, in which x< is located at x and I is the angle between forms a linear subspace of dimension N. The final physical constraint is given by a generaliza­ n«andn. The possible velocities for B at (x, t) are vectors tion of Ohm's law. A resistor component p is allowed to (v,, v.) in P x St (or £*), in which v, is the ordinary have as characteristic any non-singular curve—i.e., onevelocity of x in P and v, is angular velocity. Therefore, dimensional submanifold—A, in the (/,, u,>plane. This for this mechanical system, the state space © is the means that a physical state in K must have i„ o, compo­ product (P x S") x (P X St) of configuration and ve­ nents satisfying the condition that (i„ v.) is in A,. If the pth resistor is linear, then A, is a line through the origin locity space. A circuit with three electrical components. A series with positive slope. Thus the physical states finally satisfy connection between a resistor, an inductance, and a ca­ these further conditions, one for each resistor. The final pacitor, wired to form a closed loop, may now be con­ subspace 2 of © defined by these conditions and Kirch­ sidered. To form the space of states, each of these three hoff conditions will be a submanifold in the general case components is oriented. A state of the circuit consists of the dimension of which is equal to the number of in­ the current through each component and the voltage ductor components plus the number of capacitor com­ across each component. If i„ o, stand for the current and ponents. Miscellaneous examples. The 19th-20th-century Ital­ voltage, respectively, in the resistor, then i\, o» stand for current and voltage in the inductance and iT, o, for cur­ ian mathematician Vito Volterra has studied the growth and decline of two interacting species; for example, rab­ rent and voltage in the capacitor. For historical reasons the capacitor is conventionally oriented in the opposite bits and foxes. The population of each species can be represented by a real positive number, subject to a slight direction from the other two components. The Kirchhoff laws imply the following relations on the currents and approximation. Then the corresponding state space is the voltages. The Kirchhoff current law (ECL) asserts that full positive quadrant in the plane—i.e., the set of pairs U = i, = —i, or that the current that flows into a junc­ (r, /) of positive real numbers r and /. tion between two components equals that flowing out.

Examples frcm classical ■eeksadcs. Particlt in Euclid­ ean space P. Ordinary 3-dimensional Euclidean space is denoted by P. A particle in £* has its state characterized by its position x, a point in P, and its velocity T, a vector (or point) in P. Thus v gives the direction in which the particle is moving together with its speed, the length of v. The vector v can be thought of as being based at x. The states of this system are then all pairs (x, v), x, v each in P. This space is denoted by P x P, the Cartesian prod­ uct of P with P. Particle on a sphere. £>' is a unit ball in £*, so that D' consists of alt points the distance of which from the origin is less than or equal to one. The 2-sphere S1 is its boundary, or points of unit distance, from the origin of P. The space of states for the physical system of a particle on 5* will then be a certain subset of £* x P of the first example. There are two constraints: the first is that r must be in 5*. and the second is that v must be perpendicular to x. The first constraint follows by definition, and the second constraint follows because if the velocity were not per­ pendicular to x, then x would have to leave the 2-sphere. The set of points (x, v) that satisfies these two constraints is the state space © for this problem. © can also be thought of in the following way. If r.(S*) is the tangent space to 5* at x, so r.tS*) is the set of all vectors v in £* based at x with v tangent to the surface 5* (or perpen­ dicular to x), then ® is the union of ail these JVS*) as x varies over 5 \ In other words, © is the tangent bundle of S1, 1(5*) or the set of all (x, v) in 5* x P in which v is in

787 In Turing's models in biology, the states appear as the concentrations of a number of chemicals (or morphogens) in each cell of some structure. Sometimes, as in hydrodynamics, the space of states appears as an <*>• dimensional space (a manifold or linear space).

(350)

MANIFOLDS

(351)

x—-Km+l-x)

(352)

(<x, \.-ixJ(iJ.

(353,

^>L-,W

(354)

j-t4>}X)-X(x)

In each case of the examples, the state space is a mani­ fold. A manifold can often be thought of as a domain (or open set) in Euclidean space of some dimension. More generally a manifold is defined by glueing together such open sets; thus, for example, whUe the 2-sphere is a 2-dimensional manifold, it is not equivalent to an open set of the plane. It can be obtained though as the union of two, 2-dimensional disks that overlap in a ring. Every it-dimensional manifold can be represented by a idimensional surface in Euclidean space of some higher dimension. It is most reasonable to take a manifold—i.e., differentiable manifold—as the basic space of a dynam­ ical system. If U is an open set in Euclidean space £ and r is in C, then a tangent vector of U at x is simply a vector in £ , which can be thought of as being based at x. The tangent bundle T(f) is the product U x £. and a point (x, v) in U x E consists of base point r, vector v. More generally if x is a point in a manifold M, in which M is represented as a *-dimensional surface in some Euclidean space £, then a tangent vector of M at x is simply a vector in £ based at x and tangent to M. The space of all tangent vectors to M at x is the tangent space TJM) of M at x. Then TiM), the tangent bundle of 14, is the (disjoint) union of all these T,(M) as x varies over M, and is a manifold itself in a natural way. In all the examples of mechanical systems, the initial step was to take the manifold M of positions or con­ figurations. Then the space of states was the tangent bundle T{M) of M. A state in all of these examples con­ sists of a configuration together with a (generalized) velocity based at that configuration. This persists in much more general examples of mechanical systems.

i,-W-^;fJ«)

—+(/i-/n)y«0

vy-fVk).

vj^d,

ik. (T. o,. vv vj

cent that it may not be defined for all I going to ± °o. This system of trajectories is still called a dynamical system (or flow), and in this sense vector fields on mani­ folds are equivalent to dynamical systems. EXAMPLES OP DYHAMKAL SYSTEMS

Particle hi E' fat a fen* field. Consider the example of a particle in three-dimensional Euclidean space. It is supposed now that there is a certain vector field, a force field defined on £*, say x - * Fix), £*-►£•. Thus if the particle is at x in E", the force exerted on it is Fix), and Newton's law reads ma = Fix). Here m is a constant positive real number, the mass of the particle, and a = r"(/) (or the second derivative of x with respect to /) is the acceleration with I -* xii) a curve in £* that repre­ sents the passage of die particle in time. For an actual or physical path, Newton's equation is satisfied, or mx"(t) = Fixit)) for all t. Newton's equation is seen to be equivalent to the pair of equations (see 355) relating the derivatives of position and velocity to the velocity and the force, respectively. In this last form, the map (x, v) - » (v, Fix)/m) is a vector DYNAMICAL SYSTEMS AS VECTOR FIELDS ON MANIFOLDS field on the state space for this mechanical system On manifolds (e.g., open sets of Euclidean space) dif­ T(£*). This is the form of a vector field on the stale space ferentiation makes sense, and a differentiable map (i.e., determining the passage in time of the space. The initial transformation) from a manifold to itself with a dif­ conditions are (x, T), the position and velocity of the ferentiable inverse is called a diffeomorpbism. In these particle. terms the definition of a dynamical system can be re­ The force field x - * Fix) is called conservative if it is stated as assigning to each time t (a real number), a of the form Fix) = —grad V(x) in which V is some real diffeomorpbism *.:M->Aj with *% the identity, *.~ function on £*. Grad Vix) a defined as the vector in = *. • +., and so that *.(x) is differentiable in (>, x). Thus which the ith component is ov/9x<, for i = 1, 2, 3 (see for each r in the manifold, the trajectory f .(r) = x, 356), in terms of Cartesian coordinates. V is called the is a curve through r. The derivative of this curve potential for the system. The rlstsiral example of a force I - » ♦,(*) at / = 0 can be taken, letting X(r) be the field is that of gravitational force. If the source of this derivative of *,(x) (see 353) to obtain the velocity or force is the point located at the centre of £*, then (see the tangent vector of the curve at x. Thus Xix) is in the 357) the force is inversely proportional to the square of tangent space T.(M). The association r -* Xix) is a the distance, involving the Euclidean norm on £* (see vector field on M (or tangent vector field on M). More 358) in Cartesian coordinates. In this case. Fix) is a con­ generally, a vector field on M is any differentiable map servative force field because a potential can be chosen X:M -* TiM) with X{x) in T.ifif)- For example, if M (see 359) that is inversely proportional to the distance. is an open set in Euclidean space £, then Tff4) = M x E This gives what is sometimes called the Kepler problem and a vector field Y:M - » M x £ must be of the form because the solution curves lead to the (Kepler) elliptical l*(x) = (x, X(x)) for some map X:M -* E. In this case by orbits of the Earth about the Sun. Strictly speaking, in a slight abuse of language, X a called a vector field, so the Kepler problem configuration space is £* — 0, be­ that a vector field can be thought of as assigning to each cause the vector field describing the dynamics is not de­ point x, in M, a vector in £ based at x. fined at 0. Van der Pol equation. The particular example of an The notion of vector field on a manifold is of basic importance because among other things a dynamical electrical circuit given earlier is now considered. Physical system is characterized by its associated vector field. A laws of the inductor and capacitor give directly the dif­ converse to this statement is roughly true, and the con­ ferential equations for motion in terms of the representa­ verse could be regarded as the fundamental theorem tion of the state space found there. More precisely (see of ordinary differential equations. More precisely, if 360) the derivative of i\ is proportional to Ox and the x -> X(x) is a vector field on a manifold 14, the ordinary derivative of o, is proportional to i, (see 360), in which differential equation can be considered for * t (x), given L, the inductance, and C, the conductance, are constants by the derivative of *.(x) being equal to X(x) (see 354). of proportionality. For simplicity, L and C are taken as This equation has a solution, at least for all t satisfying constants equal to unity. Here ox = o, — /(i"0 and i, = —ix by KircfahoiTs laws and the resistor characteristic. r(x) < t < f(x) in which r, r are real functions on M defining some (perhaps maximal) open set Q in R X W The differential equations then become appropriately changed (see 361). The corresponding vector field on H" containing O x M. This solution *.(x), defined for (r, x) in Q, then has the properties of a dynamical system ex- at the point iK «,) is (o> - fiiuX - < 0 .

788 In the case in which / is linear, of the form o, — R,i„ R, > 0, then it can be deduced that all solution curves of this dynamical system tend toward the origin. The origin is a steady state and an attractor. This means that a solution at the origin stays at the origin for all time or that the vector field is zero at the origin (steady state). That the origin is an attractor means that nearby solu­ tions tend to the origin U I - > I O . Steady-state attractors play a central role in applications of dynamical systems. On the other hand, if a highly nonlinear characteristic is considered, say the derivative f (0) < 0, then the unique zero (0,1(0)) of the vector field can be shown to be a source. This means that orbits tend to (0, /(0)) as t-* — ao. If the characteristic (i\, f(i0) stays well inside the first and third quadrants for large |i»i (a natural as­ sumption even for nonlinear resistors), then as I -* °o all orbits tend toward some bounded region in 91*. This follows from a general energy theorem. From these facts, the Poincare-Bendixon theorem implies the existence of a periodic orbit, and one will be an attractor in the gen­ eral case. A periodic orbit, period w > 0 is an orbit where * . . . U ) = ♦.(*) for all (. This amounts to an oscil­ latory behaviour of the physical variables. That it is an attractor means that all nearby solutions tend toward it and thus will also oscillate, at least to a high degree of approximation. Periodic attractors are of basic impor­ tance in applications of dynamical systems. The characteristic defined (see 362) by /(i>) being equal to M'V — <0 satisfies the conditions of the previous para­ graph. Here there is a unique periodic attractor; tnc dif­ ferential equation is called van der Pol's equation. Volterra-Lotka equations. The equations describing how the populations of rabbits and foxes vary in time are called the Volterra-Lotka equations and are (sec 363) a pair of coupled first-order equations with positive constants a, j}, x, *, For example, dr/dt = ar governs bow the number of rabbits increases eating grass, without taking into account the foxes that eat them. The VolterraLotka equation can be interpreted as giving a vector field on the positive quadrant in 9f; its solution curves are in fact periodic so that the populations are expected to change in cycles under these conditions.

if, x = 0. From B a norm can be defined, ||x|| = B(x. x) and distance from x toy as||x — y\\. A Cartesian coordi­ nate system on £ can be found (see 369) so that B(x, y) is a sum of terms of the form xori. The derivative of /: U - » 9t at x in £/ is now considered. This derivative of / at x is most naturally considered to be a linear map D/(r):£ -> 91. If r is in M, the in­ ner product B on P restricts to give an inner product B.

(355)

x-v,

»-— F

on

£'x£J-r(£5)

fit

(356) (f.f.f) \Ax, dx, ix,l

G«ADIENT DYNAMICAL SYSTEMS

System* on E*. A gradient dynamical system in £* is considered. For a classical description, it is supposed /:£* - » 91 is a differentiable function and £* is provided with Cartesian coordinates (xi, xi, x,) = x. Then Jf(x) = grad /(r) is the vector field on £* that associates to the point x in £* the vector grad /(r), defined in terms of the partial derivatives of / (see 364). Thus grad /Or) can be thought of as a vector in P based at x, and it will be per­ pendicular to the level surface l'\c) at x, /(x) = c. As a particular example, take /(x) equal to the sum of three terms aa* (see 363), in which each a, is a negative con­ stant. Then the differential equation dr/dt = grad fix) can be solved directly to see that each trajectory tends to the steady-state attractor (0,0,0) = 0 in £'. Open sets oa £*. The last example can be generalized in several steps to gradient dynamical systems on ab­ stract manifolds. A further step is to the case of a real differentiable function /:(/-► 91 with V an open set of n-dimcnsioaal Euclidean space £. Given Cartesian co­ ordinates for £ (see 366). grad l(x) can be denned as be­ fore by assigning to each x in (/, the vector in £ the ith component of which is the partial derivative of / with re­ spect to Xi (see 367). Thus grad / is obtained as a map; i.e. grad F : t / - » £ . It is convenient to give a second equivalent description of grad / that extends to manifolds that leads to a mod­ ern exposition of Hamiltonian mechanics. This requires a precise definition of what is meant by Euclidean space. Euclidean space is a finite-dimensional real vector space £ together with an inner product, say B. B is first of all a bilinear form on £ so that B is a map B:£ x £ -* 91 satisfying conditions of linearity (see 368) for x, S(v). y, S(y) in £ and \ real. Moreover, B is symmetric so that B(x, y) = B(y, x). Finally, B a positive definite so that B(x, x) > 0 and B(x. x) = 0 if, and only

(357)

FW— |fp

(358)

(|x|-(ixf)'«

(359)

V(x)~-

(360)

di. dt

IU1 dt' dt

v

di. (361) do,, dt (362)

/(iV-Mii-'J.

(363)

dr --«r-V/;

df JL—gf+luf

(364)

(2-U). \dx,

7^w)-grad/(x) dstj I

(365)

/ U J - n . - r f + a j X l + flj*;

(366)

( x , . - - - . . 0 = jr

T-U). ix.

M>0

789

(367) (368)

(369)

wx)---kU)) B{x + H[x].y)~B{x.y) + B(Z[x],y) B(x.y + il[y))mBU.y) + B(x.2[yl) t?( Ax. y). KBU. y) - B(x. Xy) BU.yy-fx,*, J - l

(370)

inner product, an indefinite symmetric form oo Buclidean space can be considered. For example, if / is the bilinear form on 91' defined by being equal to the difference of Xi2(xi) and x>S(xi) (see 371), then / is symmetric and nondegenerate; and if a real differentiable P is given o a f i vector field grad iP is obtained replacing B by / in the definition. It turns out that the equations of electrical cir­ cuit theory can be expressed in this form. For example, if for P on Jt*, the function is taken to be — Uo, added to the integral from 0 to X of /(u) (see 372), then the equa­ tions of the circuit discussed earlier have the form of the derivatives of u and o, being proportional to appropriate partial derivatives of t (see 373). These equations may be expressed in terms of grad jf\x) (see 374) in which x =

£,*,(*)]

(i»,

Oj).

-O/UXgrad/W) HAMILTONIAN MECHANICS

- B{gnifU), grad/(jc)) > 0 (371)

JU.x)-x,»U,)-xfi(xJ

(372)

P{iv o,) - - < > T + J / ( « )
(373)

A. dt

(374)

f-grad^x)

(375)

*'k'

dt

»vy

8(x,2[x])-j;x,S(*,) I-I

(376)

«*.8M)-i*,su1..)-£xl,,laex1)

on the tangent space T^M). The assignment x -* B. is called a Biemannian structure on M. With this structure, the constructions of the previous section carry over. Thus for each x, D/(x) is an element of the dual space TJM)' (or r.«(M)). This map x -* Dfix) [or dftx) oftentimes] is an example of a 1-form on At". Then an isomorphism f W i V M ) -* T.'iM) is defined for each * via B. just as #• before. The inverse image of Dlix) under *»• is then grad /(x) for each r, and the previous properties of gradi­ ents follow in the same way. A special case is the unit sphere S* in £* given as the set of x such that B(x, x) = 1. Then if f-.S" -* 9t is the projection on some line—for example, the height or r-coordinatc in an (x, y , t)
Hamiltonian mechanics as developed on manifolds has the virtue of including quite general mrrhsnical systems and snows basic principles most simply. To define gradi­ ents, the inner product on Euclidean space £ was of cen­ tral importance. Hamiltonians are developed similarly with the bilinear form B, or inner product, replaced by a symplcctic bilinear form Q on £. B(x, 201) is a sum of terms x.S(x,) (see 375) for Cartesian coordinates (Xi, • • •, x.) = x on £. £ is taken to have even dimension In and a bilinear form Q on £ is defined (see 376). Not only is it seen that fl is bilinear, but it also is clearly antisym­ metric, B(x, fi[xl) = —
790

CKx.il[x])- 2 x,S£(x„.)- £ 1 . . , ^ (377)

1.1

■ x

(378) (379) (380)

(381)

f iH

I - I

»H

"Vax,.,-'i*,;

ir 1 _aH

-


dH

#/A

ax,' ' * V ._

wtcz, Lectures on Ordinary Differential Equations (1958), a

v v

r .. »,)•?,»»mv,

w

^y)-2^2>f + K w


m
gives the dynamics for this mechanical system. The in­ verse of the Legendre transformation takes trajectories of Xu back to solutions of the Euler-Lagrange equations on r(M). (S-Sm.) The reader interesteST in dynamical systems on manifolds can obtain some additional background on manifolds from SHLOMO STUNSSIO, Lectures on Differential Geometry (1961); saaoB LANO, Introduction to Differentlable Manifolds (19*2), a very basic book; and tactual SMVAX, Calculus on Manifoldj (1963), an especially readable elementary work. For more information on ordinary differential equations, in addition to the works mentioned above, lee wrrou> Huaa-

a*'

* "T''

av ~-ax^

(382)

2^2>f-

i«2*J

(383)

*„M-*n^DtfM

(384)

(DH(x)KAfw(x)). n,(X„W. * „ « )

highly recommended and leu imposing book dun most on this subject; and JOLOMON LEFJHETZ, Differential Equations: Gtomttrie Theory (1957), which discusses structural stability and the van der Pol equation. Some works on dynamical systems are STEPHEN SMALE, "What Is Global Analysis," Am. Math. Mon., 76:4-9 (1969); and "Differentiable Dynamical Systems." Butt. Am. Math. Sot., 73:747-817 (1967)—both of these include further references. For the mathematics of elec­ trical circuits, see ctuaLES A. DESOES and EENEST s. KUH,

Basic Circuit Theory (1969). cejum oaa.il. Theory ol Value (1959), gives a good background in mathematical economics. An excellent standard text with a traditional view is HxaaaaT GOLDSTEIN, Classical Mechanics (1950); some modern ap­ proaches are in LYNN H. LOOMIS and SHLOMO STEaNsaao,

Advanced Calculus (1968), see especially en. 13; ixLrK ASBAHAM, Foundations of Mechanics (1967); v.t. ABNOLO and ANDBE AVBZ, Problemes ergodiques de la micanique

dassique (1967; Eng. trans., Ergodic Problems of Classical Mechanics, 1961): and STEPHEN SMALE, "Topology and Me­

chanics," 2 pt., Inventiones Mathematicae, 10:305-331 and 11:45-64 (1970). vector field (or ordinary differential equation) corre­ sponding to the Hamiltoniaa H. As a Riemannian mani­ fold is required to define gradient vector fields, a symplectic manifold is required in order to define Hamiltonian vector fields. Conservaaoa law*. Various physical principles can be derived in this setting. For example, conservation of en­ ergy amounts to the function H being constant on tra­ jectories of Xm. This is the same thing as the assertion that for each r, DH(x) evaluated at JT«(r) is zero. But another condition holds (see 384) from the definition of Xm, and the last is lero since C is antisymmetric Liouville's theorem asserts that M«mntmii«ti systems preserve volume. In this context, that volume on the ab­ stract manifold W is the n-fold wedge product of O with itself. Without pursuing the »»">"■ «•«< definition of wedge product, it can be said that if Q is defined as above on E in coordinate, then the Liouville volume is just ordinary volume on E. It can be first checked that O itself is in­ variant under a Hamiltonian flow XM, and then it nat­ urally follows that the wedge product, the Liouville vol­ ume is preserved. This can be contrasted with the gradient case. For example, neighbourhoods of a local maximum of a gradient flow are shrunk into smaller subsets by the dynamics. Volume is strictly decreased. Simple mechanical systems, A simple mechanical sys­ tem (M, K, V) has a manifold M that is the configuration space of the system. V:M - » 9t is a function, the poten­ tial energy, and kinetic energy K. is considered to be given by some Riemannian metric on M. Thus for each x in M there is an inner product K. on TJ(hf) and JC(c) = KJ.V, o) for o in T.(M). This can be compared with the earlier mechanical system of a particle con­ strained to move on the 2-sphere in £*. In this case, JG oa T.(i*) is the restriction of inner product on E* times half the mass. Together these sum to give the energy E:T(M) -* R as a real function on the tangent bundle of hi as defined by E(o) = X(o) + V(x) for o in 7\(A»*). To put this into the sympketic perspective, it is con­ venient to consider the cotangent bundle T'(\f). There is a canonically defined symplectic structure Q defined on T*(M), which extends the earlier example of Q on £* X £*■ Furthermore, the energy £ can be transferred to T°(M) by a map called the Legendre transformation L:r(M)-*r«(AO. More precisely, if m r ^ M ) , then L(o) = *o(o) in which •>• is the usual linear map in­ duced by a bilinear form. Then the Hamiltonian H:T*(M)-*9 a defined by ff = E°Z,-\ The corre­ sponding Hamiltonian vector field X, on T*(M) = W

(I.N.S./S.Sm./Ed.)

791 LECTURE III DYNAMICAL SYSTEMS AND TURBULENCE

Steve Smale

The purpose of this talk is to present some questions and ideas in the field of dynamical systems related to problems which arise in turbulence.

We shall begin with the discussion of how

the Navier-Stokes equations define a dynamics on a certain infinite dimensional function space. Recall that the law of motion of an incompressible viscous fluid with constant density (this assumption has been made in both previous talks) is given by the Navier-Stokes Equations:

iX _ viv - (v-V)v = -Vp+ f div v = 0 v = prescribed on 3fl

where ft is a region containing the fluid, v of the fluid, p

the pressure and

f

the velocity field

the external forces,

v

is

the kinematic viscosity, or, in the way we wrote the equations, v = 1/Re , where

Re

is the Reynolds numbers.

In all our talk, 3

ft is supposed to be an open bounded set in R v:ft—► TR.

is

also assumed

to be "smooth".

with smooth boundary; As was pointed out

792 49 already in lectures one and two, we believe that the chaotieness in turbulence is intrinsically in the semiflow defined by the Navier-Stokes equations.

But when dealing with a physical prob­

lem like this, one has to define a space of states.

We consider

S — the space of states of the physical system — to be the set of all "smooth" maps prescribed

u:fi ■+ TR with boundary conditions

or tangent to

3fl .

Consider now maps

u|3£J =

u:!R-»- -S ,

where K is considered to be the time-axis and write u (x) = u(t,x). The determination of the map

u:R-> S , is given by the Navier-

Stokes Equations.

It is clear that we won't be satisfied with 5

and will consider

S

= {u €<;|div u = 0 } and maps

u:3R-»-S0.

In

this way, the Navier-Stokes equations define formally an ordinary differential equation on of

v

S. . For a discussion of the "smoothness"

as well as for candidates for

SQ (certain Sobolev spaces)

and how one copes with semiflows with unbounded generators, see the Appendix to Lecture I.

Just let us stress here once again

the idea that the existence and uniqueness theorem for the solu­ tions of the Navier-Stokes equations gives a dynamics on

S

and one works with this dynamics. There are at least two ways of attacking a discussion of the semiflow defined by the Navier-Stokes Equations; each is based on a finite dimensional approximation akin to the Galerkin Method. I.

Find an invariant finite dimensional submanifold I, preferably

low dimensional.

The following might be a way to find such a sub-

manifold: take the eigenvalue expansion of those eigenfunctions

&

and retain only

corresponding to strictly positive eigenvalues.

793 50 If there were no nonlinear term (u-V)u

one would obtain in this

way an invariant linear manifold I attracting for the semiflow

We should obtain something perturbed, "around" I the full Navier-Stokes Equations.

II.

when considering

This is a hope.

Take a finite dimensional space spanned by a finite number

of eigenfunctions of the Laplacian

On each

H

duced from

and consider a projection

one gets via this projection a dynamical system in­ S .

If

n + => one hopes to be able to approximate

the dynamics on

Sn

with those defined earlier on IR

Though mathematically challenging, both approaches miss something, namely, we should never forget that

I C S Q is in the

space of vectorfields, so any result we obtain, cannot be physi­ cally tested under this form.

The second question which arises

is how can the two methods be tied together. one of them mean something in the other?

How can a result in

This should certainly

happen, since both attempt to give information about the same physical phenomenon. These two questions are answered by the introduction of observables, which are maps

g:SQ -+3R.

For example one can consider

794 51

R defined by g (u) = u(x) for each x e P. . The main *x :S 0 quality of these maps is, that their action can be actually physi­ cally tested, so one has a certain control over the mathematical results obtained earlier.

Their second quality is that they tie

the two approaches together.

For the first one, we have a com­

mutative diagram

In the second approach one starts with maps h : R n -»■ H and obtains the observable h SQ—► Fn—**► K .

h

h

on the projections,

by the composition

The commutative diagram

shows how an observable in one approach defines an observable in the second and vice-versa. In the rest of my talk I shall be concerned only with ques­ tions of dynamical systems which inherently will appear when treat­ ing in the way described above the Naiver-Stokes Equations.

I

shall discuss two central questions related to this: stability and ergodicity. Stability.

Any reasonable mathematical model of a physical

phenomenon should be "stable", i.e. if one makes certain

795

52 perturbations, its qualitative features do not change.

One must

expect that, since any model represents an idealization of reality and hence represents itself a perturbation from the "real model", which of course is not present and nobody ever hopes to lay their hands on it.

So, if our model couldn't be "stable", the deriva­

tion from reality can be disastrous

and our model is no good!

When dealing with dynamical systems, the questions of stabil­ ity become much more precise than the very vague principle stated above and they refer to the orbits.

There are two concepts cru­

cially related to stability in dynamical systems: a)

Attractor.

We shall say that a set of orbits is an attrac-

tor or asymptotically stable if nearby orbits tend to the set as time increases; formally, an invariant closed set F.(i.e.

A

of the flow

F.A C A) is an attractor if for any neighborhood

there exists a neighborhood lim distance

V of A

such that if

U of A,

x e V, then

(F (x),A) = 0. X

t-K»

b)

Robust.

We shall call those quantities robust which persist

under slight perturbations of the system.

Here we incorporate the

various notions of stability found today in literature.

One" strong

case of robustness is structural stability or invariance of the orbit structure under slight perturbations up to a continuous change of variables. (M

Formally, two C -vectorfields X,Y € i

(M)

a compact manifold for example) are called topologic ally equi­

valent if there exists a homeomorphism

h:M -* M which sends the

orbits of X onto the orbits of Y keeping their orientation, i.e. if

m e M and

e > 0, there exists

6 > 0

such that for

0 < t < e,

796

53 0 < t' < S, where F* , F Y denote the

hF*(m) = F*,(h(m)) for some

flows of X and Y respectively.

X S X r ( M ) is said to be struc-

tually stable if there exists an open set of

Xr(M)

such that all

Y € 0

in the Cr-topology

0

are topologically equivalent to

X. Note that the concept of attractor refers to the invariance of the orbits relative to perturbations of the initial conditions whereas robutsness deals with insensitivity of the phase portrait under a perturbation of the system as a whole.

Hence the "nicest"

systems will be those which are robust in a region near an attrac­ tor not presenting qualitative changes at both types of perturbations. Example 1.

Hyperbolic equilibrium dx

e.g.:

- v

at -

x = c.e

x

with solutions

£•-»

' cl c2

6 m

y = c,e

whose phase portrait looks like

ft

>x

Let

m 6 M

be a singularity (equilibrium) of

X(m) = 0 where M is a compact manifold.

X € X (M), i.e.

We shall say that

m

is

a hyperbolic singularity or hyperbolic equilibrium if T

» X : T _ M -*• T - M m m m

has no eigenvalue with real part zero. c "

Then it is

797

54 known that the set of vectorfields which have all their singular­ ities hyperbolic is open and dense in

* (M)

(see for example

[PM], page 103). This shows that a hyperbolic equilibrium is robust since any nearby vectorfield has also only hyperbolic equilibria.

Note that in this particular example robustness does

not mean structural stability. that an equilibrium (i.e. all eigenvalues of

T X

Recall now the theorem which states

singularity) is stable if and only if

have strictly negative real parts; this

is sometimes referred to as the "principle of linearized stability". Hence we would expect that there are hyperbolic unstable equilibria which indeed is the case as our above example shows.

Example 2.

This will show that there are non-robust attractors. 2 Consider in K the flow

\

*

/

where the unit disc is the set of equilibria, all orbits outside the unit disc tend towards the unit circle. clearly an attractor, but is nonrobust,

The unit disc is

since a slight perturba­

tion will destroy the bounded set o.f fixed points. In classical differential equations one encounters two cases of compact attractors which are structually stable. classical attractor is an equilibrium

m G M

such that T X has

all eigenvalues with strictly negative real part. traits look

The first

in this case like that, for example

The phase por­

798

55

M/

•>- •*-

As concrete examples consider the equations in JR t x = c,e

dx = x

Ht

with solution

%

= y

for the first picture y = c2e

and dx dt=

it =

-

x

-y

x = e~

(oncost - c„ sint)

y = e~

(c„cost + c. sint)

with solution

x

~y

for the second picture. The second classical attractor is a stable attracting, closed orbit.

For exaple, in Van der Pol's equation

'dx

13t = y " x ^ = -

3+ x.

x

one finds a unique closed orbit, all solutions from outside

spiral towards it and all solutions from inside expand towards it, the origin being a source (for*study of Van der Pol's equation, see [HS], p. 215-228). We shall now describe a new kind of nonclassical structurally stable attractor not found in the traditional theory of ordinary differential equations and I shall call such attractors strange

799

56 attractors.

Since everything will be done by means of discrete

dynamical systems, I mention at this point that there is a canoni­ cal way to associate to each discrete dynamical system on a (com­ pact) manifold M a global flow — hence a vectorfield — on a mani­ fold M. of one dimension higher.

This then will show that we did

not restrict the generality of the example by working — easier — with discrete dynamical systems. sented at the end.

The general theory will be pre­

Our exposition follows Shub's paper [Sh].

We start off with the "expanding" map f(z) = z .

f:S

-► S

given by

If 0 denotes the full disc in 2-dimensions, denote by

R = S x D the full ring having as boundary the two dimensional torus.

For matters of convenience we shall imagine the ring R

centered at the origin of K 3 with S1 embedded in circle lying in the

as the central

R

xOy plane and having radius equal to 2.

Put

now coordinates (9,r,s) in this ring in the following way: 6 measures the angle in trigonometric sense from the Ox axis; *<

(9,r) forms a coordinate system on the annular region Rn(xOy) such that on the central circle of radius 2, r =0; (r,s) forms a coordinate system in each "slice" of the ring, s measuring the "height" of the point in the particular "slice" D(8) 2 2 so that we always have

this coordinate system, our map

f

r + s <_1. In

looks like (9,0,0) -* (26,0,0).

We want now to find an embedding

h:R -* R which models this "twice

wrapping" of the central circle.

Define

h:R ■* R by

800 57 h(8,r,s) = (28,e,cos8 + e ^ . e ^ i n e + e 2 s) is easy to check that

h

for

e 2 < e]L < 1/2; it

is in fact an embedding of R into itself.

Our "slice" D(8) is a cross-section of the ring through the point (8,0,0) on the central circle, notice now that h(D(8/2)) , h(D(8/2+ir)) C D(8). are small discs. and radii.

Actually

h(D(8(2)), h(D(8/2+ir))

We show that now and also compute their centers

Their centers are

h(8/2,0,0) = (8 .e^ose^jejSinS/^),

h(6/2 + ir,0,0) * (8,-e cos8/2,-e sin9/2) so that the distance between them - in the

D(8) plane of course - is 2e1-

To find the radius 2 2 take an arbitrary point (6,r,s) on the boundary, i.e. r + s =1 and observe that

h(8/2,r,s) = (8 , E 1 C O S 8 / 2 + e 2 r ,€., sin8/2 + e 2 s) is

at distance e. from the denter

(8,e1cos8/2,e.sin8/2). Similarly

the radius of the other disc is

e„.

It is clear that our

o-C

Of

map h wraps our initial ring R by making it thinner and longer twice around the central hole of R, inside R.

The phenomenon

that occured was: in the 9 plane two new discs of radius

e,

appear with the centers at (8 ,e,cos9/2 ,e-,sin8/2) and at (8,£^03(8/2 + ir), e,sin(8/2 + * ) ) .

Write now formally (^,e) for

(¥,ecos / , esin >/) so that with this convention our centers will be at (8/2,e.) and (8/2 + ir,e ) respectively. has the analytic expression

The second iterate

801 58

2 h (8,r,s)

2 = (48.C.COS2 + Ej^EjCOse + e 2 r

2 , e ^ i ^ S + e..e2sin8 + e 2 s ) . 2 2 As b e f o r e , a c o m p u t a t i o n shows t h a t h ( D ( 8 A ) ) , h (D( 8/i* + TT) ) C C h(D(8/2)) C D(8), h2(D(8/4 + T T / 2 ) ) , h2(D(8 A + 3IT/2) ) C C h(D(8/2+Tr)) C D(8); our picture then looks like:

The four small discs which appear have radius using our symbolical writing

—

2 e. and the centers —

at:

(f, ei ) + e 2 (i, E l ); (f, ei ) + e2(£+ir,ei) (|+1r,El) + e 2 (J*f,e 1 ); <}*T,e1) + ^ ^ ' V

.

This procedure then goes indefinitely: at each stage, for each earlier obtained disc, 2 new discs with radius shrunk by a factor of e_ will appear inside the older discs.

That gives us the idea

of looking at the whole process differently: at the first stage write pairs (8 , U

€ S «S

such that

28 2 = 8 1 ; that corresponds

to the idea that the first coordinate gives the

8

for the slice

D(8) and the second coordinate tells us for which angle, after applying

h

we still

land

in the chosen "slice". 1

1

iterate we write triples ( e ^ e ^ e ^ e S x S x S

1

with

For the second 282 = &1 ,

2 8, = 8-; the first coordinate gives the angle of the chosen slice, the second gives the angle for which the first iteration yields the 2 small discs in the chosen slice and the third coordinate shows for which angle the second iteration lands in the smaller discs. Note that the knowledge of this sequence plus the fact that at each

802 59 stage we shrink e 2 times the surface of the section and the pat­ tern of formation of the centers of the consecutive discs tells us everything we need to know in order to locate any disc obtained in this process.

Generally, we take the infinite product

S x...xS x...

and form the subset {(8., ,... ,8. , — ) | 29. + , = 8.

for all k >, 1} .

It is clear that each such sequence determines

a unique point of

f| h (R) and conversely. Our space of sequences n>.0 will be called solenoid and what we showed is that our solenoid is homeomorphic to and will be left

nn

h (R)

(the verification of continuity is easy

-° to the reader).

disc D(8) a Cantor set.

Note that

nQ

h (R) is in each

-° A = ("\ h Also, we remark that

(R) attracts

O>0 all points of R under the iterates of

h

and that it is locally a

product of a Cantor set and a one-dimensional arc. point of

A

More,at each

we have a so called "hyperbolic splitting" which has

to be understood by the fact that

h

is expanding in one direc­

tion (makes R always longer) and contracting in other two dimen­ sions (always shrinks the section by a factor of proved that

A

e

2 ^'

** *-8

is robust, in this particular case structurally

stable: small C -perturbations of the map preserve this picture. Exact mathematical statements in a general framework will be pro­ vided at the end of this lecture. Let us return now to our solenoid and present R. Williams' idea of studying such a strange attractor by inverse limits; much more about this will be said in his later talk.

Recall that our

solenoid was the set {(8-. ,... 8V,... ) 6 S x...xs x... 9 - 26. ,} IK JC ) c + 1 which is obviously a closed subspace of S x...xS x... the product topology.

endowed with

Our solenoid is actually an inverse limit of

803 60 a certain inverse system. u. . :X. ->■ X. for i] 3 i

lar

i J< j , i, j 6 W

=

V^l'W

To see this, denote X. = S

and define

1-;]

by u..(8.) = 2 e . ; in particu13 3 3

< W 2 • Then e k ,...)ex 1 x... x x k x...|u..(e.) = e i = 2 1 " D e j v i < j};

iim(x i ,u ij ) = {(81 3N

= {(8 1> ... ,8^,...) € S1x...x S1x...|28]c+1 = 9 k , Vk € J O , which is our solenoid.

f|A

via our identification becomes just the shift

(9,,...,8, ,...)'—►(28,,9,,... ,9. , . . . ) .

Even though our construc­

tion of the inverse limit here might seem a somewhat forced formal­ ism, this idea is extremely fruitful when dealing with semiflows on branched manifolds (see R. Williams' talk).

Ergodicity.

Accomplishing our first goal by giving an example

of a strange attractor, we shall concentrate on the second impor­ tant feature of attractors called ergodicity.

Loosely speaking

ergodic theory is the study of transformations and flows from the view point of measure theory.

We leave the exact definitions and

theorems for the end of the talk and pursue our previous example in order to get some feeling of what ergodic theory migh provide for the study of attractors of a diffeomorphism. We start with the remark that if n

n

d(h (x),h (y)) * 0

so that if

g:R ■+3R is a continuous map we

shall have |ghn(x) -gh n (y)| ■+ 0 proach each other: v

x,y € D(8), then

and hence the time averages ap­

, , ,n-l

-

n-1

.

|

i I.I g h ^ x ) - I gh x (y)| * 0 . 1=0

i=0

This has to be interpreted in the following way: one limit exists

804 61 if and only if the other does.

It is known in our case that these

limits exist and satisfy lim ni

n-H»

I gh^e.r.s) = [ g(e,r,s)de

J

i=o

for almost all (8,r,s) in R. Lebesgue measure on S transformation

h

Technically this says that the usual

is ergodic and invariant with respect to our

and our statement is a typical ergodic theorem.

We define now a measure

u

on our strange attractor

following way: for each continuous (8,r,s) 6 D(8)}

g:A + K put

A

in the

g(8) = min {g(8,r,s)|

and by the Riesz representation theorem we can

find a unique measure u

on A

such that

f gdu = f gd8 . Then it is S

proven that y is an invariant measure which is ergodic with re­ spect to h and for almost all (8,r,s) G R we have lim — £ gh1(8,r,s) = gdu n*» n i=o JA

.

We end these short comments about

ergodic theorems with the remark that this last statement is strongly related to our above inverse limit construction and to the fact that via this construction we transformed in a certain sequence space.

h

on the attractor to a shift

These ideas have been beautifully de­

veloped by Bowen and the results he gets — for example in [Bo] and [BoR] — using these techniques contributed very much to the under­ standing of ergodic properties of certain types of attractors. The most popular invariant in ergodic theory is entropy.

We

shall abstain here from exact definitions and only mention that the topological entropy (that's the one we are talking about here) es­ sentially gives the asymptotic exponential growth rate of the number of orbits of a certain diffeomorphism up to any accuracy and arbitrary

805

62 high period.

Very loosely speaking and disregarding an unproved

conjecture, the entropy of a diffeomorphism is bounded below by the logarithm of how many times it "wraps around" the manifold.

In

our case, the entropy will be log 2, that is log deg h and this is by no means an accident. We refer the reader to R. Bowen's talk for much more detailed information about ergodicity.

General Theory.

This section aims to present roughly the

general mathematical machinery behind our previous construction of a strange attractor.

We shall begin with an earlier promised result,

namely the explicit construction of a flow on a manifold of one dimension higher from a discrete dynamical system, it is done after [S3 page 797. Given a vectorfield X on the manifold M, a cross-section of X is a closed codimension one

submanifold

I of H

such that every

orbit of X intersects I , E is transverse to the flow of X and every orbit leaving

£

intersects

Z

in both future and past time.

If this happens, define the first return map f(x) = F.. (x), where tQ

with

F.. is the flow of

F + (x) € E .

a smooth map.

f:I ♦ I by setting

X and t. > 0

is the least

By the global smoothness of the flow, f

Note that the orbits of

to one correspondence with the orbits of

X

is

are in this way in one f, i.e.

with { ^(xJlk 6 11

compact orbits are preserved under this correspondence and hence periodic points of

f

correspond to closed orbits of X.

However,

the existence of a cross-section is not always guaranteed; for ex­ ample, X cannot have singularities and this will then restrict by

806 63 the Poineare-Hopf index theorem for compact manifolds the topological type of M . On the other hand, the converse construction is much nicer, namely, every diffeomorphism can be regarded as the first return map of a cross section of some flow. Given the discrete dynamical system defined by the diffeomorphism ex:

IK

M

+ J X M

by

f:M •* M define the map

a(s,m) = (s+1, f(m)).

operates freely on R* M

It is clear that Z if

via the action (k,(t,m)) "* a (t,m) =

(s+k , fNm)) so that the orbit space (]RxM)/Z = M. is a manifold of dimension by

1 + dim M. Note that the flow

$ (s,jn) = (s+t,m) is Z-equivariant

♦ i B u M + E x H defined i.e. °ak = a k o$ for t

all k 6 Z , t S]R, so that it induces a flow Ft:MQ -* MQ defined by

Ft(Z(s,m)) = Z(4t(s,m)) = Z(s+t,m) where Z(s,m) denotes the

Z-orbit through (s,m) e E " M .

By a general principle of passing to

quotients, F^ is smooth (see [B], page Si ) and we have defined in this way a smooth global flow of

F:ExMQ ■* MQ called the suspension

f . We shall emphasize some remarkable properties of the suspen­

sion. Note that if

IT;]RX M

+ M_ is the canonical projection — which

is a submersion — then E = ir(0 *M) is a cross-section of the flow Ft diffeomorphic via IT to M. Also, for each Z(0,m) € it(0 xM). F1(Z(0,m)) = Z(l,m) = Z(0,f_1(m)) e Z t > 0

and

t=l

is the smallest

for which this happens. Because of the commutative diagram M3m >

► f (m) e M

If

> F-, Z 3Z(0,m) 1 ^ Z ( l , i ) £ J

807 64 with vertical arrows diffeomorphisms, our initial

f

to the first return map of the new constructed flow. tion has a certain property which makes it canonical. g:£ •*■ Z

has a cross section

whose suspension is

is conjugated This construc­ If G : E x M + M

F:3RxM_ ■*• M Q

then G and F are equivalent by an orbit preserving homeomorphisms. After this lengthy digression we return to the construction of a strange attractor for a diffeomorphism on a compact manifold M. It should be pointed out that Ruelle and Takens suggested that these attractors might lead to turbulence, namely, they say that from an equilibrium via a Hopf bifurcation a closed orbit is formed and then successive higher order bifurcations (or other mechanism, as in the Lorenz equations (Lecture I)) eventually lead

to strange attractors;

reference is made to the example described before (see [RT], page 170-171). First we need some definitions and statement of theorems basic in dynamical systems. and

If E is a Riemannian vector bundle over

f:E •* E a vector bundle morphism, f

(expanding) if there exists all

will be called contracting

c > 0 , 0 < X < l ( X > l )

v e E, n € K, lf"(v) I > cXnlvl

M

such that for

(respectively If^vl > cX n lvl).

Our definition obviously depends on the norms induced by the metrics; however, these norms are equivalent locally and hence if we suppose in addition that M is compact, the property of

f

being

contracting or expanding is independent of the Riemannian metric on M.

It is also clear that the inverse of an expanding bundle auto­

morphism is contracting and vice-versa. Riemannian manifold if

Tf: TM + TM

Now, if M is a compact

f:M •+ M will be called contracting (expanding)

is contracting (expanding).

808 65 From now on

f:M ■* M

will always denote a diffeomorphism on

a compact manifold M. A closed subset

A

of a compact manifold M is hyperbolic if

f(A) = A, the tangent bundle of

restricted to A , T.M is a

T A M = E s 9 E u so that for each

continuous Whitney sum

E

Tf(E^) = E | ( x ) , Tf(E^) = f ( x ) expanding on E , i.e.

M

and

Tf

is

x e A,

contracting on E s and

there exist constants c > 0, X € (0,1) such

n

that i T f ^ v ) ! <. cX |vl for all

v6E

s

, n > 0 , ITf"n(v)l <. cXnlvl

for all

v e E , n >. 0.

c and X

do depend on the used Riemannian metric even though the

property of A

Note that in this definition the constants

being hyperbolic is independent of the metric.

We

shall come back to these constants a little bit later. A point x e M is called wandering if there is a neighborhood U of x in M such that fn(U)f~iU=* for all n> 0. The set of wandering points is clearly open and invariant under f so that 0(f), the set of nonwandering points is closed and f-invariant. if f (x)= x for some n > 0; clearly such an Axiom A. and

x

f:M •*■ M satisfies Axiom A if

x e M is periodic

is in n(f). n(f) is hyperbolic

(J(f) = {x € M|x is periodic} . A metric is adapted to an Axiom A diffeomorphism f, if fl(f)

is hyperbolic with respect to it with

c =1.

It is proved in [HP]

that every Axiom A diffeomorphism has an addpted metric so that from now on whenever we deal with Axiom A diffeomorphisms we sup­ pose to have already an adftpted metric on

Spectral Decomposition Theorem.

If

M.

f:M -► M is an Axiom A dif-

feomorphism on the compact manifold M, then there is a unique way of

809 66 writing

8(f) = Q^V. . .U £^

where the

A. are pairwise disjoint

closed sets each one of them containing a dense orbit of f »r«

is topologically transitive) and such that f-invariant).

More, Q. = X, .U.,.UX„ l

1,1

f (i.e.

f(JJ.) = 0. (i.e. Q. . with the

n^ji

X. .'s 3,1

pairwise disjoint closed sets, f(X. .) = X.., .(X„ .. 1. = X. 1.) and 1 1 n , J n_

J)

J^-LJ

^' ' -»

•'•J

f X|X. . topologically mixing. Recall that a homeomorphism

h:N •* N is topologically mixing,

if for any two open sets U,V of N, there exists for all

n

u

n _> n>

n

n

h (V) * ♦•

n Q >. 0 such that

The proof of this strong version

of the spectral decomposition theorem can be found in [Bo], page 72-71*.

Denote

Ws(fj.) = {x €M|f n (x) ->aL

that fl. is an attractor if and only if borhood of

0. .

as

n + «} and note

s

W ((i.) contains a neigh­

0. are called basic sets.

Theorem (Bowen, Ruelle). Riemannian metric on M.

Let

m

denote the measure defined by the 2 Then a basic set (1. of 3f a C Axiom A dif-

feomorphism is an attractor if and only if

m(W (fl.)) > 0.

(For

the proof see [BoR], page 195). This result coupled with the spectral decomposition theorem yields the following 2 Corollary. Every C Axiom A diffeomorphism has at least an attrac­ tor. Almost every point of M tends to an attractor under iterates of

f.

k s Also note that for Axiom A diffeomorphisms we have M = U W (0.). x i=l We shall return to the basic sets investigating measure the­ oretical properties when dealing with ergodic properties later on.

810 67 With these results in mind, we can describe the general prin­ ciple of construction of strange attractors as it is presented in [SJ, page 788.

We start off with an expanding diffeomorphism

f:M -► M of a compact manifold. dimension 1+ dim M define

g\

:Dx

and imbed

H + DxH

by

Denote by D the full unit disc of M

in D x M as

0 x M.

g^(x,y) = (Xx,y).

x

0 M e (0,y) -* (0,f(y)) e D x M; since

For 0 < X < 1

Look now at the map

dim(D x M) = 1+ 2dim M and M

is compact, the imbeddings form a dense subset of C ( O x M , D x M ) in the C -strong topology (see [H], page 55) so that our map has a C -approximation


be a tubular neighborhood of

*(M) with fibers being the various

components of T n (D x {y}) for such that now

X

y € M.

Now extend

small enough such that

h= i|iog : D « M + D x M . A

Pick

g.(DxM) C T; this can be done by

Now define for these

X's

The following facts hold: A = f|h m (DxM) has m>0

a hyperbolic structure, is a basic set for stable; A

y> to f i T + D x M

\j> is a diffeomorphism and is fiber preserving.

a compactness argument.

Let now T

h

and is structurally

is locally the product of a Cantor set and a manifold of

dimension equal to

dim M.

Williams' construction in this general setting repeats step by step what we've done in the particular case M = S . Namely, denoting X. = M, define

u^.:X. ♦ X^^ for

i < j

by

u..(m.) = ^"■'(m.) and notice that l i m ( X . , u . . ) = { ( m , , . . . , m . , . . . ) € X , x . . . x X x . . . I u . . ( m . ) = m. = < x XT x K. x n xj j x IN = f 1_:] (m. ) , Vi < j } = { ( m . , . . . ,m , . . . ) 6 Mx.. . x M x . . . | f (n>v +1 ) =

m

)c}>

811 68 the last definition of this space being extremely suggestive for our process of forming A.

Actually, the obvious map given by the construction of A , A 3 (x,m) -► (m,f 1 (m),...,f —k (m) ,...,£ J.im(X. ,u..) ] IN

establishes an isomorphism of Via this identification, h

A with our above defined solenoid.

becomes the shift:

(nu,.. . ,m^,. . .) «■+ (f(m^),m^,...,m^,...). Strongly related to this construction is the following: Theorem (Bowen, Ruelle). on

M

Let

and fl. a basic set. l

f be a C Then for

2

Axiom A diffeomorphism

m - almost all points

x e Ws(JJi) one has t

.

-, n-1

lim i I gh-'-(x) = gdu n-~> n i=0 L i

for all continuous

g:M—►IR.

Here

V

ability measure on

J5-, invariant under

denotes a certain prob­ h

which has remarkable

ergodic properties. For the construction of Vi , a proof and other related results see [BoR], page 191. Topological entropy is defined by Bowen in the following way. Let

f:M •* M

For given if for any that

be a continuous map on a compact metric space M.

e > 0, n e K, a set x,y € E

with

d(f3(x),f3(y)) > e.

x * y

E C M

is called (e,n)-separated

there is a

j,0£J^n

such

It is easily seen that such a set E must

be discrete and closed, hence finite by compactness of M.

Denote

by Z (f,e) the largest cardinality of any (n,e)-separated set in M and let

h(f,e) = lim sup - log Z n (f,e). n-H»

The topological entropy

812

of

f

i s by d e f i n i t i o n :

h ( f ) = lim h ( f , e ) .

Denote by

' cardinality of the fixed point set of ber of periodic points of

f

of period

r , i.e.

N (f) the m

N (f) is the num­

m . Then Bowen proves

(see [Bol] for a proof) h(f) = lim ._ sup r i N (f) m-»<» m m whenever

f

is an Axiom A diffeomorphism on the compact manifold

M. This theorem then justifies the entropy statement in our ex­ ample.

Indeed, in order to have

D(6) into itself by 8, the coordinates tion. n

h

h (8,r,s) = (8,r,s) we must map

and once we establish the conditions on

r and s

follow automatically from our condi­

So we really have to ask how many periodic points of period

does

S

9 z >-► z

£ S

have; the answer is clearly

remark that our diffeomorphism

2n - 1 .

Now

h

satisfies Axiom A (fl(f) = A) so 2n-l that its topological entropy equals lim = log 2. We didn't n-*« n define here the usual measure theoretical entropy which differs in general from the topological entropy; however the two concepts are strongly related.

We refer the reader to Bowen's talk for

more information on ergodicity. I finish this talk with the remark that the Lorenz attractor described in Lecture I doesn't fit into this general framework of strange attractors. systematized.

Attractors like these haven't been yet

813

70 BIBLIOGRAPHY [B]

Bourbaki, N. Varie*te*s Dif f erentiables , Fascicule des Resultats, Sl-8, Hermann, Paris

[Bo]

Bowen, R. Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Mathematics U70, Springer Verlag 1975.

[Bol]

Bowen, R. Topological Entropy and Axiom A, Proc. Symp. Pure Math., vol. 14, Amer. Math. Soc. Providence R.I. 1970, pp. 23-41.

[BoR]

Bowen, R., Ruelle, D. The Ergodic Theory of Axiom A Flows, Inventiones Math., 29 (1975) pp. 181-202.

[H]

Hirsch, M. Differential Topology, Graduate Texts in Mathematics 33, Springer Verlag 19 75.

[HS]

Hirsch, M., Smale, S.' Differential Equations, Dynami­ cal Systems, and Linear Algebra, Academic Press, 1974.

[HP]

Hirsch, M., Pugh, C. Stable Manifolds and Hyperbolic Sets, Proc. Symp. Pure Math. 14 (1970), 133-163.

[PM]

Palis, J., deMelo, W. Introducao aos Sistemas DinSmicos, Coloquio Brasileiro de Matematica Pocos de Coldas, Julho 19 75, IMPA.

[R]

Robbin, J. Topological Conjugacy and Structural Stability for Discrete Dynamical Systems, BAMS, vol. 78, No. 6, November 1972, pp. 923-952.

[RT]

Ruelle, D., Takens, F. On the Nature of Turbulence, Commun. Math. Phys. 20 (1971), pp. 167-192.

[S]

Smale, S. Differentiable Synamical Systems, B.A.M.S. 73 (1967) pp. 797-817.

[Sh]

Shub, M.

[Shi]

Shub, M. Dynamical Systems, Filtrations and Entropy, B.A.M.S. vol. 80, No. 1, January 1974, pp. 27-41.

[W]

Walters, P. Ergodic Theory — Introductory Lectures, Lecture Notes in Mathematics 458, Springer Verlag, 1975.

[Wi]

Williams, R. Expanding Attractors, Publications Mathematiques, no. 43, I HES, 1974.

Stability in Dynamical Systems,

(preprint).

814

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BULLETIN OF THE AMERICAN MATHEMATICAL SOCIETY Volume 84, Number 6, November I9?8 C Amerinn Mithemitlcel Society 1978

Catastrophe theory: Selected papers, 1972-1977, by E. C. Zeeman, AddisonWesley, London, Amsterdam, Ontario, Sydney, Tokyo, 1977, x + 675 pp., $26.50 (hard binding), $14.50 (paper binding). For the general public, catastrophe theory (or CT) has become the biggest thing in mathematics. Rene Thom and Christopher Zeeman are the two leaders of this field. L'Express (October 14-30, 1974) asserts that the "new Newton" is French (i.e. Thom). An announcement of Zeeman's lecture at Northwestern University in the spring of 1977 contains a quote describing catastrophe theory as the most important development in mathematics since the invention of calculus 300 years ago. Newsweek has given similar comparisons. Zeeman juxtaposes Newton and Thom in the volume under review (briefly ZCT), p. 623. Thom writes " . . . CT is-quite likely-the first coherent attempt (since Aristotelian Logic) to give a theory on analogy." [p. 637, ZCT). On the back cover of Thorn's book, Structural stability and morphogenesis [English translation, Benjamin, 1975 or Thorn's SSM], is the quote from the London Times review, "In one sense the only book with which it can be compared is Newton's Principia." Recently however, the importance of CT has been sharply and publicly challenged by Hector Sussman and subsequently by Sussman and Raphael Zahler [Catastrophe theory as applied to the social and biological sciences: A critique to appear in Synthese]. A critical story on CT by Gina Kolata in "Science", April 15, 1977, is headed: Catastrophe theory: The emperor has no clothes. A front page story on the New York Times, November 19, 1977 focuses on the challenges to CT. To write a review in this environment has a very personal side for me. On one hand my own work on dynamical systems is closely connected to the origins of CT. I have had a long and close personal and professional relationship with both Thom and Zeeman. More than 20 years ago I was discussing singularities of maps, transversality, and immersions with Rene Thom. Thom tried to interest me in an early draft of chapters of his book Structural stability and morphogenesis in 1966. On the other hand I have remained skeptical and aloof from CT, perhaps due to my conservatism in science. While my colleagues and students were showing enthusiasm for CT, I gave critical lectures, one at the University of Chicago in 1974, one at the Aspen Institute of Physics in 1975. More recently I have been quoted negatively in the "Science" and New York Times references above. This is the first time I have written on the subject, and I should warn the reader of this negative bias, far from shared by many of my fellow mathematicians. Some of the mathematics underlying CT, especially transversality, and singularities of maps, has played a constructive role in outside disciplines, and is destined to play an ever increasing role. On the other hand I feel that CT itself has limited substance, great pretension and that catastrophe theorists have created a false picture in the mathematical community and the public as

815 BOOK REVIEWS

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to the power of CT to solve problems in the social and natural sciences. It is to the credit of Sussman and Zahler that they have seriously challenged this false picture. On this matter of warning to the reader, there is also the problem of my "quoting out of context". In fact the quotes I have made above and will make below are often imbedded in more prosaic language, perhaps tempering the brief and sometimes sharp sounding statements I refer to. At this point, I would like to acknowledge my debt to J. Guckenheimer, C Zeeman, H. Sussman, and M. Hirsch among others for very useful conversations on the subject here. Besides the work of Thorn and Zeeman, on Catastrophe Theory, I have also learned much from that of Sussman and Zahler. Just what is Catastrophe Theory? Thom writes that CT " . . . has to be considered as a theory of general morphology" 633, ZCT and Zeeman "CT is a new mathematical method for describing the evolution of forms in nature." (l.ZCT). However the new mathematics associated to CT really is contained in what is called elementary catastrophe theory (ECT) by Thom and Zeeman. Very briefly, ECT studies a smooth real valued map/ as a function (often called a "potential function") of a state x and parameter p. Here the state lies in some Euclidean space and p also varies in a (usually low dimensional) Euclidean space. The problem is to find local canonical forms for "generic" /, using a smooth change of coordinates in the variables x,\k,y— /(x, y). In this case, "generic" means for an open dense subset in a suitable function space. A local canonical form is defined in a neighborhood of a pair (JCQ, /IQ) at which/ is singular. ECT solves this problem when ft is in a low (< 5) dimensional space. In particular, when y. lies in a 4 dimensional space (e.g. space-time), Thom found seven canonical forms. We will shortly describe the case of the cusp catastrophe where x is in R and p. in R2. Is CT much more than ECT? Here Thom and Zeeman differ. Thom writes " . . . that Christopher's criticisms arise basically from a strict dogmatic view of CT which he identifies with E C T . . . " 633, ZCT. Zeeman replies (same page) that his emphasis on ECT has been mainly because of its usefulness in applications. In fact, the book under review, although titled CT, deals almost completely with ECT (its mathematics and applications). In my mind, when CT goes beyond ECT it loses pretty much any direct touch with mathematics. It is true that Thom refers to nonelementary CT as "generalized catastrophes, composed map catastrophes, (7-invariant catastrophes" etc. 633, ZCT. But here the relationship of the mathematics to nature in the CT literature becomes tenuous and infrequent For example in Thom (SSM), the words "generalized catastrophe" are used to describe situations where no mathematical model is proposed. More particularly under the picture of "feather buds on a chicken embryo", Thom writes "a generali­ zed catastrophe and example of symmetry in biology" (Figure 21, Thorn's SSM). He labels a picture of "the Crab nebula, the remains of the explosion of a supernova" with "a partially filament catastrophe in astrophysics" (Figure 17, Thorn's SSM). Thom described the filament catastrophe as a

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BOOK REVIEWS

certain "codimension two" type of generalized catastrophe, but the mathematics is not specified. No doubt it is in this spirit Thorn writes (p. 189 in Structural stability, catastrophe theory and applied mathematics, SIAM Rev., April, 1977) "The truth is that CT is not a mathematical theory, but a body of ideas, I dare say a state of mind." In ECT, one can take the gradient of the "potential" function with respect to the state variable to obtain a dynamical system x' - grad^(x) parame­ terized by n, /,,(*) - f(x, n). It is in this context that Thorn introduced ECT in his book SSM, Chapter 5. He takes ft to be in R4 or space-time. Elementary catastrophes are defined there as the points p of space-time for which the qualitative dynamics of x' - grad^(x) changes in a neighborhood of the attractors (stable equilibria) of that dynamics. Generalized catastrophes are defined analagously where the gradient dynamical systems are replaced by a more general class of dynamical systems. However the mathematical theorems that Thorn states (later proved by Mather) actually give the classification of elementary catastrophes according to the theory of singularities of maps in contrast to the theory of dynamical systems. Guckenheimer subsequently pointed out (Bull. Amer. Math. Soc. 79 (1973), 878-890, and in Bifurcation and catastrophe, Dynamical Systems (ed, Peixoto) Academic Press, 1973) that the two classifications differ already when n is in R3 (and certainly for y. in R4). I believe this weakens much of the scientific or philosophical foundations of CT which is a theory based on dynamical systems (Thorn's SSM). If even the elementary catastrophes don't correspond to the dynamical classification, what of the generalized catastrophes where the mathematics in SSM deals in only a few known examples? In fact ECT seems much closer to what economists call comparative statics than to dynamical systems. One can sense a corresponding change of perspective in the work of the catastrophe theorists. As we have noted, the starting point of CT in Thorn's SSM was the "bifurcations" of dynamical systems parameterized by spacetime. In Zeeman's work, these parameters changed from "space-time" to "control parameters". Now in his SIAM article (cited above) Thorn also speaks of control parameters and seems to de-emphasize the dynamical system foundations of CT, especially relative to space-time. The cusp catastrophe is the most important example of a catastrophe and much if not most of the book under review (ZCT) centers around die cusp catastrophe and its applications. Zeeman with Isnard (ZCT, 329 in Some models from catastrophe theory in the social sciences) describes the canonical model as follows. In 3 dimensional space R3, variables (a, b, x\ let M be the cubic surface defined by x3 - a + bx. Here a, b are horizontal axes and are the controls; x is the vertical state variable. The fold curve F is where the vertical lines are tangent to M and is given by 3x2 - b. The projection of F onto the control space is called the bifurcation set B. The equation of B is 27a2 — 4b3 and has a cusp at the origin. It is supposed that M is given by dP/dx — 0 where P is some (probability or potential) function P: R 3 -»R

817 BOOK REVIEWS

1M3

and that a sociologically (or physically) meaningful subset G of M a the set of local maxima (or local minima) of P. The state of the system stays on G and is controlled by the choice of (a, b). Thus when (a, b) crosses B, the state may be forced to make a discontinuous jump to remain on G. This is all laid out clearly in the following Figure 330, ZCT from Zeeman and Isnard.

FIGURE 11*

In the model under discussion, a is a numerical representation of a threat, b the cost and x represents military action. The discontinuity, representing a jump in military action could be thought of as a declaration of war. Thus what Zeeman and Isnard have given us is a model for the study of a nation deciding upon its level of action in some war. I consider this paper the most developed of Zeeman's papers on CT and the social sciences and he writes "And I believe that sociology may well be one of the first fields to feel the full impact of this new type of applied mathematics,..." (627, ZCT). Sussman and Zahlcr have discussed this model in detail. Here I would like to focus on the question of its justification as a model of military decision •Reprinted from the chapter entitled "Some models from catastrophe theory in the social sciences" by C A. Isnard and E C Zeeman in The UseofModeb in the Social Sciences (Social Issues in the Seventies Series) edited by Lyndhurst Collins, Tavistock Publications, London, copyright O 1976 Seminan Committee of the Faculty of Social Sciences of the University of Edinburgh, Scotland.

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making; in this I believe the authors have failed. Their efforts in this direction lie on the mathematical side. Zeeman and Isnard write "... we shall intro­ duce sociological hypotheses, and translate them into mathematics. The deep theorems of CT will enable us to synthesize the mathematics. We can then translate the synthesis back into sociological conclusions. It is not immediately apparent, without the use of the intervening mathematics, that the sociological hypotheses imply the sociological conclusions, and that is the purpose of using catastrophe theory.*' (314, ZCT). The trouble is with the sociological hypotheses. Evidence for them, or for the model, should be given, for example from military history, studies in decision making, or sociological, political studies in general. There is much theory and data from history and social sciences relevant to the model of Zeeman and Isnard. None of this flnds its way into the paper directly or indirectly save for brief references to Tolstoy on calculus in War and peace and Lorenz, On aggression. Let us look at some of the sociological hypotheses that Zeeman and Isnard in fact do make. Summarized into mathematics, they express these hypotheses graphically by:

threat FIGURE 6, 316,

ZCT

cost large b « constant

threat

FIGURE 7,317, ZCT

819 BOOK REVIEWS

IMS

Zeeman and Isnard then ask how does Figure 6 evolve into Figure 7. "The main theorem of CT tells us that qualitatively there is only one way for this evolution to occur" [p. 318, ZCT], and they deduce Figure 11, cited above. One trouble is the definition for "small" cost and "large" cost that Zeeman and Isnard use. They mean there exists b0 such that "small b" means b < b0 and "large b" means b > b^, the same b0 for "small" and "large". Thus Figures 6 and 7 (the sociological hypotheses) already describe the model for all * save b0; Figure 6 applies if b < b^ Figure 7 if b > b0 [p. 331, ZCTJ. No evidence or justification is given for this sociological hypothesis, that such a b0 exists (there are arguments given earlier however to justify a "delay rule" vs. "Maxwell's rule"). In an earlier paper by Zeeman alone on a model for the stock exchange, (paper 11 in ZCT), Zeeman uses the words small and large in the same way as above [p. 364, ZCT]. Thus his hypotheses already give, without using any mathematics at all, the structure of the model as a surface in E3 save for a 2-dimensional plane described by one control parameter being constant Nevertheless Zeeman refers to these hypotheses as "disconnected" and "local". He states: "Summarising: we insert seven disconnected elementary local hypotheses into the mathematics, and the mathematics then synthesises them for us and hands us back a global dynamic understanding." [p. 362, ZCT] and elsewhere in that paper, "We now use the deep classification theorem of Thorn to synthesise the information acquired so far into a 3-dimensional picture of the surface 5 . . . " (p. 366, ZCT]. Note that Zeeman is referring to the same theorem that Thorn credits to Whitney in "Nature", December 22, 1977. Thorn referring to Sussman and Zahler writes: "I would also like to point out a misquotation by the authors. The classification theorem for the "Cusp catastrophe" erroneously quoted as "Thorn's theorem" is in this specific case due to H. Whitney." No justification for the hypotheses of Zeeman's stock exchange model is given in terms of existing data and/or theory of stock exchanges, price theory, etc. In fact no reference to any economic literature is given in this paper. A defense might still be made that even though Zeeman doesn't justify the stock exchange or war models, still it is good that he has proposed them. Doing so introduces topology (or geometry) into the social sciences and with luck sociologists or economists will be able to develop the models, test them, verify them, etc. Indeed it is important for scientists, social or otherwise, to be aware of mathematical possibilities for models. A positive aspect of all the publicity given to CT is that it may have increased this kind of awareness. On the other hand good mathematical models are not generated by mathematicians throwing models to sociologists, biologists, etc. for the latter to pick up and develop. Both Thorn and Zeeman seem to fit this caricature sometimes in their work or when they give their views on the future of CT in science. Good mathematical models don't start with the mathematics, but with a deep study of certain natural phenomena. Mathematical awareness or even sophistication is useful when working to model economic phenomena for example, but a successful model depends much more on a penetrating study and understanding of the economics.

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On the other hand around CT, not only does mathematics come first but one sees a sort of mathematical egocentricity; understanding the world is a mathematical (even geometrical) problem. Thorn's position on this is clear; e.g. " . . . Eliminate the "obvious" meaning and replace it by the purely abstract geometrical manipulation of forms. The only possible theoretisation is Mathematical." (638, ZCT) or " . . . I agree with P. Antonelli, when he states that theoretical biology should be done in Mathematical Departments; we have to let biologists busy themselves with their very concrete-but almost meaningless-experiments; in developmental Biology, how could they hope to solve a problem they cannot even formulate?" (636, ZCT). Along with this mathematical egocentricity there is a kind of mystification of the subject that is being created by both Thom and Zeeman. Zeeman does this when he speaks of the "deep classification theorem of Thom" as above and elsewhere in his papers. Presenting this picture to nonmathematicians and even nontopologists has an intimidating effect Thom does this by using technical mathematical terms without explanation when addressing nonmathematical audiences, and often writing obscurely. Then Zeeman deepens the mystifying power of CT by explaining Thorn's obscurities with: "When I get stuck at some point in his writing, and happen to ask him, his replies generally reveal a vast new unsuspected goldmine of ideas" (622, ZCT). Some defenders of CT may accuse me of discussing very special examples not characteristic of the hterature of the subject I feel that the problem of lack of justification discussed above, is also found in Zeeman's other models. Furthermore Thorn's models are even less specific and less developed. On the other hand, Thorn's work in CT covers many subjects; in this connection Zeeman writes in his Scientific American article, April 1976, p. 65: "The method has the potential for describing the evolution of form in all aspects of nature, and hence it embodies a theory of great generality." Corresponding to the universality of CT is a certain superficiality, almost in a dual sense! But one can find even more local maxima in nature than cusps. It is Thom and Zeeman who have brought CT to the attention of the scientific community with their studies mainly in biology and the social sciences. Thus I can't go along with Ian Stewart's assertion, in "Nature", December 1, 1977, p. 382, T h e case in favour of CT rests not on speculative models in the social sciences, but on successful applications to the physical sciences." I would like to make it clear that I find merit in the Catastrophe theorists use of modern calculus and geometric techniques in models in science. In particular discontinuities can often best be understood via this kind of mathematics. For example it would be important to find a calculus oriented model for the computer, a machine which is intrinsically discrete. Such a calculus model would not be exact but it could give great insight to automata theory. There is much value to science in some of the underlying mathematics of CT. I think especially of transversality theory and the theory of singularities of maps. The idea of transversality goes back in history, but took a good development with Pontryagin and Whitney. Thom had used transversality in

821 BOOK REVIEWS

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his early work and by 1956 was putting the concept into a powerful systema­ tic use with his theorem of transversality of jets. In its modem form, transversality for a smooth map/: V-+ W, relative to a submanifold M of W means that whenever f(x) « y 6 M, TX(W)~ Ty(M) + Df(x)(Tx(V)). In other words the derivative of / at x has image complementing the tangent space of M at y. In this case the inverse function theorem implies that / " \M) is a submanifold of the expected dimension. Transversality has many ramifications, especially when / itself is a deriva­ tive map (or jet map) of some order of another map. These ideas give form to the study of singularities of maps. Again it was Thorn who, after fundamental work of Whitney, substantially enriched that theory. Now transversal maps have the property of being an open and dense subset of an appropriate function space. The openness of transversality often can be used to show that corresponding models have the important property of robustness. A model is robust if the properties under study remain after perturbation. The approxi­ mation properties of transversality imply that one might expect to find transversality present in smooth models. For these reasons it is important for mathematically oriented scientists to be aware of transversality and such examples as cusps. In particular, workers in bifurcation theory from a classical point of view should be (and to some extent are) studying these modern ideas. Elementary catastrophe theory studies a particular class of maps, (the target space is 1-dimensional, but parameters are allowed) from the above point of view. Sometimes, as in classical mechanics, this situation arises naturally. For example, statics in a simple mechanical system (M, K, V) studies the local minimum of the potential. Here K is kinetic energy or a Riemannian metric on configuration space M while V: A/-»R is the potential energy. Suppose that V depends on a parameter n in R* (i.e., comparative statics). Then ECT is natural for such a study as in ZCT, # 17, A Catastrophe model for the stability of ships. While this kind of mathematics has a legitimate and even important place in physics, I must object to Zeeman's assertion in the introduction of his ship article (442, ZCT): "it (this example SS) is a prototype revealing catastrophe theory as a natural generalization of Hamiltonian dynamics." More generally the theory of singularities of maps has a constructive role to play in the physical, biological and social sciences. For example comparative statics, as in Paul Samuetson's Foundations, studies economic equilibrium prices/* satisfying f(p, (i) =■ 0 where for each parameter value p,p -*f{p, /t) is a map from R" to R". The problem is: how does a solution vary with-p, even for small p. Since economic theory has shown pretty clearly that the excess demand (in some form) / is not derived from any potential function, CT itself is not relevant. On the other hand, the problem naturally fits into the theory of singularities of maps. We end this review by a remark on history. Catastrophe theorists often speak as if CT (or Thorn's work) was the first important or systematic (CT is "systematic" as a certain study of singularities, but not as a study of discontinuous phenomena) study of discontinuous phenomena via calculus mathematics. My view is quite the contrary and in fact I feel the Hopf

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bifurcation (1942) for example lies deeper than CT. The Hopf theory shows how a stable equilibrium bifurcates to a stable oscillation in ordinary differential equations. Moreover, there is the reference Theory of oscillations by Andronov and Chaiken, 1937, with English translation in 1949 published by the Princeton University Press which is never referred by Thorn or Zeeman. This book besides giving an early account of structural stability, gives a good account of dynamical systems in two variables with explicit development of discontinuous phenomena, quite close to Zeeman's use of the cusp catastrophe. Examples from physics and electrical engineering are studied in some depth. STEPHEN SMALE

823

ON THE PROBLEM OF REVIVING THE ERGODIC HYPOTHESIS OF BOLTZMANN AND BIRKHOFF Dedicated to the memory of Rufui Bowen

I would like to develop the idea that by introducing a dissipation/forc­ ing term into Hamilton's equations of physics, one might be able to revive the ergodic hypothesis of Boltzmann and Birkhoff. The compelling fact is that with one main exception (end of section 1), Hamiltonian systems are not ergodic; furthermore other qualitative features of these systems don't seem amenable to study. On the other hand, large classes of general ordinary differential equations have ergodic structurally stable attractors with mixing and other properties. Ergodic theory was developed originally in response to problems arising from Hamilton's ordinary differential equations; but it seems now to have a more natural home in general dynamical systems (at least to the extent it relates to ordinary differential equations). The goal of this note is to pose the problem of finding a nonHamiltonian perturbation of a given Hamiltonian system to produce an c-dense ergodic attractor. Section 1. We give a very brief review of the origins of ergodic theory. But see, e.g., Birkhoff and Koopman in [4], Khinchin [9], or Lanford [10] for a more extended account. The word ergodic was first used by Boltzmann about 100 years ago, at the time he introduced the first version of the "ergodic hypothesis." Towards the understanding of thermodynamics, Boltzmann was con­ cerned with the statistics of a system of a large number of particles moving according to the laws of mechanics. Thus state space (or phase space) of the system consists of the possible positions and momenta of each particle and the dynamics is described by Hamiltonian ordinary differential equa­ tions. Energy is a function on state space and "conservation of energy" asserts that energy is constant under the motion of a state. The flow is thus defined on each surface VE of constant energy E. Moreover, there is a natural measure on state space (Liouville) and on each surface of constant energy as well. The flow leaves these measures invariant. Boltzmann's "ergodic hypothesis" asserted that the trajectory of a single point in phase space passed through every point on that energy surface. Written for the "International Conference on Non-linear Dynamic*** of the New York Academy of Science, Dec. 1979.

824

S. SMALE Before long that was revised to the "quasi-ergodic hypothesis," namely that almost all trajectories were dense on the energy surface. The quasi-ergodic hypothesis was often accepted under additional con­ ditions on the system; e.g., if the system has no constants other than energy and if the energy surface is bounded. For example, there is the 1923 paper of Fermi, "Proof that a mechanical normal system in general is quasi-ergodic" [8]. The situation was clarified by Birkhoff s ergodic theorem in 1931 (which in turn was partly inspired by Von Neumann's mean ergodic theorem). THEOREM (BIRKHOFF ERGODIC THEOREM [4]). Suppose X is a space with a finite measure and a flow X X R-*X, (x, t)^*xt, which leaves the measure invariant {e.g., x, is given as the solution of an ordinary differential equation on X so that x,j r _ 0 = x G X is the initial condition). Let f:X-*R be an integrable function {e.g., X -» VE is an energy surface in phase space and f is a "phase function"). Then except for a set of initial x of measure 0,

r-»oo i Jo exists and f:X-*R

is integrable and invariant under the flow.

Thus "time averages" exist almost everywhere. The flow above is called ergodic (or metrically transitive in the terminol­ ogy of Birkhoff) provided every invariant integrable function is constant (almost everywhere). If the flow on X is ergodic, then according to the ergodic theorem of Birkhoff, / is constant and

lim 1 [Tf(x,)dt-f f T-»oo / Jo

JyE

is independent of x: time averages equal space averages. Thus an observ­ able (phase function) has a time average independent of initial condition, The above considerations suggest a new version of the ergodic hypothe­ sis or what Birkhoff called the hypothesis of metric transitivity. We quote from the 1932 article of Birkhoff and Koopman, in [4], p. 465. It may be stated in conclusion that the outstanding unsolved problem in the ergodic theory is the question of the truth or falsity of metrical transitivity for general Hamiltonian systems. In other words the QuasiErgodic Hypothesis has been replaced by its modern version: the Hypothe­ sis of Metrical Transitivity. This hypothesis played an important role in Birkhoffs later work. He not only believed it but part of his work is written assuming that it is true.

825 THE ERGODIC HYPOTHESIS OF BOLTZMANN AND BIRKHOFF

Let me quote from his famous "pontifical memoir," "Nouvelles Recherches . . . " in [4], p. 632. J'ai reussi plus recemment i demontrer meme l'existencc des regions annulaires d'instabiliti, ce qui indique que Phypothese de transitivity est presque certainement remplie dans le cas general. D'ailleurs cette hypothese a toujour ete employee par les physiciens avec grand profit dans la theorie de la mecanique statistique. Cest pourquoi nous allons nous occuper de la theorie generate d'un tel systeme transitif . . . These beliefs held sway in mathematical physics until KolmogorofFs famous Amsterdam Congress paper in 1954 (see [1]) and subsequent work of Arnold and Moser in 1961-1962 (see [3J, [15]). The work of Kolmogoroff, Arnold, and Moser, KAM, showed that near "elliptic" closed orbits of a general Hamiltonian system on an energy surface, ergodicity failed. In that case there exist families of invariant tori of positive measure. Furthermore these elliptic orbits occur frequently in Hamiltonian sys­ tems. Thus the hypothesis of metrical transitivity is false in a definite way (see [1], [13D. In spite of the work of Kolmogoroff, Arnold, and Moser, there seems to be some kind of ergodicity in physical systems. For example Lebowitz writes [11] p. 41: Now I believe that almost all real physical systems are "essentially" ergodic. Indeed this is necessary for understanding why equilibrium statistical mechanics, which includes a description of fluctuations in thermal equilibrium works so well in the real world. There is one exceptional case of some importance where the system has no elliptic points and ergodicity has been proved. Geometrically, this happens when a particle moves on a manifold of negative curvature (Anosov) [2]. On the physical side is the work of Sinai [19] on ergodicity of a system of elastic spheres. In these cases the ergodicity proofs came out of the framework of the general differentiable dynamical systems of the next section; ideas of E. Hopf were used. Section 2. Parallel to the growth of interest in Hamiltonian systems, there has been a development of general dynamical systems. Part of the motivation for this development has been from the physical considerations of dissipative and driving forces which destroy the Hamiltonian character as in the Navier-Stokes equations or the Rayleigh dissipation function. Another part of the motivation has to do with the mathematical simplicity and geometry of dealing with unconstrained ordinary differential equa­ tions, as the Poincare-Bendixon theorem. A basic feature of these general

826

S.SMALE dynamical systems (or sometimes simply "dynamical systems," or "differentiable dynamical systems" or the "qualitative theory of ordinary differential equations") is the notion of asymptotic behavior of a (typical) solution. This leads to certain invariant sets and in particular to attractors of the dynamical system. Dynamical systems have two kinds of classical attractors which persist under small perturbations of the differential equations. These are the stable equilibria and the stable nontrivial periodic solutions or oscillators. An important development of recent times is a new kind of attractor which is robust in the sense that its properties persist under perturbations of the differential equation (it is structurally stable). These new attractors are sometimes called strange attractors. The under­ lying phenomena was seen by Poincare, Birkhoff, Cartwright and Littlewood, Hadamard, Morse, and others. But more recently a systematic mathematical basis has been laid for their understanding. And comple­ menting that base has been the work of Lorenz [12], May [14], and others on the applied side. We will give a little of the mathematical side as it applied to our problem here; but see [20], my article in [17], Bowen [5], [6], Ruelle [18], and the cited references for a more complete picture. Axiom A no-cycle dynamical systems, or for short, Axiom A systems enjoy some satisfying features. First they form a large class and second they are robust and admit analysis. This class includes all the simplest examples which are robust (gradient systems, etc.) and, at the other extreme, those Hamiltonian systems known to be ergodic and mixing. But it also includes a large spectrum of examples in between, dynamical systems with horseshoes, strange attractors, etc. Some important examples are not Axiom A, however, and it is an important problem to expand Axiom A systems to include, e.g., the Lorenz attractor. But the wealth of interesting and useful examples included make Axiom A systems an attractive class. On the other hand, properties of Axiom A systems persist under pertur­ bation; they are "fl-stable," i.e., structurally stable on the set of all recurrent solutions. This leads to a number of other good properties. For the definitions and development, the reader may consult the above refer­ ences. Furthermore, there is the fundamental Bowen-Ruelle theorem, which goes as follows: First recall that a closed invariant set A is called an attractor if all nearby solutions lead to A as /-»oo. The basin of A is the union of all solutions which tend to A as f-»oo. An Axiom A system has a finite number of attractors. THEOREM (BOWEN-RUELLE THEOREM). For an Axiom A system, except for an initial set of Lebesgue measure 0, time averages exist for continuous

827

THE ERGODIC HYPOTHESIS OF BOLTZMANN AND BIRKHOFF phase junctions. More precisely, except for this initial set, a solution t-*xt tends to some attractor A. The at tractor A has a canonical invariant ergodic measure \a and if is continuous function on the space, then

lim ^ (T
r-*oo / Jo

JA

It is important to note that the theorem applies to almost all solutions with respect to ordinary Lebesgue measure on the space; but the asymp­ totic behavior is described by a new measure on A. In the last 25 years, there has been a turnabout in the relationship of ergodic theory to ordinary differential equations. Previously, ergodicity was associated only to constrained ordinary differential equations as Hamiltonian (or volume preserving at least). Nowadays it is not the Hamiltonian, but the general dynamical systems where ergodicity is fitting more naturally. We would conclude that theoretical physics and statistical mechanics should not be tied to Hamiltonian equations so absolutely as in the past. On physical grounds, it is certainly reasonable to expect physical systems to have (perhaps very small) non-Hamiltonian perturbations due to friction and driving effects from outside energy absorbtion. Today also mathemati­ cal grounds suggest that it is reasonable to develop a more nonHamiltonian approach to some aspects of physics. This leads to the problem we pose in Section 3. Section 3. Let us formalize the problem of reviving the ergodic hypothe­ sis via the introduction of a dissipative/forcing term. We will do this by posing an abstract idealized mathematical problem, and then discussing some variations and ramifications. In particular the dissipative/forcing term in our statement is any perturbation which balances the energy. We will call an attractor A ergodic if it satisfies the conclusion of the BowenRuelle theorem, with the measure positive on open sets of the attractor. A subset A of X is called (.-dense if every point of X is within c of some point of A. MAIN PROBLEM. Given a Hamiltonian system dx/dt « £ with energy surface VE and « > 0, is there a new dynamical system dx/dt — i) on VE, not necessarily Hamiltonian, e-close to £ (uniformly with derivatives up to order r), with an ergodic attractor A c-dense in VE such that almost every point of VE tends to A as r -> oo? More broadly, to what extent can such a system dx/dt — TJ be found?

This problem asks for a perturbation of a Hamiltonian system with very strong properties and it might well happen that somewhat less could be

828

S.SMALE

true and still be interesting for the goal of reviving the ergodic hypothesis. On the other hand more could be true. The following remarks elucidate these questions. Remarks: (1) S. Newhouse has pointed out to me that his well-known example can be used to show that a Hamiltonian system cannot always be perturbed to an Axiom A system. Of course if the Hamiltonian system is the geodesic flow of a compact Riemannian manifold with negative curvature, no perturbation is necessary. One would like a solution of the problem with good robustness proper­ ties. And also one would want the attractor to be mixing as well as ergodic. (2) The results of Newhouse, Ruelle, and Takens [16] are related here. They show that non-Hamiltonian perturbations of uncoupled harmonic oscillators may produce Axiom A strange attractors. Their argument seems to yield more; one obtains an Axiom A system with just one attractor which is c-dense in a single invariant torus of the original Hamiltonian system. One might be able to extend their argument to get the attractor c-dense in the energy surface. But still this would work only for the uncoupled oscillators. The coupled oscillators seem much more difficult as in (3) below. Also for the case of the uncoupled oscillators, a nonstrange attractor could probably be obtained. (3) A focus for the main problem could be the KAM picture of an area-preserving diffeomorphism of the 2-disk which centers about a gen­ eral elliptic point (see, e.g., [3], p. 77 or [1], p. 585). Can this diffeomorphism be perturbed to a system with a single c-dense attractor? The same question applies to general 2-dimensional area-preserving diffeomorphisms. (4) One way of relaxing the problem in a natural way is to not fix the energy surface, but consider perturbations in an c-neighborhood of a given energy surface. In fact there is no reason to demand that the dissipative effort and the driving effect exactly balance to preserve the energy. (5) The physical side of these matters might constrain the nonHamiltonian dissipative/forcing perturbation to having a special form. If the original Hamiltonian, for example, came for a simple mechanical system (A/, K, V) (M a manifold, K kinetic energy, and V a potential) then especially it would make sense to consider only non-Hamiltonian perturba­ tions with some special structure. Furthermore in this case one might want to study large perturbations. (6) Another important example is the Navier-Stokes and Euler equa­ tions of fluid dynamics (a complicating feature is that state space is infinite dimensional). The Navier-Stokes equations can be regraded as the Hamil­ tonian Euler equations with an added term which gives a non-Hamiltonian perturbation. Moreover it is often believed that the Navier-Stokes equa-

829 THE ERGODIC HYPOTHESIS OF BOLTZMANN AND BIRKHOFF tions model turbulence. Turbulence according to Ruelle and Takens (see [17] or [18]) corresponds to a strange attractor of the perturbed Hamilto­ nian system. Furthermore Chorin in his numerical scheme [7] for the Navier-Stokes equations takes a random non-Hamiltonian perturbation of Euler equations and finds statistics that could be explained by the presence of a strange attractor. While not giving rigorous mathematical backing for our main idea, I find the above considerations supporting the idea of the ergodic hypothesis lying in the framework of non-Hamiltonian perturbations of Hamiltonian systems. REFERENCES

[1] Abraham, R. and Marsden, J., Foundations of Mechanics, 2nd Edition, Benja­ min, Reading, 1978. [2] Anosov, D., Geodesic flows on compact riemannian manifolds of negative curvature, Proc. Steklov Inst. of Math. 90 (1967). [3] Arnold, V. I. and Avez, A. Theorie Ergodique des Systemes Dynamique, Gauthier-Villars, Paris, 1967. [4] Birkhoff, G., Collected Math. Papers, Vol. II, Amer. Math. Soc., N.Y., 1950. [S] Bowen, R., Equilibrium States and the Ergodic theory of Anosov Diffeomorphisms, Springer-Verlag, N.Y., 1975. [6] , On Axiom A Diffeomorphisms, Amer. Math. Soc., Providence, 1978. [7] Chorin, A., Numerical study of slightly viscous flow, J. Fluid Meek 57 (1973), 785-7%. [8] Fermi, E., Beweis, dass ein mechanisches Normalsystem in allgemeinen quasiergodisch ist, Phys. Z. 24 (1973), 261-264. [9] Khinchin, Statistical Mechanics, Dover, N.Y., 1949. [10] Lanford, O., Erogodic theory and approach to equilibrium for finite and infinite systems, Ada Physica Austriaca, Suppl. X. (1973), 619-639. [11] Lebowitz, J., Hamiltonian flows and rigorous results in non-equilibrium statis­ tical mechanics, in Proceedings of the 6th IUPAP Conference on Statistical Mechanics, Rice, Freed, and Light (eds.), University of Chicago, 1972. [12] Lorenz, E., Deterministic non-periodic flow, J. Atmos. Sci. 29 (1963) 130-141. [13] Markus, L. and Meyer, K., Generic Hamiltonian Dynamical Systems are neither Integrable nor Ergodic, Am. Math. Soc., Providence, 1974. [14] May, R., Simple mathematical models with very complicated dynamics, Na­ ture 261 (1976), 459-467. [15] Moser, J., Stable and Random Motions in Dynamical Systems, Princeton Univ. Press, Princeton, 1973. [16] Newhouse, S., Ruelle, D., and Takens, F., Occurence of strange Axiom A extractors near quasi-periodicflowson Tm, m > 3, preprint.

830

S. SMALE [17] Ratiu, T. and Bernard, P. (eds.), Turbulence Seminar, Berkeley Springer-Verlag, N.Y., 1977.

1976/77,

[18] Ruelle, D., The Lorenz attractor and the problem of turbulence, in Turbulence and Navier Stokes Equations, R. Teman (ed.), Springer-Verlag, N.Y., 1976. [19] Sinai, Y., Dynamical systems with elastic reflections, Russ. Math. Surveys 25 (1970), 137-189. [20] Smale, S., Differentiable dynamical systems, Bull. Am. Math. Soc. 73 (1967), 747-817.

831

ON HOW I GOT STARTED IN DYNAMICAL SYSTEMS* 1959-1962

1. Let me first give a little mathematical background. This is conve­ niently divided into two parts. The first is the theory of ordinary differen­ tial equations having a finite number of periodic solutions; and the second has to do with the case of infinitely many solutions, or, roughly speaking, with "homoclinic behavior." For an ordinary differential equation in the plane (or a 2nd-order equation in one variable), generally there are a finite number of periodic solutions. The Poincare-Bendixson theory yields substantial information. If a solution is bounded and has no equilibria in its limit set, its asymptotic behavior is periodic. The Van der Pol equation without forcing is an outstanding example which has played a large role historically in the qualitative theory of ordinary differential equations. There was some systemization of this theory by Andronov and Pontryagin in the 1930's when these scientists introduced the notion of structural stability. There is a school now in the Soviet Union, the "Gorki school" which reflects this tradition. Andronov's wife, AndronovaLeontevich, was a member of this school, and I met her in Kiev in 1961 (Andronov had died earlier). A fine book (translated into English) by Andronov, Chaiken, and Witt gives an account of this mathematics. The work of Andronov-Pontryagin was picked up by Lefschetz after World War II. Lefschetz's book and influence helped establish the study of structural stability in America. The second part of the background mathematics has to do with the phenomenon of an infinite number of periodic solutions (persistent under perturbation) or the closely related notion of homoclinic solutions. Again, Poincare wrote on these problems; Birkhoff found a deeper connection between the two concepts, homoclinic solutions and an infinite number of periodic points for transformations of the plane. From a different direc­ tion, Cartwright and Littlewood, in their extensive studies of the Van der Pol equation with forcing term, came across the same phenomena. Then Levinson simplifed some of this work. 2. Next, I would like to give a little personal background. I finished my thesis in topology with Raoul Bott in 1956 at the University of Michigan; and that summer I attended my first mathematics conference, in Mexico City, with my wife, Clara. It was an international conference in topology and my introduction to the international mathematics community. There I * Partly baaed on a talk given at a Berkeley seminar circa 1976.

832

S. SMALE

met Rene Thom and two graduate students from the University of Chi­ cago, Moe Hirsch and Elon Lima. Thom was to visit the University of Chicago that fall, and 1 was starting my first teaching job at Chicago then (not in the mathematics department, but in the College). In the fall 1 became good friends with all three. I attended Thorn's lectures on transversality theory and was happy that Thom and Hirsch became interested in my work on immersion theory. My main interests were in topology throughout these years, but already my thesis contained a section on ordinary differential equations. Also at Chicago, I used an ordinary differential equation argument to find the homotopy structure of the space of diffeomorphisms of the 2-sphere. Having both Dick Palais and Shlomo Steinberg at Chicago was very helpful in getting some understanding of dynamical systems. 3. It was around 1958 that I first met Mauricio Peixoto. We were introduced by Lima who was finishing his Ph.D. at that time with Ed Spanier. Through Lefschetz, Peixoto had become interested in structural stability and he showed me his own results on structural stability on the disk D2 (in a paper which was to appear in the Annals of Mathematics, 1959). I was immediately enthusiastic, not only about what he was doing, but with the possibility that, using my topology background, I could extend his work to n dimensions. I was extremely naive about ordinary differential equations at that time and was also extremely presumptuous. Peixoto told me that he had met Pontryagin, who said that he didn't believe in structural stability in dimensions greater than two, but that only increased the challenge. In fact, I did make one contribution at that time. Peixoto had used the condition on D2 of Andronov and Pontryagin that "no solution joins saddles." This was a necessary condition for structural stability. Having learned about transversality from Thom, I suggested the generalization for higher dimensions: the stable and unstable manifolds of the equilibria (and now also of the nontrivial periodic solutions) intersect transversally. In fact, this was a useful condition, and I wrote a paper on Morse inequalities for a class of dynamical systems incorporating it. However, my overenthusiasm led me to suggest in the paper that these systems were almost all (an open dense set) of ordinary differential equations! If I had been at all familiar with the literature (Poincare, Birkhoff, Cartwright-Littlewood), I would have seen how crazy this idea was. On the other hand, these systems, though sharply limited, would find a place in the literature, and were christened Morse-Smale dynamical sys­ tems by Thom. This work gave me entry into the mathematical world of ordinary differential equations. In this way, I met Lefschetz and gave a lecture at a conference on this subject in Mexico City in the summer of 1959.

833 ON HOW I GOT STARTED IN DYNAMICAL SYSTEMS

We had moved from Chicago in the summer of 1958 to the Institute for Advanced Study in Princeton; Peixoto and Lima invited me to Rio to finish the second year of my NSF postdoctoral fellowship. Thus, Clara and I and our two kids, Nat and Laura, left Princeton in December, 1959, for Rio. 4. With kids aged \ and 2 { , and most of our luggage consisting of diapers, Clara and I stopped to see the Panamanian jungles (from a taxi). I had heard about and always wanted to see the famous railway from Quito in the high Andes to the jungle port of Guayaquil in Ecuador. So, around Christmas of 1959, the four of us were on that train. Then we spent a few days in Lima, Peru, during which we were all quite sick. We finally took the plane to Rio. I still remember well arriving at night and going out several times trying to get milk for the crying kids, always returning with cream or yogurt, etc. We learned later that milk was sold only in the morning in Rio. But with the help of the Limas, the Peixotos, and the Nachbins, life got straightened out for us. In fact, it happened that just before we arrived the leader of an abortive coup, an air force officer, had escaped to Argentina. We got his luxurious Copacabana apartment and maids as well. Shortly after arriving in Rio my paper on dynamical systems appeared, and Levinson wrote me that one couldn't expect my systems to occur so generally. His own paper (which, in turn, had been inspired by work of Cartwright and Littlewood) already contained a counter-example. There were an infinite number of periodic solutions and they could not be perturbed away. Still partly with disbelief, I spent a lot of time studying his paper, eventually becoming convinced. In fact this led to my second result in dynamical systems, the horseshoe, which was an abstract geometrization of what Levinson and Cartwright-Littlewood had found more analytically before. Moreover the horseshoe could be analysed completely qualitatively and shown to be structurally stable. For the record, the picture I abstracted from Levinson looked like this:

VZZZZZZZZZZZZZ2A

V7/////////////77, 7ZZZZZZZZZZZZZZ

834

S. SMALE

When I spoke on the subject that summer (1960) at Berkeley, Lee Neuwirth said: "Why don't you make it look like this?"

V////////////A V, //

'/////////////A 'A

I said "fine" and called it the horseshoe. I still considered myself mainly a topologist, and when considering some questions of gradient dynamical systems, I could see possibilities in topol­ ogy. This developed into the "higher-dimensional Poincare conjecture" and was the genesis of my being quoted later as saying I did my bestknown work on the beaches of Rio. In fact, I often spent the mornings on those beaches with a pad of paper and a pen. Sometimes Elon Lima was with me. In June I flew to Bonn and Zurich to speak of my results in topology. This turned out to be a rather traumatic trip, but that is another story. I had accepted a job at Berkeley (at about the same time as Chern, Hirsch, and Spanier, all from Chicago) and arrived there from Rio in July, 1960. Except for a few lectures on "the horseshoe," I was preoccuped the next year with topology. But, in the summer of 1961, I announced to my friends that I had become so enthusiastic about dynamical systems that I was giving up topology. The explicit reason I gave was that no problem in topology was as important and exciting as the topological conjugacy problem for diffeomorphisms, already on the 2-sphere. This conjugacy problem represented the essence of dynamical systems, I felt. 5. During this year I had an irresistible offer from Columbia University, so we sold a house we had just bought and moved to New York in the summer of 1961. But before taking up teaching duties at Columbia, I spoke at a conference on ordinary differential equations in Colorado Springs and then in September, 1961, went to the Soviet Union. At a meeting on nonlinear oscillations in Kiev, I gave a lecture on the horseshoe example, "the first structurally stable dynamical system with an infinite number of periodic solutions." 1 had a distinguished translator, the topologist, Postnikov, whom I had just met in Moscow. Postnikov agreed to come to Kiev and translate my talk in return for my playing go with him. He said he was

835 ON HOW I GOT STARTED IN DYNAMICAL SYSTEMS

the only go player in the Soviet Union. My roommate in Kiev was Larry Markus. I met and saw much of Anosov in Kiev. Anosov had followed the Gorki school, but he was based in Moscow. After Kiev I went back to Moscow where Anosov introduced me to Arnold, Novikov, and Sinai. I must say I was extraordinarily impressed to meet such a powerful group of four young mathematicians. In the following years, I often said there was nothing like that in the West. I gave some lectures at the Steklov Institute and made some conjectures on the structural stability of certain toral diffeomorphisms and geodesic flows of negative curvature. After I had worked out the horseshoe, Thorn brought to my attention the toral diffeomorphisms as an example with an infinite number of periodic points which couldn't be perturbed away. Then I had examined the stable manifold structure of these dynamical systems. 6. After teaching dynamical systems the fall semester at Columbia, I was off again, with Clara and the kids, this time to visit Andre Haefliger in Lausanne for the spring quarter. Besides lecturing in Lausanne, I gave lectures at the College de France, Urbino, Copenhagen, and, finally, Stockholm, all on dynamical systems and emphasizing the global stable manifolds, which I found more and more to lie close to the heart of the subject. In Stockholm, at the International Congress, I saw Sinai again and he told me that Anosov had proved all the conjectures I had made the preceding year in the Soviet Union. In Lausanne I had begun to start thinking about the calculus of variations and infinite-dimensional manifolds, and this preoccupation took me away from dynamical systems for the next three years.

836 Physica D 51 (1991) 267-273 North-Holland

Dynamics retrospective: great problems, attempts that failed* Steve Smale Department of Mathematics / Economics, University of California, 2120 Oxford Street, Berkeley, CA 94720, USA

Ten problems of dynamical systems are stated. A short discussion of each is given, with some references.

This is an account of some problems of dynam­ ics which I believe to remain major challenges. My period of activity in this subject for the most part ended before 1970; thus the reader should consider this note as reflecting a voice out of the past. The problems selected are those with which I have had some experience, one way or another. In several instances I devoted much time toward their solution with little to show for it. There are ten problems and the first eight have a yes or no answer. The differential equations in R3 of Lorenz [23] may be written as i - -lOx + lOy, y = 28JC — y — xz,

z=-lz+xy.

(L)

Problem 1. Is the dynamics of (L) described by the "geometric Lorenz attraction" of Williams, Guckenheimer and Yorke? The precise definition of the geometric Lorenz attractor (inspired by the numerical work of Lorenz) can be found in refs. [15, 41]. The "de­ scription by" entails a homeomorphism preserv­ ing solution curves. Many physicists may feel that problem 1 is too academic an exercise, since the numerical work 'Partially supported by the National Science Foundation and the Science and Engineering Research Council, UK.

seems so convincing. Yet a solution of problem 1 could yield new concepts or perhaps important general methods for demonstrating strange attractors in physical systems. As long as it remains unsolved, there will be a missing link in one of the most important examples of chaotic dynam­ ics. Moreover, problem 1 is just the tip of an ice­ berg. It is a special case of the general problem of establishing and analyzing strange attractors in differential equations of engineering and physics. Here is a very brief account of some background mathematics related to problem 1. CD. Birkhoff constructed some kinds of strange attractors but did not analyze their dynamics. I constructed a structurally stable, well-analyzed strange attractor [38, 36] for a differential equation of four vari­ ables called a DE or a solenoid which is used by Ruelle and Takens [34]. Yet the differential equa­ tion was not inspired by any physical system. Rychlik [35] and Robinson [33] have proved the existence of a strange attractor for a system of differential equations not far removed from the Lorenz equations. Benedicks and Carleson [1] and Mora and Viena [26] have similar results for the Henon attractor. Chaotic behavior for physi­ cal systems had been shown by proving the exis­ tence of horseshoes by Melnikov, Marsden and Holmes [15]. I should say something about the meaning of the term "strange attractor" used above. That will take us into a little detour into describing the

0167-2789/91/S03.50 O 1991 - Elsevier Science Publishers B.V. (North-Holland)

837 268

S. Smalt / Dynamics: grtat problems

Table 1 Axiom A (hyperbolic) structurally stable? systems understood? definition of strange attractor

Beyond hyperbolicity

yes no pretty much in principle, no clear hazy a basic attracting set, neither an equilibrium nor periodic positive Lyapunov, exponent; sensitive dependence; homoclinic point; robust

dichotomy, axiom A dynamical system and "be­ yond hyperbolicity", which will be helpful later. Table 1 helps to communicate the contrast I want to make. We may suppose the dynamical systems re­ ferred to in the chart as discrete and reversible with a compact state space (a manifold) M. Thus the dynamics is represented by a diffeomorphism T: M-»M or TeDirKM). All differentiability under consideration will be of class C , some r=.l,2 See ref. 138] for a general reference. Also a brief description follows. Say that a fixed point x e M of T is hyperbolic if the derivative DT(x) (i.e. matrix of partial derivatives) has no eigenvalue of absolute value 1. For a hyperbolic fixed point x, the stable mani­ fold W * ( x ) - {y e M| 7"'( y) -►* as i - »}

and unstable manifold V/u(x) = [yeM\Ti(y)-*xm>i

— -°°}

are cells (one to one immersed) in M of comple­ mentary dimensions. A periodic point x e M of period k is a fixed point of r*, and definitions of hyperbolicity etc., carry over. One can extend the notion of hyperbolicity to invariant sets of 7 as well. Such a set relevant for our purposes is the set ft = 1X7"), of non-wander­ ing points (Birkhoff) defined as follows. Say x «E fl if there is a neighborhood N of x all of whose iterates {7"*N]AeZ are disjoint.

T e DifHM) is said to satisfy axiom A if the periodic points are dense in fl and fl is a hyper­ bolic set. (We assume also a "nocycle condition".) The work of many people, e.g. Mane, have essen­ tially identified structurally stable with axiom A. The basic ideas of axiom A extend to continu­ ous time (ordinary differential equations) and one dimensional endomorphisms (discrete, not neces­ sarily invertible). Problem 2. Can the Navier-Stokes equations on the two-dimensional torus be dynamically nontrivial? In other words consider two dimensions, the Navier-Stokes equations with special periodic boundary conditions (two equal periods), with time independent forcing term. Must there be a unique equilibrium with every solution tending to it as time goes to °°? Alexandre Chorin first suggested that problem 2 could have a negative answer, even likely so. Foias and Temam (see e.g. ref. [10]) have shown that solutions must tend to a finite dimensional set and have given estimates (upper bounds) on the dimension of this set. Moreover Foias (June 1990) told me that while problem 2 is still un­ solved, Babin and Visik have given examples with a second equilibrium in the case of a rectangular torus (rather than a square one). Quite some number of years ago working on this problem, 1 believed that I could show that triviality of the dynamics of problem 2 for the truncation of the first twelve terms of the Fourier series. The following is taken from my notes of that time.

838 S. Smale / Dynamics: great problems

The primary function is taken to be the vorticity w(x), x = (*,, x2), so expanding w(x,l)=

wk(t)e2"''^',

Y. * - ( * , , * 2 )e/ 2 cR 2

In terms of the wk, the equations of motion are wk= -v\\k\\2wk + E

W^.^+Z*.

i +j - k

k<EZ2,

k*(0,0)

where fk comes from the expansion of the exter­ nal force, fix) = Y.k fk e 2 "* * and bk is linear in each wt,Wj. The first twelve terms are grouped as llfcll2 = 1, k = (0,1), (1,0), (0,-1), and (-1,0); ||*||2 = 2, it = (1,1), (1,-1), (-1,1), and ( - 1 , - 1 ) ; finally ||*||2 = 4, A: = (0,2), (2,0), (-2,0) and (0,-2). Let al,a2,ai each of the corresponding func­ tions, say as maps R4 -»C 4 . The equations of motion now take the form a, = - i / o , + c 1 ( a 1 , a 2 ) + / I , a 2 = -2va2 +

c2(a2,aA)+f2,

d 4 = -4i>a 4 +/ 4 , using the properties of the above bk. From the last set of equations one obtains the global stabil­ ity. I hope this is correct! See also ref. [11]. Problem 3. Let T: M -» M be Anosov (i.e. glob­ ally hyperbolic). Is T topologically the same as the Lie group model of John Franks? The reference to the problem with technical terms defined is Franks in ref. [4]. But let me explain the problem and its importance. As I have said above a certain understanding of the axiom A systems has been reached. There are the irreducible building blocks called basic sets,

269

partially ordered by the dynamics according to a "no cycle condition". Still challenging problems remain as to what is the nature of these basic sets and how they are put together. Problem 3 repre­ sents the prototype of (the first of) these ques­ tions. Here the non-wandering set of T is supposed to be the manifold M with a hyperbolic structure for T defined on M. An example (Lie group induced) is helpful. Let T0: R" -» R" be a linear map, integer co­ efficients, determinant ±1. Then 7"„ induces a diffeomorphism T: M -» M, M = R"/Z" where I" is the integer lattice in W. If no eigenvalue of T0 has absolute value 1, then T has a hyperbolic structure on M defined by that on R". The same construction works for nilpotent Lie groups in place of R" (considered as an Abelian Lie group). If additionally one takes into account a finite group of automorphisms, one obtains the general Lie group model of Franks. We are ask­ ing if every hyperbolic T as above is related to such a Lie group model by a homeomorphism preserving the dynamics. In the case of an expanding map T: M -» M (not invertible, but expanding in every direction, as z -> z" for complex numbers of absolute value 1), an important theorem of Gromov [14], answer­ ing a question of Shub [38] gives a positive answer to the problem. The corresponding problem for ordinary dif­ ferential equations has been proposed by Tomter (see ref. [4]) and answered negatively in ref. [16]. Further papers on the problem of understand­ ing basic sets of axiom A dynamical systems include ones by Farrel and Jones, Hirsch, Manning, Newhouse, Williams referred to in refs. [38, 13]. The problem of how the basic sets of an axiom A dynamical system are related, via some general form of Morse theory, is discussed with a survey of the literature in refs. [12, 36]. The next three problems each ask if a certain property of dynamical systems is a generic prop­ erty.

839 270

5. Small / Dynamics: great problems

A Baire set of a complete metric space is the intersection of a countable number of open dense sets. It is a characteristic fact (Baire) that a Baire set is dense. A generic property is a property true for diffeomorphisms belonging to some Baire set of DifRM), C topology, some r - 1,2,3. In the case of one dimensional dynamics we will replace DifRM) by End(M) the space of all C maps g: M -»M where M is one dimensional. The key step in proving that a property is generic is an approximation. An example of a generic property (due to Kupka and myself, see e.g. ref. [38]) of / e DifRM), M a compact manifold is: (a) every peri­ odic point is hyperbolic (b) for each pair of peri­ odic points p . g e M , W*(p) and Wu(
There is an easy argument which works in the C° case and it was Peixoto who observed that this argument failed for C' approximations. He proved the C case of problem 5, all r, in case of one dimensional manifolds. Pugh and Robinson [32] prove the C1 version of problem S for the Hamiltonian case. In this situation a compact energy surface consists only of non-wandering points and thus for C1 Hamil­ tonian dynamics on compact energy surfaces genericalty the period points are dense. This, to a great extent, justifies Poincari's focus on periodic solutions; yet the C problem, r > 1, Hamiltonian case, remains unresolved. For continuous-time dynamical systems (ordin­ ary differential equations) on two dimensional orientable manifolds, Peixoto (see ref. [38]) has given an affirmative answer to problem S. Problem 6. Is it a generic property that the centralizer Z(D of 7" e DifRM) consists of only the iterates of Tl Here Z(D consists of all S e DifRM) which commute with T: Z(T) ~{Se

Diff(M)| ST= TS).

Thus, "is ZiT) = {Tk)k e z a generic property?"

Problem 5. Is it a generic property of T e DifRM) ( C , r > 1) that the periodic points are dense in the non-wandering points?

I suggested this problem [38] inspired by work of N. Kopell. Since then, the main successes on the problem have been achieved by Palis and Yoccoz (see ref. [28D. They have found an almost complete affirmative answer in the case of axiom A diffeomorphisms. I find this problem interesting in that it gives some focus in the dark realm, beyond hyperbolicity, where even the problems are hard to pose clearly. The work of Palis and Yoccoz uses heav­ ily structures implied by axiom A.

This problem is a form of the "closing lemma", which in the case of C , r = 1, has been solved by Pugh [30].

Problem 7. Let P(x, y), Q(x, y) be real polyno­ mials in two variables and consider the differen-

840 S. Smalt /Dynamics: great problems

tial equation on R (i) Is there a bound K on the number of limit cycles of the form K i nq, where n is the maximum of the degree of P and Ql This can be regarded as the second half of Hilbert's sixteenth problem (cf. ref. [3]). Problem 7 seems to be the most elusive of Hilbert's prob­ lems. In fact, the progress of recent decades is backwards. Dulac [8] asserts the result that (1) has always a finite number of limit cycles. Subse­ quently Landis and Petrovski [20] give the bound on the number of limit cycles as nq (q - 3). Later an error in ref. [20] was found and ac­ knowledged. Still later a counterexample to one of their explicit bounds was found by Songling [39]. More recently, IFyashenko [17] has found an error in the work of Dulac [8]. On the other hand, yet not fully published, Il'yashenko [17], as well as Ecalle, Martinet, Moussu and Ramis [9] have given a proof of Dulac's result. See also refs. [22, 31]. I spent much time on the following special case of problem 7, corresponding to Lienard's equa­ tion:

f(x) a real polynomial, /(0)-0.

various talks I raised the question of estimating this number of fixed points. In response Linz, de Melo and Pugh [21] found the lower bound [ j d e g / ] - 1, and conjectured this as an upper bound. Yet still an upper bound of the form (deg/)* has not been found. On other occasions I spoke on what I called the "Pugh problem", x=h(x,t)-x" +h2(t)x"-2

+

hl(t)xn'i

+ ...,

Oirsl,

(3)

to estimate the number of solutions Kin) with the boundary condition x(0)=Jt(l). In response, L. Nirenberg wrote me (Septem­ ber 20, 1976) of his C° example of n - 4, Kin) 6. But upper bounds with ht polynomials (or C") are still unknown. Problem 8. Given any set of masses, / « „ . . . , m„ > 0 in the n-body problem of celestial me­ chanics, is the number of relative equilibria fi­ nite? The problem was raised by Wintner [42]. A relative equilibrium is a solution of Newton's equation which is given by a rotation. One may see ref. [37] for details and where the following equivalence is shown. Proposition. For a given set of n masses, the relative equilibria correspond exactly to the criti­ cal points of the function

sr-y-n*). dy dt

271

(2)

(If f(x) is x3 —x, then (2) is van der Pol's equa­ tion with one limit cycle.) It can be seen that all solutions must circle around the origin in a clock­ wise direction (save for the unique equilibrium at (0,0)). Thus there is a "cross-section" T: R*-» R+, R+ being the positive y axis, defined by following a solution curve. The limit cycles of (2) are pre­ cisely the fixed points of the analytic map 7. In

V: (SK - A)/SO(2) -» R. Here

SK~ (*e(R2)1 £>,*,. = 0, i2>,iu,ii 2 -u A - lx e(R) m | JC, -Xj, some i *j\,

841 272

S. Smale / Dynamics: great problems

and ,
equation Z(p) = 0 has a solution, yielding a price system for which supply equals demand, all mar­ kets. I worked on problem 9 several years, feeling that it was the main problem of economic theory. Some more discussion of the problem and refer­ ences to some relevant literature can be found in ref. [38]. I suggest problem 10 on the foundations of quantum mechanics rather hesitantly. My idea comes more from emotion than extensive think­ ing or a solid background in quantum mechanics. My feelings here also are based on a projection of my philosophy about classical mechanics in which Hamiltonian dynamics is less central that is usu­ ally the case in physics (see ref. [38], chapter "On the problem of reviving the ergodic hypothesis of Boltzmann and Birkhoff').

Problem 9. Extend the mathematical model of general equilibrium theory to include price ad­ justments.

Problem 10. Extend the dynamics of quantum mechanics in a way which contains the successful features of the existing theory, and permits tran­ sitions between equilibrium states.

There is a well-established static theory of eco­ nomic equilibrium prices. Problem 9 asks for a dynamical theory with states as price vectors (or maybe enlarged to include other economic vari­ ables). This dynamic theory should meet certain conditions, such as having its equilibria coinciding with the equilibria of the existing theory. The time development of prices should be determined by the actions of economic agents, producers and consumers. For the classic theory one can see ref. [5]; here are a few words of description. There is a func­ tion Z(p) - D(p) - S(p) from the space of price vectors of commodities with values in commodity space. Here Z is called the excess demand func­ tion and D the demand defined by aggregation over the demands of the individual consumers of the economy. The supply function S is also ob­ tained by aggregation. Economics justifies condi­ tions on individual behavior which lead to axioms on Z. The existence theorems assert that the

By "dynamics permitting transitions between equilibrium states", I mean the existence of solu­ tions <J>('), -oo < / < oo; of the differential equa­ tion which satisfytfr(»)-» tfr„ as t -» +oo, and IA0,IAI are stationary states ("eigenvectors"). A way of achieving the above could go by adding some dissipation and/or forcing terms to the existing Hamiltonian or the Schrodinger equation. On a slightly more technical level, consider as the space of states the infinite dimensional com­ plex projective space P„ as a quotient of some L2 Hilbert space, dividing out the non-zero complex numbers. The usual quantum dynamics is that induced by t'H'ij), H an appropriate differential operator. By adding to H a term icA, c a real constant and A self-adjoint (as the Laplacian), one does get the solutions
842 S. Smale / Dynamics: great problems

ble manifold structure of the Hamiltonian system modified. The closest reference 1 know to the above is ref. [6].

References 111 M. Benedicks and L Carleson, The dynamics of the Hlnon map, preprint (1989). [2) P. Blanchard, Complex analytic dynamics on the Riemann sphere, Bull. Am. Math. Soc. 11 (1984) 85-141. 13] F. Browder, ed.. Mathematical Developments Arising from Hilbert Problems (Am. Math. Soc., Providence, RI, 1976). [4] S.S. Chern and S. Smale, Global Analysis (Am. Math. Soc., Providence, RI, 1970). 15] G. Debreu, Theory of Value (Wiley, New York, 1959). 16] D. Crespin, Stability of dynamical systems and quantum mechanics I, preprint, Dept. de Matem., Univ. Central de Venezuela (1983). [7] A. Douady and J. Hubbard, Etude dynamique des polynomes complex, I 84-2, II 84-4, Publ. Math. d'Orisay, Univ. de Paris-Sud. Dept. de Math. Orsay, France (1984-1985). [8) H. Dulac, Sur les cycles limites. Bull. Soc. Math. France 51 (1023)45-188. [9) J. Ecalle, J. Martinet. R. Moussu and J.P. Ramis, Non accumulation des cycles limite, Univ. L. Pasteur, Stras­ bourg, Publ. de I'lnst. de Recherche Math. Avancee 329 (1987) 182-192. [10) C. Foias, B. Nicolaenko and R. Temam, The connection between infinite dimensional and finite dimensional dy­ namical systems, Contemporary Math., Vol. 99 (Am. Math. Soc., Providence, RI, 1989). 111) V. Franceschini and C. Tebaldi, Breaking and disappear­ ance of tori, Commun. Math. Phys. 94 (1984) 317-329. 112) J. Franks, Homology and dynamical systems, CBMS Ser. 49 (Am. Math. Soc., Providence, RI, 1982). 113) J. Franks and R. Williams, Anomotous Anosov flows, in: Lecture Notes in Mathematics, Vol. 819 (Springer, New York, 1980) pp. 158-174. [14] M. Gromov, Groups of polynomial growth and expanding maps, IHES Publ. Math. 53 (1981) 53-78. [15] J. Guckenheimer and P. Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, Appl. Math. Science 42 (Springer, New York, 1983). [16) M. Handel and W. Thurston, Anosov flows on new three manifolds, Invent. Math. 59 (1980) 95-103. [17] J. Il'yashenko, Dulac's memoir "On limit cycles" and related problems of the local theory of differential equa­ tions, Russ. Math. Surv. VHO (1985) 1-49. [18] M. Jakobson, On smooth mappings of the circle onto itself. Math. USSR Sb. 14 (1971) 161-185. [19] R. Kuz'mina. An upper bound for the number of central

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configurations in the plane n-body problem, Sov. Math. Dokl. 18 (1977) 818-821. [20] E. Landis and I. Pctrovski, On the number of limit cycles of the equation djr/dy ~ Pix,y)/Q(x,y) where P and Q are polynomials. Math. Sb. 43 (1957) 149-168. [21) A. Linz, W. de Melo and C. Pugh, in: Geometry and Topology, Lecture Notes in Mathematics, Vol. 597 (Springer, New York, 1977). [22] N. Lloyd, Limit cycles of polynomial systems - some re­ cent developments, in: New Directions in Dynamical Systems, London Math. Soc. Lecture Note Series 127 (Cambridge Univ. Press, Cambridge, 1989) pp. 192-234. [23] E. Lorenz, Deterministic non-periodic flow, J. Atmosph. Sci. 20 (1963) 130-141. [24) W. de Melo, Lectures on one-dimensional dynamics (IMPA, Rio de Janeiro, 1989). [25] R. Moeckel, Relative equilibria of the four-body prob­ lem, Ergod. Theory Dynam. Systems 5 (1985) 417-435. [26] L. Mora and M. Viena, Abundance of strange attractors, IMPA, Rio de Janeiro preprint (1989). [27) F. Moullon, The straight line solutions of the problem of N bodies, Ann. Math. 12 (1910) 1-17. [28] J. Palis, Centralizers of diffeomorphisms, in: Workshop on Dynamical Systems, eds. Z. Coelho and E. Shiels (Longman/Wiley, Essex/New York, 1990). [291 J. Palmore, Measure of degenerate relative equilibria I, Ann. of Math. 104 (1976) 421-429. [30] C. Pugh, An improved closing lemma and a general density theorem, Am. J. Math. 89 (1967) 1010-1022. [31] C. Pugh, Hilbert's 16th problem; limit cycles of polyno­ mial vector fields in the plane, in: Dynamical Systems - Warwick 1974, ed. A. Manning, Springer Lec­ ture Notes in Mathematics. Vol. 468 (Springer, New York, 1975). [32] C. Pugh and C. Robinson, The C' closing lemma includ­ ing Hamiltonians, Ergod. Theory Dynam. Systems 3 (1983)261-313. [33] C. Robinson, Homoclinic bifurcation to a transitive attractor of Lorenz type, Nonlinearity 2 (1989) 495-518. [34] D. Ruelle and F. Takens, On the nature of turbulence, Commun. Math. Phys. 20 (1971) 167-192. [35] M. Rychlik. Lorenz attractors through Sil'nikov-type bi­ furcation, Part 1, Ergodic Theory Dynam. Systems, to appear. [36] M. Shub. Global Stability of Dynamical Systems (Springer, New York, 1987). [37) S. Smale, Topology and mechanics I and II, Invent. Math. 10 (1970) 305-331 and 11 (1970) 45-64. [38) S. Smale, Mathematics of Time (Springer, New York, 1980). [39] Shi Songling, On limit cycles of plane quadratic systems, Sci. Sin. 25(1982)41-50. [40] C. Sparrow, The Lorenz Equations (Springer, New York, 1982). [41) R. Williams, The structure of Lorenz attractors, Publ. Math. IHES 50 (1979) pp. 101-152. [42) A. Wintner, The Analytical Foundations of Celestial Mechanics (Princeton Univ. Press, Princeton, 1941).

843

WHAT IS CHAOS? STEVE SMALE In an attempt to explain my own part in chaos theory, let me start with a few words on the role of mathematics in science. There are at least three (overlapping) ways that mathematics may contribute to science. The first, and perhaps me most important, is this: Since the mathematical universe of the mathematician is much larger than that of the physicist, mathematicians are able to go beyond existing frameworks and see geometrical or analytic structures unavailable to me physicist Instead of using the particular equations used previously to describe reality, a mathema­ tician has at his disposal an unused world of differential equations, to be studied with no a priori constraints. New scientific phenomena, new dis­ coveries, may thus be generated. Understanding of the present knowledge may be deepened via the corresponding deductions. Those who anticipated general relativity illustrate my point. It was the mathematicians who first saw that geometry did not have to be Euclidean, that there could exist, in principle, different possible worlds. These discov­ eries were crucial for Einstein's great achievements. Another example of this kind of contribution is found in the history of chaos. Physicists (and many mathematicians as well) were slow to recognize chaotic phenomena because they were oriented towards solving particular equations and analyzing particular solutions. But even 100 years ago, Poincare" had seen the possibility of homoclinic points by transcending that methodology. (My feeling is that homoclinic points lie at the heart of chaos; later I will describe them and defend that point of view.) A more recent development of the theory of homoclinic points occurred when mathematicians took the bolder step of looking at the completely arbitrary differential equation, no longer tied to the physical world. This gave the freedom to create geometric constructions, unhampered by specifics. In this broader picture, the centrality of homoclinic phenomena became clear and an appropriate analysis was carried out. That analysis was then applied to the traditional equations of engineering systems. The second way in which mathematicians contribute to science has to do with the consolidation of new physical ideas. One may express this as the proof of consistency of physical theories. Examples here include the

844

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STEVE SMALE

mathematical foundations of quantum mechanics with Hilbert space, its operator theory, and corresponding differential equations. Closer to the subject of our talk is the proof of the Feigenbaum conjectures by Lanford and the development of the geometric Lorenz attractor by Williams, Guckenheimer, and Yorke. The third way in which mathematicians contribute to science is by describing reality in mathematical terms, or by simply constructing a mathematical model. I feel that this purely phenomenological approach has less importance for science than the first two ways I have described. My criticism of catastrophe theory* was along these lines; and now I feel some chaos and fractal practitioners are turned too much in this direction. Let us turn to the question, "What is chaos?" Certainly the typical answer, "sensitive dependence on initial conditions," is reasonable. But to be relevant to physics, more is required: the property should be robust or preserved under perturbations of the system. A response that involves a mechanism and meets the above criteria is: Chaos is the presence of a (transversal) homoclinic point (One would allow some borderline case where this might not be exactly right.) As a step towards understanding this statement, the mathematical meaning of homoclinic point needs to be given. Ahomoclinic point is a state which tends to a certain equilibrium in both forward and backward time under the dynamics. Let us be more precise. All of the following considerations apply to n-dimensional dynamics, but for simplicity of ex­ position we take states to be in a plane. Consider dynamics with time discrete and reversible. Thus time t takes values 0 (now), 1 (1 unit forward), 2,3,... and for the past t = -1,-2,... . For a state z in the plane, z/ denotes the state at time t which was initially z; so z = zo. Then z/+; =T(n) where 7/is the transformation of the plane onto itself giving the dynamics. * Bulletin of the American Mathematical Society 84 (1978), pp. 1360 -1368. Re­ printed in Steve Smale, The Mathematics of Time, Springer - Verlag, New York, 1980—Ed.

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WHAT IS CHAOS?

91

As an example let T(x,y) = (2x,(l/2)y), where (x,y) are the coordinates of a point z in the plane. The dynamical situation can be represented by the diagram

\f

(x,y) - z

•

T(x,y) = 2 j

•22

\

Note especially how, under time, states are contracted in the vertical direction toward the jc-axis and simultaneously expanded away from the vertical axis. Any state on die y-axis remains there for all time and the same is true for the jc-axis. This property may be stated as "me jr-axis (to be called W") and y-axis (to be called Ws) are invariant under the dynamics." A state on the y-axis tends to the origin as time gets larger and larger. In this example the dynamics, or T, is linear, and no chaos can be present. Yet understanding mis simple example is a good prelude to understanding the deeper examples to come. Returning to the general case, we define a state p in the plane to be an equilibrium of a dynamics r provided that T(p) -p. Thus.p is stationary in time. In our linear example p = (0,0) is the unique equilibrium. A crucial development in the recent theory of dynamical systems is the emphasis on

846

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STEVE SMALE

the set of all states approaching an equilibrium p as time increases indefi­ nitely (i.e.,p such that z,->p as t-**>). Call this set the stable manifold Ws of p. The linear example above has the vertical axis as its stable manifold. Similarly one defines the unstable manifold Wu of p as the set of points which tend top as time goes backward indefinitely (jt-*p as r-*-<»). In our example, W* is the horizontal axis. From the definitions it follows that both Ws andW" are invariant under the dynamics. A point q different from p which belongs to both Ws and ft is a homoclinic point. Thus q will tend to p in both forward and backward time. In our example it is clear that there are no homoclinic points. This is an illustration of the relation between chaos and homoclinic points (lack of both!). Now imagine that the linear dynamics T is modified a bit to a non­ linear transformation of the plane onto itself. Our diagram might change to:

ku

The stable manifold Ws of the fixed point p is no longer straight, but still remains a curve nicely imbedded in the plane. The same applies to W*. There are no homoclinic points.The next step is to imagine a larger modification of T, so large in fact that Ws and W* cross as in the following diagram.

847

WHAT IS CHAOS?

93

This crossing point q is a homoclinic point. A technical condition, neces­ sary for the subsequent analysis, is that q is a transversal crossing point: Ws and Wu arc not tangent at q. Let us explore some consequences that follow merely from the existence of this transversal homoclinic point q. By the invariance of Ws and W*, qj lies on Ws and W*, where qj is q at one unit of time later.

This forces the picture:

848

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STEVE SMALE

Note that another homoclinic point h must be present Similarly going backwards in time we see:

Eventually in time, the curve on W* between h and q, becomes stretched along W* , as between hj and qi in the next diagram.

This part of W* is forced to meet up with Ws , die part of Ws between h.j and q.j to create an additional homoclinic point k. The above analysis can be applied to k. One can begin to understand PoincanS's statement in 1899, referring to homoclinic points [5, p. 839]: "On sera frapp6 de la complexity de cettefigure,que je ne cherche m6me pas a tracer. Rien n'est plus propre a nous dormer une id6e de la complication du

849

WHAT IS CHAOS?

95

probleme des trois corps et en g&idral de tous les problemes de Dynamique..." ** Now that we have seen some of die complications that the existence of a homoclinic point forces upon us, we turn to the question, "Is there some way of making an analysis of die dynamics in this environment?" Toward that end consider the following curved rectangle containing part of Ws.

The dynamics contracts along Ws and stretches along W u , and also diis contracting and stretching applies to points in die neighborhoods of Ws and Wu, respectively. Thus at a certain time later the curved rectangle ABCD has become die new curved rectangle A B C D in the next diagram.

"One will be struck by the complexity of this figure, which I am not even going to attempt to draw. Nothing can better give us an idea of the difficulty of the three-body problem, and of dynamics problems in general.."—Ed.

850

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STEVE SMALE

Next we straighten out Ws to get the following diagram containing die dynamics of the homoclinic point

f This dynamics is called the horseshoe, and we have argued that a homoclinic point implies the horseshoe dynamics. The converse can also be proved. Note also we might have given a direct geometric construction of the horseshoe, not assuming the existence of a homoclinic point. The horseshoe is quite amenable to a detailed analysis. We will indicate quite briefly how that can be done. Consider die set of states which at one unit of time later lie on the intersection of the horseshoe and die square. This is shaded in the next diagram.

■

1

m

1

o

I

i

851

WHAT IS CHAOS?

97

The two vertical strips we have labeled 0 and 1. If a state z has the property that z/ is in the square for all t=0, ± 1 ,±2,... then we assign to it a sequence ofO's and l's with "a marker," e.g 0111.011100 ... = ... S-JSOSJS2 ... by S{ =0ifzj e KQ. $ = 7ifz,- eKj. Here K0 is the horizontal strip labeled 0 and Kj the horizontal strip labeled 1 in the diagram. It may be shown that this assignment is a 1-1 correspon dence between the states and the sequence of symbols. The corresponding dynamics on the sequences moves the period forward one place. This allows the original dynamics of the homoclinic situation to be studied via the combinatorial problem of moving the period in the space of sequences of zeros and ones. This construction, the horseshoe, has some consequences. First, it yields the fact that homoclinic points do exist and gives a direct construction of them. Second, one obtains such a useful analysis of a general transversal homoclinic point that many properties follow, including sensitive depen­ dence on initial conditions — "a large number," anyway. Third, one can prove robustness of the horseshoe in a strong global sense (called structural stability). Summarizing, we have indicated why homoclinic points imply chaos. The converse is less formalizable, but I believe it is basically true. However, there is a stronger notion of chaos, which we might call full chaos, where the set of initial conditions with sensitive dependence has positive or even full measure. In this case the corresponding homoclinic point could well be associated with a chaotic attractor, such as the Lorenz attractor. We stop short of pursuing the discussion in that direction here. LetnieerKitheserKXesbyrrakmgsorrKremafcar (1) A history oflhe horseshoe may be found in [4] and [7]. ThernarhenTancsofrhehorseshoe istbundin[3],[6],[7]. (2)Scmeofltemamoutstandmgpioblemsm in [8]. (3) My own work in recent years has been in the areas of complexity and computatioa But this isrelatedtochaos through such question ihe Mandelbrot set See [1] and [2]. (4) My favorite book mat gives an introduction to chaos is [3].

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References 1. Blum, L.; Shub, M. and Smale, S., "On a theory of computa­ tion and complexity over the real numbers: NP-completeness, recursive functions and universal machines," Bulletin of the American Mathematical Society 21 (1989), 1-46. 2. Blum, L. and Smale, S., "The G5del incompleteness theorem and decidability over aring,"Preprint, Berkeley (1990). 3. Devaney, R., An Introduction to Chaotic Dynamical Systems, Addison -Wesley, Redwood City, California, 1989. 4. Gleick,]., Chaos, Viking, New York, 1987. 5. Poincarf, H., Les Methodes Nouvelles de la Micanigue Cileste, Vol I-m, Gauthier-Villars, Paris, 1899. 6. Smale, S., "Diffeomorphisms with many periodic points," in Differential and Combinatorial Topology (ed. S. Cairns), Princeton Univ. Press, Princeton, New Jersey (1965). 7.

, The Mathematics of Time, Springer, New York, 1980.

8.

, "Dynamics retrospective: great problems, attempts mat failed," for Non-linear Science: The Next Decade, Los Alamos, May 1990.

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DISCUSSION GLEICK: I wonder whether I was the only person who was disappointed when Steve Smale stopped short of offering his opinion on whether he feels that current developments in chaos and fractals have been too much descriptive and phenomenological. In case I wasn't, maybe you could take the opportunity to say. SMALE: It's a little complicated to do that in a short time. The simple answer is, some are and some aren't It may be more fruitful to talk about those developments that are more enduring. I could talk on that for a while, but otherwise, I'm not sure what to say. Certainly, I do see a lot of literature on chaos and fractals I would regard as kind of superficial. Not a lot, but a certain amount, I would say is pretty superficial. I see a lot of beautiful things happening, too. I see both. PRIGOGINE: Professor Smale is, I understand very much your emphasis on homoclinicity, but, why reduce chaos to homoclinicity? There may be other mechanisms like the one which I discussed this morning, resonances, which are all mechanisms for chaotic behavior as seen on the computers and as confirmed in the sense of sensitivity to initial conditions. SMALE: All I can say is from my experience in this kind of dynamics I do not see resonances as a key thing to chaos. Resonances certainly play a big role, especially in the Hamiltonian formulation, which I think is not the place to lookfirstof all for chaos. It's more in the dissipative or dissipative with forcing kind of a dynamics. In that kind of dynamics, at least from experience and examples, I would say resonances are not so clearly con­ nected with chaos. That's my opinion. PRIGOGINE: Mark my disagreement. In other words, I believe that it's very important to go from a Newtonian dynamics to dissipative chaos, and in a Newtonian dynamics it's a matter of fact that you find chaos or sensitivity to initial conditions is related to resonances and then to overlapping reso­ nances. There's a lot of things to be done. I would not brush aside the problem of resonance.

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SMALE: Let me respond once more. I think in Hamiltonian mechanics, Hamiltonian dynamics, one does, of course, have lots and lots of resonance. But also one has typically lots of homoclinic points. And to me it's the homoclinic points mat are related to the sensitive dependence on initial conditions as I would define it, exponential increase of error. MANDELBROT: In your introduction on the role of mathematics, I was amused to hear Poincare' quoted as a mathematician. After all, he was a professor of theoretical physics and a journalist who just dabbled in mathematics. The second person you mentioned, who provided the foun­ dations of quantum mechanics, was also a very peculiar mathematician since he was, in a certain sense, hounded out of the mathematics profession. Therefore, I think what you did was to describe the influence on chaos of two very great men of the past, and those two men should not be appropriated by mathematics. QUESTION FROM THE AUDIENCE: We know from your work on the Poincare- conjecture how dependent topology is on dimension 5 or higher. Does chaos theory display a similar dependence on dimension? SMALE: I would say primarily, no. There are some theorems in dynamical systems, some theoremsrelatedto chaos, where topology plays an especially big role. In those theorems PoincanS's conjecture for dimension 5 does play a role, but I would say primarily for the dynamical phenomena, no. QUESTION FROM THE AUDIENCE: On mathematics and physics do you believe that all of physics is determined in some yet unseen level by mathematical logic, i.e., does mathematics describe physics, or does physics describe the inevitable result of mathematics? SMALE: WelL I don't have a lot of answers to that question, but I would surely say that it's not true that physics can bereducedto mathematics. That's for sure. Or, in any kind of a version of that statement, I would say no.

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PRIGOGINE: I think you said something very deep and very beautiful at the beginning of your lecture when you said that the world of mathematics is in a sense a less constrained world. In other words, there are many possible worlds and the role of the physicist is in a sense to make choices between all the possibilities which are intellectually possible. For example, you can perfectly imagine a world formed only by integrable systems without chaos. It's a perfectly coherent world. You can imagine a world formed by atoms which are randomly colliding, and that's aH. But our world is not an arbitrary world, and therefore creation is, in a sense, the role of the physicist, who constrains the various possibilities opened by mathematics and from time to time tries to use some mathematical notions which had never been used before. FEIGENBAUM: I'd like to make a comment loosely related to that. One of the points that Professor Smale made at the beginning of his talk was the ranking as #3 of the role of description, and in some way that's not necessarily the circumstance, in my mind, in physics. If one thinks of the development of mechanics, there are two fundamentally distinct things. One is called kinematics, and the other is called dynamics. The invention of kinematics by Galileo is by far the most profound part of the subject, and one of the ways that a physicist often thinks is that it's important to figure out the right setting in which you describe something and then how you should describe it It's a profound statement to say that an object is a set of pieces, each of those, loosely speaking, made out of a point which lives in a certain 3-dimensional continuum and propagates according to a continuous time. That's a profound statement of description, and the realization that that sort of description was what was at stake in describing things is one of the profound accomplishments in erecting the structure of physics. So in some ways description could be arbitrarily important in the development of the science. SMALE: I said these were overlapping characterizations of the ways in which mathematicians can contribute. A description can also carry great insights, a description can also be a discovery at the same time. By saying "purely descriptive," I meant to exclude that kind of insight Benoit Mandelbrot, at lunch yesterday, gave me a very good example of something

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that was descriptive: Kepler's use of ellipses for planetary orbits. That was descriptive, but also it was an elegant, simple description. It was a description and a great discovery at the same time, a great insight. So I do say that those ways are overlapping. GLEICK: I'd like to ask Professor Smale whether he thinks it's worthwhile making a distinction, within the concept of proof, between proofs that deliver some insight and proofs that don't? That is, one would hope that a proof that Lorenz's equations actually contain the geometric version of the Lorenz attractor would, as you suggested, offer some insight as well as simply confirming the suspicions of a thousand physicists. On the other hand, when one thinks of the proof of, for example, the four-color map theorem, a proof that involves thousands of pages of computer analysis and can't be comprehended by any individual, I think one would concede that that's a proof that doesn't provide any particular insight. Is that a useful distinction, and is it a distinction that mathematicians recognize? Is one sort of proof considered more desirable? Are there any criteria by which one can sort out insight-providing proofs from less useful proofs? SMALE: Oh, I don't know about an answer to the last question about criteria for distinguishing proofs, at least any kind of formal criteria. Of course, I agree with you that an answer to the question I asked about the Lorenz attractor is likely to give us something important. I think it's likely just in view of the context of that problem. It's likely to give us some kind of mechanism for establishing chaos in more general systems. The problem has put up resistance and it's been around for so long that there must be some kind of stumbling block. In dealing with that, confronting that, we're likely to be led to some kind of more general conceptual criteria for saying that this or that physical system will be chaotic. And I think that could be very, very important for science. It's not sure. It could be a very superficial proof without insight. It's possible, but I think not so likely. And then there are degrees of insight, too, so it's not too simple a question. PEITGEN: I think that James Gleick has touched a very important point about mathematics. I think that proof, in the first place, does not desire to fmd out whether something is right or wrong. That's not the role of mathematics. I think the role of proof is to find out why something is right

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or wrong, and in that sense, of course, proof means to deliver insight into something. In that sense, of course, proof means much more than just die mechanical decision by a computer whether a certain number is of that kind or another kind. I think that it's essentially the soul and role of mathematics to somehow bring order into the complexity of the universe of models which we make all the time. When we do science, we always make a model of what we see. When we are in front of a typewriter, atfirstwe don't understand how the typewriter works, but soon we find out how it works because we make an internal model in our brain. And there are so many models floating around that we need order to somehow compare die models and not get lost, and I diink that's the primary role of mathematics and of proof. SMALE: Fine. Makes sense. MANDELBROT: When a problem has remained open for a very long time, it does not at all follow diat the proof brings insight. There's an example which I like very much, the problem of minimal surfaces due to Plateau. A friend of mine proved Plateau's conjecture. A hundred and fifty years after me fact, she got great acclaim for it because it proved to be very difficult, but she says on every occasion her belief in the result didn't change in any fashion. And she learned nothing from her proof. But she learned a great deal, in contrast, for making computer experiments on minimal surfaces, and diat's one part which you didn't discuss. You spoke of the influence of Poincare" and von Neumann on physical problems. And again, I don't count them as mathematicians, eidier of them. But you didn't speak of the influence of physics on mathematics, and I think the two are very strongly linked. In the case of minimal surfaces, the problems were not reopened because somebody solved a very long-standing open question, but because somebody had die good sense of using the computer very intelligendy and discovering minimal surfaces and new conjectures about them. The field revived because of this second act. The same person did bom, so there's no issue of personalities, but die fruitful thing was die second. SMALE: First of all, I'll say diat diere are many, many, many important mings I didn't say in my talk. So lots of mese questions, like die influence of physics on mamematics, I didn't try to deal wim. On die omer hand, I

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certainly wouldn't make any absolute statements about the existence of a problem for a long time making its solution more important. But I would say mere's a tendency in that direction. One of the first persons to get the Field's medal, Douglas, won it for his solution of Plateau's problem. In fact, his work has been very influential and conceptual. PRIGOGINE: The problem with proof is, of course, such an enormous problem mat it is difficult to make any comments, because we know mat there are problems which are undecidable. Therefore, essentially so far as I can see, the problem of proof is to relate a specific property to a more general sense of those things which you admit. Therefore, when you cannot prove something in the frame of these general things, then the question is, can you generalize? Can you change the initial conditions from which you have started? I tried to give you an example in my lecture. Some systems are nonintegrable, and you can prove that they are not integrable—Poincare* has proved it. But you can then ask, can you not change your perspective to make them integrable in a different sense. And it's the same story about roots of algebraic equations and so on. Therefore, the question of proof is really of great importance in enlarging the intellectual horizon in which you are working. QUESTION FROM THE AUDIENCE: Just how does one turn a sphere inside out without cutting or folding it? SMALE: That's one question I will not try to answer at all. There's a movie on turning a sphere inside out, which is fairly well known. It gives the answer.

859 STEVE SMALE

Finding a Horseshoe on the Beaches 1 of Rio What Is CfcaoaT A mathematician discussing chaos is featured in the movie Jurassic Park. James Gleick's book Chaos remains on the best-seller list for many months. The characters of Tom Stoppard's celebrated play Arcadia discourse on the mean­ ing of chaos. What is the fuss about? Chaos is a new science that establishes the omnipres­ ence of unpredictability as a fundamental feature of com­ mon experience. A belief in determinism, that the present state of the world determines the future precisely, dominated scientific thinking for two centuries. This credo was based on me­ chanics, where Newton's equations of motion describe the trajectories in time of states of nature. These equations have the mathematical property that the initial condition determines the solution for all time. This was taken as prov­ ing the validity of deterministic philosophy. Some went so far as to see in determinism a refutation of free will and hence even of human responsibility. At the beginning of this century, with the advent of quan­ tum mechanics, the untenability of determinism was ex­ posed. At least on the level of electrons, protons, and atoms, it was discovered that uncertainty prevailed. The equations of motion of quantum mechanics produce solu­ tions that are probabilities evolving in time. In spite of quantum mechanics, Newton's equations gov­ ern the motion of a pendulum, the behavior of the solar system, the evolution of the weather, many macroscopic situations. Therefore the quantum revolution left intact many deterministic habits of thought For example, well af­ ter the Second World War, scientists held the belief that long-range weather prediction would be successful when computer resources grew large enough.

In the 1970s the scientific community recognized another revolution, the theory of chaos, which seems to me to deal a death blow to the Newtonian picture of determinism. The world now knows that one must deal with unpredictability in understanding common experience. The coin-flipping syndrome is pervasive. "Sensitive dependence on initial conditions" has become a catchword of modem science. Chaos contributes much more than extending the do­ main of indeterminacy, just as quantum mechanics did more than half a century earlier. The deeper understand­ ing of dynamics underlying the theory of chaos has shed light on every branch of science. Its accomplishments range from analysis of electrocardiograms to aiding the construction of computational devices. Chaos developed not from newly discovered physical laws, but by a deeper analysis of the equations underlying Newtonian physics. Chaos is a scientific revolution based on mathematics—deduction rather than induction. Chaos takes the equations of Newton, and uses mathematical analysis to establish the widespread unpredictability in the phenomena described by those equations. Via mathemat­ ics, one establishes the failure of Newtonian determinism by using Newton's own laws!

In 1960 in Rio de Janeiro 1 was receiving support from the National Science Foundation (NSF) of the United States as a postdoctoral fellow, while doing research in an area of mathematics which was to become the theory of chaos. Subsequently questions were raised about my having used U.S. taxpayers' money for this research done on the beaches of Rio. In fact none other than President Johnson's science adviser, Donald Homig, wrote in 1968 in Science:

'iris K an flgqwdeo' version of a paper to appear w\ the prooeadnge of The Intemetionai Congress of Science and Technotogy-45 year* of the National neeenrrh Could of Brad.

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860 This blithe spirit leads mathematicians to seriously pro­ pose that the common man who pays the taxes ought to feel that mathematical creation should be supported uiith public funds on the beaches of Rio ... What happened during the eight years between the work on the beaches and this national condemnation? The 1960s were turbulent in Berkeley where I was a pro­ fessor, my students were arrested, tear gas frequently filled the campus air; dynamics conferences opened under cur­ few; Theodore Kaczynsld, the suspected Unabomber, was a colleague of mine in the Math Department The Vietnam War was escalated by President Johnson in 1966, and I was moved to establish with Jerry Rubin a confrontational antiwar force. Our organization, the Vietnam Day Committee (VDC), with its teach-in, its troop train demonstrations and big marches, put me onto the front pages of the newspapers. These events led to a sub­ poena by the House Unamerican Activities Committee (HUAC), which was issued while I was en route to Moscow to receive the Fields Medal in 1966. The subsequent press conference I held in Moscow attacking U.S. policies in the Vietnam War (as well as Russian intervention in Hungary) created a long-lasting furor in Washington, D.C. Let me hark back to what actually happened in that Spring of 1960 on those beaches of Rio de Janeiro. Plyfnsj DOVPH ts> Ma*

In the 1950s there was an explosion of ideas in topology, which caught the imagination of many young research stu­ dents such as myself. I finished a Ph.D. thesis in that do­ main at the University of Michigan in 1966. During that sum­ mer I, with my wife, Clara, attended in Mexico City a conference reflecting this great movement in mathematics, with the world's notables in topology present and giving lectures. There I met a Brazilian, Elon Lima, who was writ­ ing a thesis in topology at the University of Chicago— where I was about to take up the position of instructor— and we became good friends. A couple of years later, Elon introduced me to Mauricio Peixoto, a young visiting professor from Brazil. Mauricio was from Rio, although he had come from a northern state of Brazil where his father had been governor. A goodhumored pleasant fellow, Mauricio, In spite of his occa­ sional bursts of excitement, was conservative in his man­ ner and in his politics. As was typical for the rare mathe­ matician working in Brazil at that time, he was employed as teacher in an engineering college. Mauricio also helped found a new institute of mathematics (IMPA), and his as­ pirations brought him to America to pursue research in 1967. Subsequently he was to become the President of the Brazilian Academy of Sciences. Mauricio was working in the subject of differential equa­ tions or dynamics and showed me some beautiful results. Before long I myself had proved some theorems in dy­ namics. In the summer of 1968, Clara and I with our newborn son, Nat, moved to the Institute for Advanced Study GAS)

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in Princeton, New Jersey. I was supposed to spend two years there with an NSF postdoctoral fellowship. However, due to our common mathematical interests, Mauricio and Elon invited me to finish the second year in Rio de Janeiro. So Clara and I and our children, Nat and newly arrived Laura, left Princeton in December, 1969, toflydown to Rio. The children were so young that most of our luggage con­ sisted of diapers, but nevertheless we were able to realize an old ambition of seeing Latin America. After visiting the Panamanian Jungle, the four of us left Quito, Ecuador, Christmas of 1969, on the famous Andean railroad down to the port of Guayaquil. Soon we were flying into Rio de Janeiro, recovering from sicknesses we had acquired in Lima. I still remember vividly, arriving at night, going out several times trying to get milk for our crying children, and returning with a substitute, cream or yogurt We later teamed that, in Rio, milk was sold only in the morning, on the street At that time Brazil was truly part of the "third world." However, our friends soon helped us settle into Brazilian life. We arrived just after a coup had been attempted by an air force colonel. He fled the country to take refuge in Argentina, and we were able to rent, from his wife, his lux­ urious 11 -room apartment in the district of Rio called Leme. The U.S. dollar went a long way in those days, and we were even able to hire the colonel's two maids, all with our fel­ lowship funds. Sitting In our upper-story garden veranda, we could look across to the hill of the favela (called Babylonia) where Black Orpheus was filmed. In the hot humid evenings pro­ ceeding CamavaL we would watch hundreds of the favela dwellers descend to samba in the streets. Sometimes I would join their wild dancing, which paraded for many miles. In front of our apartment, away from the hill, lay the fa­ mous beach of Copacabana. I would spend my mornings on that wide, beautiful, sandy beach, swimming and body surfing. Also, I took a pen and paper and would work on mathematics. MsjtlMMMSlAS _ _ Mtg. SjAAflll

Very quickly after our arrival in Rio, I found myself work­ ing on mathematical research. My host institution, Instituto da Matematica, Pure e Apttcada (IMPA), funded by the Brazilian government, provided a pleasant office and work­ ing environment Just two years earlier IMPA had set up its own quarters, a small colonial building in the old sec­ tion of Rio called Botafogo. There were no undergraduates and only a handful of graduate mathematics students. There were also a very few research mathematicians, no­ tably Pelxoto, Lima, and an analyst named Leopoldo Nachbin. There was also a good math library. But no one could have guessed that in less than three decades IMPA would become a world center of dynamical systems, housed in a palatial building, as well as a focus for all Brazilian science. In a typical afternoon I would take a bus to IMPA and soon be discussing topology with Elon or dynamics with Mauricio, or be browsing in the library. Mathematics re-

861 search typically doesn't require much—a pad of paper and a ballpoint pen, library resources, and colleagues to query. I was satisfied. Especially enjoyable were the times spent on the beach. My work was mostly scribbling down ideas and trying to see how arguments could be put together. I would sketch crude diagrams of geometric objects flowing through space, and try to link the pictures with formal deductions. Deep in this kind of thinking and writing on a pad of paper, I was not bothered by the distractions of the beach. It was good to be able to take time off from the research to swim. The surf was an exciting challenge and even sometimes quite frightening. One time when lima visited my "beach of­ fice," we entered the surf and were both caught in a current which took us out to sea. While Eton felt his life fading, bathers shouted the advice to swim parallel to the shore to a spot where we were able to return. (It was 34 years later, just before CamavaL, that once again those same beaches al­ most did me in. This time an oversized wave bounced me so hard on the sand it injured my wrist and tore my shoul­ der tendon; and then that same big wave carried me out to sea. I was lucky to get back using my good arm.)

At that time, as atopotogist, I prided myself on a paper that I had just published in dynamics. I was delighted with a conjecture in that paper which had as a consequence that (in modem terminology) "chaos doesn't exist"! This euphoria was soon shattered by a letter I received from Norman Levinson. I knew him as coauthor of the main graduate text in ordinary differential equations and as a sci­ entist to be taken seriously. Levinson wrote me of an earlier result of his which ef­ fectively contained a counterexample to my conjecture. His paper in turn was a clarification of extensive work of the British mathematicians Mary Cartwright and J. L Littlewood done during World War II. Cartwright and Littlewood had been analysing some equations that arose in war-related studies involving radio waves. They had found unexpected and unusual behaviour of solutions of these equations. In fact Cartwright and littlewood had found signs of chaos, even in equations that arose naturally in engineering. But the world wasn't ready to listen. I never met littlewood, but in the mid-sixties, Dame Mary Cartwright, then head of a women's college (Girton) at Cambridge, invited me to high table. I worked day and night to try to resolve the challenge to my beliefs that the letter posed. It was necessary to translate Levinson's analytic arguments into my own geo­ metric way of thinking. At least in my own case, under­ standing mathematics doesn't come from reading or even listening. It comes from rethinking what I see or hear. I must redo the mathematics in the context of my particu­ lar background. And that background consists of many threads, some strong, some weak, some algebraic, some vi­ sual My background is stronger in geometric analysis, but following a sequence of formulae gives me trouble. I tend to be slower than most mathematicians to understand an

argument The mathematical literature is useful in that it provides clues, and one can often use these clues to put together a cogent picture. When I have reorganized the mathematics in my own terms, then I feel an understand­ ing, not before. I eventually convinced myself that indeed Levinson was correct, and that my conjecture was wrong. Chaos was al­ ready implicit in the analyses of Cartwright and Littlewood. The paradox was resolved, I had guessed wrongly. But while teaming that, I discovered the horseshoe! i its NOvwvnov The horseshoe is a natural consequence of a geometrical way of looking at the equations of Cartwright-Iittlewood and Levinson. It helps understand the mechanism of chaos, and explain the widespread unpredictability in dynamics. Chaos is a characteristic of dynamics, that is, of time evolution of a set of states of nature. Let me take time to be measured in discrete units. A state of nature will be ide­ alized as a point in the two-dimensional plane. I will start by describing a non-chaotic linear example. The idea is to take a square, Figure 1, and to study what happens to a point on this square in one unit of time, un­ der a transformation to be described. The vertical dimension is shrunk uniformly towards the center of the square and the horizontal is expanded uni­ formly at the same time. Figure 2 shows the domain ob­ tained by this process, A*B*D*C*, superimposed over die original square ABDC. I have also shaded in the set of points which don't move out of the square in this process. The second of our three stages in understanding is the perturbed linear example Now the square is moved into a bent version of the elongated rectangle of Figure 2. Thus Figure 3 describes the motion of our square obtained by a small modification of Figure 2. The horseshoe is the fully non-linear version of what happens to points on the square, by an extension of the process expressed in Figures 2 and 3. This is the situation when motion makes a qualitative departure from the lin­ ear model. See Figure 4.

A

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862 "visual motion." The results in the next section concern vi­ sual motions. In summary, the horseshoe is a fully non-linear motion. In the next section, I will show how chaos comes out of this picture. T I M HoVB#WIO# M

The horseshoe is the domain surrounded by the dotted line. Instead of a state of nature evolving according to a math­ ematical formula, the evolution is given geometrically. The full advantage of the geometrical point of view is beginning to appear. The more traditional way of dealing with dy­ namics was with the use of algebraic expressions. But a description given by formulae would be cumbersome. That form of description wouldn't have led me to insights or to perceptive analysis. My background as a topologlst, trained to bend objects like squares, helped to make It possible to see the horseshoe. The dynamics of the horseshoe is described by moving a point in the square to a point in the horseshoe according to Figure 4. Thus the comer marked A moves to the point marked A* in one unit of time. The motion of a general point x in the square is a se­ quence of points XQX\XI ... Here xo = x is the present state, Xi is that state a unit of time later, X2 that state two units of time later, etc. Now imagine our visual field to be just the square itself. When a point is moved out of the square we will discard that motion. Figure 6 shades in the points which don't leave the square in one unit of time. I will call a sequence which never leaves the square a

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The laws of chance, with good reason, have traditionally been expressed in terms of flipping a coin. Guessing whether heads or tails is the outcome of a coin toss is the paradigm of pure chance. On the other hand it is a deter­ ministic process that governs the whole motion of a real coin, and hence the result, heads or tails, depends only on very subtle factors of the initiation of the toss This is "sen­ sitive dependence on Initial conditions." A coin-flipping experiment is a sequence of coin tosses each of which has as outcome either heads (H) or tails (T). Thus it can be represented in the from HTTHHlTlTri... . 2 A general coin-flipping experiment is thus a sequence so Si «2 . . . where each of »o, «i, «2 • • • is either H or T. Here is the result of the horseshoe analysis that I found on that Copacabana beach. Consider all the points which, under the horseshoe mapping, stay in the square, i.e., don't drift out of our field of visioa These "Visual motions* cor­ respond precisely to the set of all coin-flipping experi­ ments! This discovery demonstrates the occurrence of un­ predictability in fully non-linear motion and gives a mechanism of how determinism produces uncertainty. The correspondence is the following. To each visual mo­ tion there is an associated coin-flipping experiment If xo x\ X2 ... is the visual motion, at time i = 0, 1, 2, 3 , . . . as­ sociate H or heads if xt lies in the top half of the square and T or tails if it lies in the bottom half. Moreover, and this is the crux of the matter, every pos­ sible sequence of coin flips is represented by a horseshoe motion. Therefore the dynamics is as unpredictable as coin-flipping. In the natural one-one correspondence Xo Xi X2 . . • —* *0 8l *2 • - • ,

xo x\ xj . . . is a motion lying in the square and so »\ *a . . . is a sequence of ffs and T"s. On the left is a determlnisticaUy generated motion and on the right a coin-flipping experiment We have seen complete unpredictability pop up within deterministic motion, the horse­ shoe. This is chaos.

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T t » HUMwi Orttfna t* As chaos is a mathematically based revolution, it is not sur­ prising that a mathematician first saw evidence of chaos in dynamics.

*To grve a complete picture n tr-MS section, one needs to reverse lime and consider sequences of heads end tala wticn go beck In time as weL

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Henri Poincare' was (with David Hilbert) one of the two foremost mathematicians in the world active at the end of the last century. I heard of him first as an originator of topol­ ogy, who had written an article claiming that a manifold with the same algebraic invariants as the n-dimensional sphere was topologically identical to the n-dimensional sphere. Then he found a mistake in his proof. Restricting himself now to 3 dimensions, he formulated the assertion as a prob­ lem, now called Poincarg's Conjecture, still one of the three or four great unsolved problems in mathematics today. More germane to my present story is his contribution to the study of dynamics. Poincare' made extensive studies in celestial mechanics, that is to say, the motions of the planets. At that time it was a celebrated problem to prove the solvability of those underlying equations, and in fact Poincarl at one time thought that he had proved it Shortly thereafter, however, he became traumatized by a discovery which not only showed him wrong but showed the impossibility of ever solving the equations for even three bodies. This discovery he christened "homoclinic point"

3

A homoclinic point is a motion tending to an equilibrium as time increases and also to that same equilibrium as time recedes into the past See Figure 6. Here p is an equilib­ rium and h marks the homoclinic point The arrows rep­ resent the direction of time. This definition sounds harmless enough but carries amazing consequences. Poincare wrote concerning his discovery^ One will be struck by the complexity of this figure, which I won't even try to draw. Nothing can more clearly give an idea of the complexity of the three-body problem and in general of all the problems of dynamics ... In addition to showing the impossibility of solving the equations of planetary motion, the homoclinic point has turned out to be the trademark of chaos; it is found in es­ sentially every chaotic dynamical system. It was in the first half of this century that American mathematics came into its own, and traditions stemming from Poincare in topology and dynamics were central in this development G.D. Birkhoff was the most well-known

My c w i transition ftwti the French.

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American mathematician before World War II. He came from Michigan and did his graduate work at the University of Chicago, before settling down at Harvard. Bukhoff was heavily influenced by Poincar6's work in dynamics, and he developed these ideas and especially the properties of homoclinic points in his papers in the 20s and 30s. Unfortunately, the scientific community soon lost track of the important ideas surrounding the homoclinic points of Poincar6. In the conferences in differential equations and dynamics that I attended in the late 60s, there was no awareness of this work. Even Levinson never showed in his book, papers, or correspondence with me that he was aware of homoclinic points. It is astounding how important scientific ideas can get lost, even when they are aired by leading mathematicians of the preceding decades. I learned about homoclinic points and Poincar6's work from browsing in Biukhoffs collected works, which I found in IMF'A's library. It was because of the recently discovered horseshoe that the homoclinic landscape was to sink into my consciousness.In fact there was an important relation between horseshoes and homoclinic points. If a dynamics possesses a homoclinic point then I proved that it also contains a horseshoe. This can be seen in ELgure 7. Thus the coin-ilipping syndrome underlies the homoclinic phenomenon, and helps to comprehend it. The Thlnl Forco

I was lucky to find myself in Rio at the contluence of three different historical traditions in dynamics. These three strains, while dealing with the same subject, were isolated from each other, and thi4 isolation obstructed their development. I have already discussed two of these forces, Cartwright-Littlewood-Levinsonand PoincarCBukhoff. The third had its roots in Russia with the school of differential equations of A. Andronov in Gorki in the 1930s. Andronov had died before the firsttime I went to the Soviet Union, but in Kiev, in 1961,I did meet his wife, AndronovaLeontovich, who was stlll working in Gorki in differential equations. In 1937 Andronov teamed up with the Soviet-mathematician L. Pontryagin. Pontryagin had been blinded at the age of 14, yet went on to become a pioneering topologist. The pair described a geometric perspective of differential equations they called "rough," subsequently called structural stability. Chaos, in contrastto the two previouslymentioned traditions, was absent in this development because of the resMcted class of dynamics. m e n years later the great American topologist Solomon Gfschetz became enthusiastic about Andronov and Pontryagin's work. Lefschetz had also suffered an accident, that of losing his arms, before turning to mathe rnatics, and this perhaps generated some kind of bond between him and the blind Pontryagin. They first met at a topology conference in Moscow in 1938, and again after the war. It was through Lefschetz's influence, in particular from an article of his student De Baggis, that Mauricio Peixoto in Brazil learned of structural stability.

Peixoto came to Princeton to work with Lefschetz in 1967, and this is the route which led to our meeting each other through Elon. After thismeeting, I studiedLefschetz's book on a geometric approach to differential equations, and eventually came to know Lefschetz in Princeton. Thanks to Pontryagin and Lefschetz, there was the specter of topology in the concept of structuralstability of ordinaxy differential equations. I believe that was why I listened to Mauricio. Good Luck

Sometimes a horseshoe is considered an omen of good luck. The horseshoe I found on the beach of Rio certainly seemed to have such a property. In that spring of 1960I was primarily atopologist, mainly motivated by the problems of that subject, and most of all driven by the great unsolved problem posed by Poincar6. Since I had started doing research in mathematics, I had produced false proofs of the 3dimensional Poincar6 Conjecture, returning again and again to that problem. Now on those beaches, within two months of tinding the horseshoe, I found to my amazement an idea which seemed to succeed provided I returned to Poincard's original assertion and then restricted the dimension to 5 or more. In fact the idea not only led to a solution of Poincar6's Conjecture in dimensions water than 4 but it gave rise to a large number of other nice results in topology. It was for this work that I received the Fields Medal in 1966. Thus ".. . the mathematics created on the beaches of Rio ..." (Hornig) was the horseshoe and the higherdimensional Poincar6 Conjecture.

865

The Work of Curtis T. McMullen Steve Smale DEPARTMENT OF MATHEMATICS CITY UNIVERSITY OF HONG KONG KOWLOON, HONG KONG JULY 2,

1998

Curtis T. McMullen has been awarded the Fields Medal for his work in dynamics as well as for his contributions to the theory of computation, complex variables, geometry of three manifolds, and other areas of mathematics. I limit myself here to a brief discussion of some of his results. The search for understanding of solutions of a polynomial equation has had a central and glorious place in the history of mathematics. Already the ancient Greek mathematicians had approximated the square root of two, ie., the solution of x2 = 2 by what is now called Newton's Method. Providing a solution for equations such as x2 + 1 = 0 led to the intro­ duction of complex numbers in mathematics. Group theory was introduced to understand which polynomial equations could be solved in terms of radicals. Earlier there had been such Formulas for degrees 2 (the quadratic formula taught in high school), 3 and 4. For degrees greater than 4 there are no such formulae. Instead of formulae, algorithms have been developed which produce (perhaps by complex routines) a sequence of better and better approximations to a solution of a general polyno­ mial equation. In the most satisfactory case, iteration of a single map, Newton's Method, :onverges to a zero for almost all quadric polynomials and initial points; it is a "generally :onvergent algorithm". But for degree 3 polynomials it converges too infrequently. Thus I was led to raise the question as to whether there existed for each degree such a generally convergent algorithm which succeeds for all polynomial equations of that degree. McMullen answers this question in his thesis, under Dennis Sullivan, where he shows that no such algorithm exists for polynomials of degree greater than 3, and for polynomials Df degree 3 he produces a new algorithm which does converge to a solution for almost all polynomials and initial points. Thus McMullen "finished the job" since this work answers, in degree 3, "yes", and degree greater than three, "no"; it is complete. This indicates his depth of understanding of the situation and is characteristic of his later work.

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For the proof of his result McMullen establishes a rigidity theorem for full families of rational maps of C into C with no attracting cycles other than fixed points. Members of such families are conjugate by a linear fractional (Moebius) transformation. The attracting cycles condition is implied by the general convergence. One obtains radicals by Newton's method applied to the polynomial f(x) = xd — a, starting from any initial point. In this way solution by radicals can be Been as a special case of solution by generally convergent algorithms. This fact led Doyle and McMullen to extend Galois Theory for finding zeros of polynomials. This extension uses McMullen's thesis together with the composition of generally convergent algorithms (a "tower") and the introduction of finite Moebius groups. They showed that the zeros of a polynomial could be found by a tower if and only if its Galois group is nearly solvable, extending the notion of solvable with the inclusion of the Moebius group A5 (the alternating group). As a consequence, for polynomials of degree bigger than 5 no tower will succeed. For degree 5, Doyle and McMullen construct an algorithm following some ideas dating to Felix Klein's famous lectures on the quintic and the icosahedron, and using the classi­ cal theory of invariant polynomials. Thus the power of the tower of generally convergent algorithms is found. Quite beautifull! T. Y. Li and Jim Yorke introduced the word "chaos" into dynamics in connection with the map of population biology, LT : [0,1] —> [0,1], LT(x) = n ( l — 1). Bob May had been intrigued by this map because there was an infinite sequence of period doubling parameters r, converging to s = 3.57.... Soon thereafter, Mitch Feigenbaum's work (with similar results due to Coullet-Tresser) demonstrating the universality properties of this map, helped establish the acceptance by physicists of the new discipline of dynamical systems. The sequence (r,— r,_i)/(r 1+ i — n)) has a limit, a number which is independent of the period doubling map! Key to Feigenbaum's work was the concept of renormalization and the convergence of the renormalizations of an iterate of the Feigenbaum map L, to a fixed point F of the renormalization operator. Let us see what renormalization means for the second iterate I? of L = LT for some 2 < r < 4. So L([0,1]) C [0,1] as above, and L has a second fixed point q = (r — l)/r. Define p by the conditions 0 < p < q and L(p) = q. Thus I? acts on \p,q] (with a sign reversal) something like L on [0,1]. If LJ([p, q]) C [p,q] the conditions for renormalization are present. Let A be the map Ax = (x — q)/(p — q), sending [p,q] onto [0,1]. The renormalized 1? is given by RL{x) = ALiA~1(x), where R is the renormalization operator acting on L. For certain r one may be able to repeat this process. If one can do it indefinitely then L is called infinitely renormalizable. This is a very special situation but occurs for the

867 Feigenbaum map L, above. Lanford found computer assisted proofs of the conjectures of Feigenbaum and subse­ quently Sullivan put them into a broader, detailed, conceptual framework, finding important relations between 1-dimensional dynamics and parts of classical function theory as Kleinian groups. Yet the proof of fast (exponential) convergence of the renormalizations, a basic ingredient in this program, was missing until McMullen's beautiful work was published in the second of his two Annals of Math Studies in 1996. The fast convergence was necessary to yield the crucial rigidity of the theory ("C 1+ ° conjugacy"). With the notation as above, McMullen's result for the Feigenbaum map may be expressed by the estimate: \RkL,(x) - F(x)\ < c/9*,0 < 1. Complex one dimensional dynamics is the study of the iterates of a polynomial map P : C —» C. This has become the most advanced and the most technical part of dynamics. Yet one simple problem may be singled out as giving some focus to this subject. Among polynomial maps of a given degree d, are the hyperbolic ones dense? A polynomial is called hyberbolic (sometimes axiom A), if the orbits of its critical points tend under time to an attracting cycle ("including infinity"). I naively gave this as a thesis problem in the 1960's. Today it is still unsolved even for d = 2, but there are a number of partial results. Quadratic dynamics may be studied for polynomials in the normalized form Pc{z) = a2-f c, with parameter c € C. The unique critical point is zero and if it tends to oo under iteration, the dynamics is well understood in terms of symbolic dynamics. The Mandelbrot set M is defined as the set of c € C for which this is not the case. This often pictured set can be thought of as a "tree with fruit", the fruit being the components of its interior. McMullen proves in the first of his Annals of Math Studies: If c is in a component of the interior of the Mandelbrot set which meets the real axis. then Pc is hyperbolic. As McMullen writes, "if one runs the real axis through M, then all the fruit which is skewered is good". Earlier Yoccoz had done an important special case, and I am ignoring here much other earlier fundamental work in complex (and real) dynamics such as Fatou, Julia, Douady and Hubbard. I am also ignoring the later work of Lyubich and Graczyk-Swiatek. Again the ideas of renormalization play a big role in the proof but now in the context of complex maps. To describe more precisely these ideas, the idea of a quadratic-like map is useful. A quadratic polynomial map C —» C is a proper map of degree 2. A holomorphic proper map

868

f : U —* V of degree 2, with the closure of U a compact subset of V, and having a critical point q in U, is called quadratic-like. Here U, V are supposed simply connected open sets of the complex numbers. For example, an iterate of a quadratic polynomial restricted to an appropriate neighborhood of its critical point is often quadratic like. If moreover, the critical point of / doesn't escape (all the iterates of q are well defined), then according to DouadyHubbard, this map is topologically conjugate to a quadratic map of the form Pc(z) = z2 + c, for some c in the Mandelbrot set M. The map Pc(z) = z 2 + c, c € M, is said to be renormalizable if /** is quadratic-like, the critical point 0 € U and 0 doesn't escape. Pc is called infinitely renormalizable if there are infinitely many values of such n. For the problem of density of hyperbolic polynomials in degree two, the case of finitely renormalizable points had been dealt with earlier by Yoccoz. McMullen's work is on the problem of infinitely renormalizable points in M. It contains an intricate analysis of the dynamics of these maps. Moreover in these two books McMullen establishes new results in complex function theory and the geometry of 3-manifolds. Another important result of McMullen is his proof of Kra's "Theta conjecture". Let X be a compact Riemann surface with a finite number of points removed and its associated Riemannian curvature constant at — 1, in other words a hyperbolic surface. Its universal covering, A — ► A\ has as its group of covering transformation, G, the fundamental group of X. Let Q(A) be the space of holomorphic quadratic differentials with finite norm given by \\<j>\\ = / \4>\ and similarly define Q(X). To 4> €
871

45

Steve Smale and Geometric Mechanics

JERROLD E. MARSDEN*

1. Some Historical Comments In the period 1960-1965, geometric mechanics was "in the air." Some key papers were available, such as Arnold's work on KAM theory and a little had made it into textbooks, such as Mackey's book on quantum mechanics and Sternberg's book on differential geometry. In this period, Steve was work­ ing on his dynamical systems program. His survey article (Smale [1967]) contained important remarks on how geometric mechanics (specifically Hamiltonian systems on symplectic manifolds) fits into the larger dynamical systems framework. In 1966 at Princeton, Abraham ran a seminar using a preprint of the survey article and it was through this paper that I first en­ countered Smale's work. After he visited the seminar, the importance of what he was doing was obvious; also, it became evident that there was great power in asking simple, penetrating, and sometimes even seemingly naive questions. I should add that in the mathematical physics seminar at Princeton that I also had the good fortune of attending, Eugene Wigner had a remarkably similar aura. Smale's dynamical systems work suggested developing similar ideas like structural stability, dynamic bifurcations, and genericity in the context of mechanics. Structural stability aspects were developed by Abraham, Buchner, Robinson, Robbin, and others. The generic bifurcations of equilibria, relative equilibria, Hamiltonian-Krein-Hopf bifurcations that can occur in Hamil­ tonian systems has been studied by Williamson, Arnold, Meyer, van der Meer, Duistermaat, Cushman, Golubitsky, Stewart, and others. See Abraham and Marsden [1978], Marsden [1992] and Delliniz, Melbourne and Marsden [1992] for further information and references. In 1966-1967, two important personal events occured. First, Abraham gave his lectures on mechanics at Princeton from which our book Foundations

* Research partially supported by NSF Grant DMS-8922704 and DOE Contract DE-FGO3-88ER25064.

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of Mechanics arose. Second, Arnold's [1966] paper on rigid-body mechanics and ideal fluid mechanics appeared. The latter paper influenced me pro­ foundly; it showed how the dynamics of these systems could be interpreted as geodesic flow on SO{3) with a left-invariant metric and on DiffVol(ft)—the volume-preserving diffeomorphism groups of, say, a region Q in R3—with the right-invariant metric defined by the kinetic energy of the fluid. This paper, together with the emerging work of Kostant and Souriau on the role of symmetry groups and the momentum map, laid important ideas latent in the traditional approach to mechanics, in clear and concise geometric terms. Smale gave a course of lectures on mechanics in the fall of 1968 at Berke­ ley, the semester I arrived. This led to his two-part paper Topology and Mechanics (Smale [1970]). In the same period, I completed a paper with Ebin (Ebin and Marsden [1970]) in which we put Arnold's work on fluid mechanics in the context of Sobolev (H') manifolds and showed the remarkable fact that Arnold's geode­ sic flow on H'-DuT^Q) (the volume-preserving diffeomorphisms of O to itself of Sobolev class H') comes from a smooth geodesic spray. This fact is remarkable because it allows one to solve the initial value problem using only Picard iteration and ordinary differential equations theory on TH'Diffv0|(ft) and one would not expect this since the Euler equations for fluid mechanics are rather nasty PDEs, not ODEs! This led to a number of inter­ esting analytic and numerical developments in fluid mechanics. Around this same time, I began work with Fischer on the Hamiltonian structure of gen­ eral relativity (see, for example, Fischer and Marsden [1972]). At this stage I had only a rough idea how these two topics might be related to Smale's work. Already around 1971, with so much happening in geometric mechanics, Abraham and I started work on the second edition of Foundations of Me­ chanics with the help of Ratiu and Cushman. Doing so prompted thoughts about how all of these ingredients might fit together. Especially interesting was the question of how Smale's papers might be linked with Arnold's. Smale used symmetry ideas in the context of tangent and cotangent bundles of configuration spaces with Hamiltonians of the form kinetic plus potential energy; that is, he dealt with simple mechanical systems. The examples and some of the theory were concerned with abelian symmetry groups. In this context, Smale's work contained some of the essential ideas of what we now call reduction theory. It was natural to attempt to put Smale's ideas and those of Arnold in the more general and unifying context of symplectic manifolds. Doing so led to the paper with Weinstein (Marsden and Weinstein [1974]) that was completed in early 1972. Some of these ideas were found indepen­ dently by Meyer [1973] whose paper appears in the proceedings of the 1971 conference organized by Peixoto that was, in effect, a large conference on Steve's work on dynamical systems as a whole. This effort led to the now fairly well-developed area of reduction theory. There are expositions of this subject available in the 10 or so texts and monographs that are currently devoted to geometric mechanics (Abraham

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and Marsden [1978], Guillemin and Sternberg [1984], and Arnold [1989] are examples). We will come back to a description of one of the reduction theorems later (the cotangent bundle reduction theorem) and give an indica­ tion of why this approach made so much fall beautifully into place. Briefly, if one starts with a cotangent bundle T*Q, and a Lie group G acting on Q, then the quotient {T*Q)/G is a bundle over T*(Q/G) withfiberg*, the dual of the Lie algebra of G One has the following structure of the Poisson reduced space (T*Q)/G (its symplectic leaves are the symplectic reduced spaces of Marsden and Weinstein [1974]).

T*Q

Phase space i

G 1

Reconstruction and geometric phases

Reduction 1

]

'

l

Reduced phase space

(T*Q)/G

Arnold

9 j Shape spac e \' / T*(Q/G)

T Smale FIGURE 1

Thus, one can say—perhaps with only a slight danger of oversimplifica­ tion—that reduction theory synthesises the work of Smale, Arnold (and their predecesors of course) into a bundle, with Smale as the base and Arnold as the fiber. This bundle has interesting topology and carries mechanical con­ nections (with associated Chern classes and Hannay-Berry phases) and has interesting singularities (Arms, Marsden, and Moncrief, Guillemin and Stern­ berg, Atiyah, and others). We will describe some of these features later.

2. Highlights from Topology and Mechanics One of Smale's main goals was to use topology, especially Morse theory, to estimate the number of relative equilibria in a given simple mechanical sys­ tem with symmetry, such as the n-body problem, and to study the associated

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bifurcations as the energy and momentum are varied. This approach proved to be quite successful and has been carried on by Palmore, Cushman, Fomenko, and others to give more detailed information in the n-body prob­ lem and basic information in other problems like vortex dynamics, rigidbody mechanics, and other integrable systems. One should also mention that this paper of Smale did a lot for the subject itself. The paper attracted worldwide attention and brought many excellent young people into the field. The idea of using topology and geometry in a classical subject to bring new insights and fresh ideas must have been quite appealing. Smale's strategy was to study the topology of the level sets of the energymomentum map H x J : ? - » R x g* on a given phase space P with a given Hamiltonian H and a symplectic group action having a momentum mapping J: P -»g*. He lays out a program for studying bifurcations in the level sets of the energy-momentum mapping as the level value changes. In doing so, he sets out the basic equivariance properties of momentum maps, apparently independent of the other people normally credited with introducing the mo­ mentum map, namely, Lie, Kostant, and Souriau. (An interesting historical note is that Lie had most of the essential ideas, including—according to Weinstein—the fact that the momentum map is a Poisson map, its equivari­ ance, the symplectic structure on the coadjoint orbits, and more, all back in 1890!) Smale also made the important observation that a point zeP is a regular point of J iff the symmetry (isotropy) group of z is discrete. This idea surfaces again in the study of the solution space of the Einstein or YangMills equations, as we shall see below. One of the most important objects that Smale introduced was the amended potential K„ that plays a vital role in current developments in stability and bifurcation of relative equilibria. The amended potential is a geometric gener­ alization of the classical construction of the "effective potential" in the (planar) two-body problem, which is obtained by adding to the given poten­ tial V(r), the centrifugal potential at angular momentum value \i; in this simple case, H2 V = V+ — In this situation, reduction corresponds to the elimination of the angular variable 0 (division by the group G = S 1 ) and the replacement of the poten­ tial V by the potential V„. As we shall see later, in special situations, such as the abelian case, the reduction of a simple mechanical system is again a simple mechanical system, but the general situation, even for groups like the rotation group, is more complicated. The fact that the abelian reduction of a simple mechanical sys­ tem is again a simple mechanical system is essentially contained in Smale's paper, but it has a long history with a surprising amount contained in the work of Routh around 1860 in his books on mechanics. See, for example,

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Routh [1877]. In fact, Routh's work contains results that get rediscovered from time to time in the modern literature, but that is another story. For problems like rigid-body mechanics and fluid mechanics, one has to deal with Lie-Poisson structures on g* (the "Arnold fiber") and magnetic terms on T*(Q/G) (the "Smale base"). The magnetic terms modify the canoni­ cal symplectic structure on T*(Q/G) with the addition of the /i-component of the curvature of a connection on Q -* Q/G called the mechanical connection. This connection, defined below, is given implicitly in Smale's paper (it is introduced in §6 of his paper). Smale studies relative equilibria by applying critical point theory to V^. The function V^ contains much of the information of E x J through the general fact proved by Smale that a point of Q is the configuration of a relative equilibrium if and only if it is a critical point of V^. (Some of these concepts are recalled in the next section.) Smale's examples deal with the abelian case (when the "Arnold fiber" has trivial Poisson structure); in this situation, K„ defines a function on Q/G, and on this quotient space, one expects the critical points to be generically nondegenerate. In the general case (such as a rotating rigid body with internal structure), one has to carefully synthesize the analysis of Arnold and Smale to get the sharpest information. This basic theory, and some simple but very informative examples, com­ prise part I of "Topology and Mechanics." Part II is concerned with the planar n-body problem in which G = S 1 is the planar rotation group and Q is R2", minus collision points. The study of relative equilibria is done by determining the global topology of the level sets of E x J and their quotients by S1—Theorems A and B of the paper. Theorem C relates this to critical points of VM and Theorem D determines the bifurcation set. Theorem E re­ lates the topology of the rduced phase space to that of the configuration space. Corollaries give more details for n = 2 and n = 3. An interesting con­ sequence of these results is Moulton's theorem stating that there are n!/2 classes oicolinear relative equilibria. The fact that one is looking for colinear relative equilibria enables one to reduce the problem to one of finding critical points of a function on real projective n — 1 space minus collisions. In fact, a combinational argument shows that this space has n!/2 components and the corresponding function has a single nondegenerate maximum on each com­ ponent. There have, of course, been many important contributions to the n-body problem since 1970, such as those of Palmore, McGehee, Mather, Meyer, and others. The book of Meyer and Hall [1991] can be consulted for some of the relevant literature.

3. A Glimpse at Reduction Theory In this section we will focus on some of the ways Smale's paper is connected with some of the current research in geometric mechanics. No attempt is made at thoroughness here—the focus is on selected topics of personal inter-

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est only! In particular, we focus on some aspects of reduction theory, one of the most fruitful outgrowths of Smale's and Arnold's work. Of course, others also deserve much credit for setting the foundations, especially Lie, Kostant, Kirillov, and Souriau. Let P be a symplectic manifold and G be a group acting symplectically on P. Let g be the Lie algebra of G and 9* its dual. Let G act on g* by the coadjoint action and let J: P -»g* be an equivariant momentum map; that is, J is equivariant and J generates the group action in the sense that for each {eg, ■*<J.«> = Zr>

where Xf is the Hamiltonian vector field determined by the function / and £r is the infinitesimal generator of the action on P. If G„ is the isotropy subgroup of fie g*, the reduced space at n is Equivariance guarantees that G„ acts on J~x(n), so the quotient makes sense. If n is a (weakly) regular value and the quotient is nonsingular, the reduction theorem states that P„ is a symplectic manifold and that G-invariant Hamil­ tonian systems on P decend to Hamiltonian systems on P. Given a G-invariant hamiltonian H on P, a relative equilibrium (in the terminology of Poincare) is a point in P whose dynamic orbit equals a oneparameter group orbit. Relative equilibria correspond to critical points of if x J and to (dynamically) fixed points on P„. There are three interrelated special cases. First, if P = T*G, then the re­ duced space at fi is the coadjoint orbit through fi and its reduced symplectic structure is that of Kirillov, Kostant, and Souriau. Second, if n = 0 and p = T*Q (with the canonical cotangent structure), then P0 = T*(Q/G) with the canonical symplectic structure. Finally, if G is abelian (or G = G„), then P„ = T*(Q/G) although the structure on T*(Q/G) need not be canonical. We need to also recall that the coadjoint orbits 0 c g* are the symplectic leaves in the Lie-Poisson structure

where one uses "—" for left actions and "+" for right actions. Here SF/Sfi e g is the generalized functional derivative defined by SF

\

^ - , V ) = DF(A<)-V

for all v € g*. As is well-known [or follows using Poisson reduction in the form (T*G)/G s g*], this bracket makes g* into a Poisson manifold. This term "Lie-Poisson" structure was coined by Marsden and Weinstein [1983] since this expression occurs explicitly in Lie's work around 1890. If G is the coadjoint orbit through /*, then following Marie [1976] and

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Kazhdan, Kostant, and Sternberg [1978], one finds a symplectic identification P„ s J_1(
where < •, • > denotes the natural pairing and « • , • » is the metric pairing. If the action is locally free, then \(q) is a positive definite symmetric tensor. Define at: TQ -> g by «(»,) = 1(9)-'J(FL(rt», where t;, e TqQ and FL: TQ -» T*Q is the Legendre transformation deter-

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mined by the metric. The above formula for a is equivalent to the one given by Smale. The first person to note and exploit the fact that a: TQ -+g is a Gconnection on the bundle Q -»Q/G seems to have been Kummer [1981]. The /^-component of the curvature of at is added to the canonical symplectic structure on T*(Q/G) in the reduction process. This was observed, without the language of connections, and the phrase "magnetic term" coined by Abraham and Marsden [1978]. The picture of Q -*■ Q/G as a G-bundle carrying the connection a is the beginning of the story of the "gauge theory of deformable bodies'1 and the remarkable work of Wilczek, Shapere, and Montgomery on the link between optimal control and the motion of a colored particle moving in the YangMills field a. See Shapere and Wilczek [1989], Montgomery [1990] and references therein. The ^-component of a defines a one-form aM: Q -► T*Q. One of the prop­ erties of a connection translates to

which is the way Smale thought of a. The mechanical connection was ex­ plicitly used by Smale to describe the amended potential as the composition of the Hamiltonian with a„. The momentum shift T*Q -» T*Q taking a covector pa at q to the covector Pq — v-M) therefore maps J(/i) to J"l(0). This, in effect, replaces reduction at H by reduction at 0, which gives T*(Q/G). It is by this means that the cotan­ gent bundle reduction theorem is proved. Reduction has its counterpart, reconstruction, which is part of the theory. This concerns how one constructs dynamic trajectories in J - 1 (/i) <=■ P given the reduced dynamic trajectory in P„. It turns out that a induces a connection on the G„-bundle J~l(n) -* P„, and the horizontal lift and holonomy of this connection play a basic role in reconstruction and in the interpretation of geometric phases (Hannay-Berry phases). See Marsden, Montgomery, and Ratiu [1990] for details. In particular, in this reference one will find a beauti­ ful formula of Montgomery for the phase shift of a rigid body—when a rigid body undergoes a periodic motion in its reduced (body angular-momentum space), then the actual body does not return to its original position, but undergoes a rotation about the constant spatial angular momentum vector through an angle given by

where A is the solid angle on the sphere of radius || n\\ enclosed by the trajec­ tory in body angular-momentum space, £ is its energy, and T is the period. The first term, the geometric phase, is the holonomy of the canonical one form regarded as an S 1 connection on the Hopf bundle J _1 (^) -»S 2 . This type of

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formula proves to be useful in a number of problems, such as the control of the attitude of a rigid body with internal rotors; see Bloch, Krishnaprasad, Marsden, and Sanchez de Alvarez [1992]. Geometric phases come up in a variety of other problems as well, and the ideas of reduction and reconstuction can be useful for understanding them. Some of these are described in Marsden, Montgomery, and Ratiu [1990] and Montgomery [1990]. Many other applications involve integrable systems— one of these is the phase shift that one sees when two solitons interact. This is described in Alber and Marsden [1992]. Others that seem likely are phe­ nomena like Stokes' drift in fluid mechanics.

4. Other Directions Here we describe a few other recent research directions, each of which has a specific link with Smale's paper.

4.1. General Methodology Most academics get judged on specific contributions—in mathematics, one is ideally judged on specific theorems. In many circumstances, this is a sound procedure, but what often turns out to be more valuable for science as a whole is the point of view or pedagogical approach that is developed. Smale (along with Poincare, Arnold, Atiyah, Singer, and a few others) gave us the valuable and influential point of view of dynamical systems and, more gener­ ally, of global or geometric analysis. This view has profoundly influenced a whole generation of workers and has had a pervasive effect, often taken for granted. It has also indirectly influenced areas Steve never worked in. For instance, global analysis ideas have proved useful in nonlinear elasticity, even to the point of designing better numerical codes. The book of Marsden and Hughes [1983] is typical of many works showing this impact.

4.2. Stability of Relative Equilibria The context of the cotangent bundle reduction theorem provides a setting for another synthesis of the works of Arnold [1966] and Smale [1970]. This concerns explicit (computable) criteria for the dynamic stability of relative equilibria. The main recent works on this point are Simo, Posbergh, and Marsden [1990] and Simo, Lewis, and Marsden [1991]. The Lie-Poisson reduction methods of Arnold are built around the fact, mentioned above, that (T*G)/G s g*. In this case, Arnold worked out ex­ plicit stability criteria and applied them to rigid bodies and fluids. Working in the context of fluids, to overcome technical difficulties with the PDEs

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involved, he developed what is now known as the Arnold method or the energy-Casimir method in Arnold [1969] and related references. This tech­ nique was developed by a number of authors and was applied to a variety of fluid and plasma problems; Holm et al. [1985] contains a fairly complete survey and bibliography to that date. In the general case, Smale's work suggests that one should test for stability by looking at the second variation 82Vll at a critical point in Q. However, should one view it on 6/G„ or on g* x Q/G? How does the Arnold stability criterion fit in? This is an interesting point because the philosophies of the two approaches are rather different. From the point of view of Smale, things are considerably simpler in the abelian case and here the criterion is clear— test 62VI, for positive deflniteness on Q/G. A more general suggestion is to test <52K„ for deflniteness on Q/GM (this criterion is an exercise in Foundations of Mechanics but is implicit in Smale's paper). Note that if Q = G, then Q/G,, is the orbit through n, so one can expect d2Vpi to correspond to the Arnold criterion on a coadjoint orbit. However, Arnold's philosophy was different: the energy-Casimir method is more tractable if one relaxes the restriction to orbits in the spirit of the Lagrange multiplier theorem. Namely, we add to the Hamiltonian a function that Poisson commutes with every other function, that is, with a Casimir function. (As an aside, we note for amusement only that some would like to call such a Casimir function a "Casimirian," so it would sound just like a Hamiltonian or a Lagrangian. Unfortunately, En­ glish is neither logical nor perfect—we also do not call a "Green's function" a "Greenian," even though it probably is more correct to do that.) At this point in the history of geometric mechanics, it was not clear whether the energy-Casimir method of Arnold or the second variation method sug­ gested by Smale's work was the more appropriate. A motivation for looking more deeply into this problem came from nonlinear elasticity. Here, the com­ plexity of the orbits of g* means that Casimir functions are difficult or impos­ sible to And. Arnold already realized this for three-dimensional ideal flow (where the only known Casimir is the helicity) and this fact surely was a discouragement for the method. Abarbanel and Holm [1987] made some progress on this problem by working directly in material representation, before reduction. (It would be interesting to return to this and related ques­ tions in plasma physics studied by Morrison [1987] in the light of the block diagonalization work described below, and the work in progress of Bloch, Krishnaprasad, Marsden and Ratiu on instability criteria with the addition of dissipation obtained using Chetaev's method.) All of these factors led to the development of the energy-momentum method (or block-diagonalization, or reduced energy-momentum method). The key to this method is the development of a synthesis of the Arnold and Smale methods. One splits the space of variations of (a concrete realization of) Q/G„ into variations in G/GM and variations in Q/G. With the appropriate split­ ting, one gets the block-diagonal structure

881

45. Steve Smale and Geometric Mechanics <52F„ =

Arnold form

0

0

Smale form

509

where the Smale form means S2 V^ computed on Q/G. This method turns out to be an extremely powerful one when applied to specific systems such'as spinning satellites with flexible appendages. Perhaps even more interesting is the structure of the linearized dynamics near a relative equilibrium. That is, both the augmented Hamiltonian Ht = H — < J , 0 and the symplectic structure can be simultaneously brought into the following normal form: Arnold form

0

0

0

Smale form

0

0

0

Kinetic energy > 0

2

S H( =

and Coadjoint orbit form Symplectic Form =

—*

* Magnetic (coriolis)

0

-/

0 / 0

where the columns represent the coadjoint orbit variable (G/G„), the shape variables (Q/G), and the shape momenta, respectively. The term * is an interac­ tion term between the group variables and the shape variables. The magnetic term is the curvature of the /i-component of the mechanical connection, as we described earlier. For G = S0(3), this form captures all the essential features in a wellorganized way: centrifugal forces in K„, coriolis forces in the magnetic term, and the interaction between internal and rotational modes. In fact, in this case, the splitting of variables solves an important problem in mechanics: how to efficiently separate rotational and internal modes near a relative equilibrium.

4.3. Bifurcation and Symmetry Breaking Smale realized, as pointed out earlier, that the symmetry group of a point in phase space determines how degenerate it is for the momentum map. Correspondingly, one expects, from the work of Golubitsky and co-workers, that these symmetry groups will play a vital role in the bifurcation theory of relative equilibria and its connections with dynamic stability theory. The beginnings of this theory has started and it will be tightly tied with the normal form methods of Subsection 4.2. Smale concentrated on the topology of the level sets of H x J and their associated bifurcations as the level sets vary. However, in many problems one also wants to vary other system parameters as well.

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A simple example will perhaps help here. Consider the dynamics of a parti­ cle moving without friction in a rotating circular hoop, as in Fig. 2.

g _ acceleration due to gravity

FIGURE 2. A

ball in a rotating hoop.

As the angular velocity Q> of the hoop increases past y/g/R, a Hamiltonian pitchfork bifurcation occurs near the central equilibrium point, as in Fig. 3.

Hamiltoniin pitchfork bifurcation as FIGURE 3.

(0 T Vg/R

The Hamiltonian bifurcation for the ball in the rotating hoop.

The stability of the central point, which has Z 2 symmetry, gets transferred to the bifurcating solutions, for which the Z 2 symmetry is lost. Related ideas appear in the work of Golubitsky and Stewart [1987] and in the study of a rotating planar liquid drop (with a free boundary held with a surface tension r) in Lewis, Marsden, and Ratiu [1987] and Lewis [1989]. In the latter, a circular drop loses its circular symmetry to a drop with Z 2 x Z 2 symmetry as the angular momentum of the drop is increased (although the stability analysis near the bifurcation is somewhat delicate). There are also interesting stability and bifurcation results in the dynamics of vortex patches,

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especially those of Wan in a series of papers starting with Wan and Pulvirente [1984].

4.4. Discrete Symmetries In mechanics, the time-honored discrete symmetry is reversibility—the antisymplectic involution {q,p)-+(q, —p). However, there are many interesting discrete symmetries that are symplectic—the spatial Z 2 symmetry of the ball in the hoop in Fig. 2 being a simple example. In bifurcation theory with symmetry, the Golubitsky school shows that discrete symmetries (and their corresponding fixed point sets, etc.) play an important role in the theory. A similar thing is true in the Hamiltonian case. For instance, discrete and con­ tinuous symmetries play a key role in the wonderful work of Bobenko, Reyman, and Semenov-Tian-Shansky [1989] that puts the integrability of the Kowalewski top into a reduction-theoretic framework. These ideas have been put into a general framework of discrete reduction by Harnard, Hurtubise, and Marsden [1991]. It would be of interest to go back to Smale's program with these discrete symmetry ideas to see their effect.

4.5. Singularity Structures in Solution Spaces We already noted that Smale observed that singular points of J are points with symmetry. This is a simple but a profound observation with farreaching implications. Abstractly, it turns out that level sets of J typically have quadratic singularities at its singular (= symmetric) points, as was shown by Arms, Marsden and Moncrief [1981]. In the abelian case, the images of these symmetric points are the vertices, edges, and faces of the convex poly­ hedron J(P) in the Atiyah-Guillemin-Sternberg-Kirwan convexity theory. (See Atiyah [1982] and Guillemin and Sternberg [1984].) These ideas apply in a remarkable way to solution spaces of relativistic field theories, such as Einstein's equations of general relativity and the YangMills equations. Here the theories have symmetry groups and, appropriately interpreted, corresponding momentum maps. The relativistic field equations split into two parts—Hamiltonian hyperbolic evolution equations and ellip­ tic constraint equations. The solution space structure is determined by the elliptic constraint equations, which, in turn, say nothing other than the mo­ mentum map vanishes. A fairly long story of both geometry and analysis is needed to really estab­ lish this, but the result is easy to understand in the terms we have given: The solution space has a quadratic singularity precisely at those field points that have symmetry. For further details, see Fischer, Marsden, and Moncrief [1980] and Arms, Marsden and Moncrief [1982]. Whereas these results were motivated by perturbation theory of classical solutions (gravitational waves as solutions of the linearized Einstein equa­ tions, etc.), there is some evidence that these singularities have quantum im-

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plications. For example, there appears to be evidence that, in the Yang-Mills case, wave functions tend to concentrate near singular points (see, for exam­ ple, Emerich and Romer [1990]). It would be of interest to explore these ideas further using the theory developed by Sjamaar [1990] and Sjamaar and Lerman [1991],

4.6. Mechanical Integrators With Steve's more recent interests in computation, it might be appropriate to note that there is quite a bit of activity in developing numerical codes that respect the underlying structure of a mechanical system with symmetry. For example, one can develop codes that preserve exactly the energy-momentum map H x J or that preserve the symplectic structure and J (it turns out that one cannot do all of these; see Ge and Marsden [1988]). There are too many references to adequately survey here, but the one just cited, Channel! and Soovel [1990], references therein, and recent works of Feng, Krishnaprasad, and Simo, will give one a start. To obtain an integrator preserving J is related to finding an algorithm F^,: P -* P that is consistent with the symmetry. To get one that preserves H, one can base the analysis on a discretization of the variational principle, and to get one preserving the symplectic structure, one can discretize the Hamilton-Jacobi equation. One of the interesting things about these integrators is that they seem to perform better than conventional ones (such as Runga-Kutta schemes) in long-term integrations where chaotic dynamics becomes important.

4.7. Homoclinic Chaos Of course, Smale is noted for the famous "horseshoe" that is associated to homoclinic tangles. The technical way that this is handled is via what is usually called the Birkhoff-Smale theorem, which associates an invariant Cantor set having a well-understood symbolic dynamics to a homoclinic tangle. This phenomena had its origins in the work of Poincare on the threebody problem and led to the Poincari-Melnikov-Arnold technique for ex­ plicitly finding homoclinic tangles in specific systems. (See Wiggins [1988] for a thorough account of these topics.) We note that the method has proved effective in establishing homoclinic chaos for PDEs by using infinitedimensional versions of the Poincare-Melnikov-Arnold theorem and the Birkhoff-Smale theorem; see Holmes and Marsden [1981]. It is also interesting to note that horseshoes and reduction fit nicely to­ gether and this is needed when one proves that various systems with symme­ try (such as rigid bodies with attachments) have homoclinic chaos (see Holmes and Marsden [1983]). Here ones sees that Smale's work on chaos in dynamical systems and his work on symmetry and mechanics fit together in a mutually suportive way. This is, of course, just one example of the

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many connections running through different parts of Smale's work when it is viewed on a global scale.

Epilogue In this paper, I have sketched only some of Smale's involvement with me­ chanics. He was also interested in problems such as rotating fluid masses and provided much good advice about such problems. He was also interested in elementary particles, and using topology to help classify them—see Abraham [I960]. This subject is of course now in vogue with people like Witten, who is a good example of someone who has a blend of analysis, geometry, and topology in the Smale spirit. A curious twist in Smale's work involves his recent work and the work of others on linear programming and computational complexity described else­ where in this volume. We are now witnessing the beginnings of deep links between this work and mechanics by people like Deift, Brockett, Bloch, Flaschka, Ratiu, and others. For example, efficient ways of diagonalizing martices can be done by following the dynamics of integrable Hamiltonian systems (for instance, of Toda type) on appropriate spaces of matrices. This is one of many nice illustrations of the conference theme "unity and diversity" that runs through Smale's work and the approach he takes to his topics. Whereas there is a broad diversity in the subject matter, there is a deep unity, not only the obvious one of using global analysis methodology throughout his work, but nonobvious ones, like the preceeding link between computa­ tional techniques and mechanics, that repeats in unexpected yet beautiful ways in each of the subjects that he treated.

References Abarbanel, H.D.I., and D.D. Holm [1987] Nonlinear stability analysis of inviscid flows in three dimensions: incompressiblefluidsand barotropicfluids.Phys. Fluids 30,3369-3382. Abraham, R. [1960] Lectures of Smale on Differential Topology. Mimeographed notes, Columbia University. Abraham, R., and J. Marsden [1978] Foundations of Mechanics. Addison-Wesley Publishing Co., Reading, Ma. Alber, M., and J.E. Marsden [1992] On geometric phases and the phase shift of solitons, Comm. Math. Phys. 149, 217-240. Arms, J.M. [1981] The structure of the solution set for the Yang-Mills equations. Math. Proc. Comb. Phil Soc. 90, 361-372. Arms, J.M., J.E. Marsden, and V. Moncrief [198 lj Symmetry and bifurcations of momentum mappings. Commun. Math. Phys. 78,455-478. Arms, J.M., J.E. Marsden, and V. Moncrief [1982] The structure of the space solu­ tions of Einstein's equations: II. Several Killings fields and the Einstein-YangMills equations. Ann. Phys. 144,81-106.

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Arnold, V.I. [1966] Sur la geometric differentielle des groupes de Lie de dimenson infinie et ses applications a 1'hydrodynamique des fluids parfaits. Ann. Inst. Fourier, Grenoble 16, 319-361. Arnold, V.I. [1969] On an a priori estimate in the theory of hydrodynamical stability. Amer. Math. Soc. Transl. 79, 267-269. Arnold, V.I. [1989] Mathematical Methods of Classical Mechanics. Second Edition. Springer-Verlag, New York. Atiyah, M. [1982] Convexity and commuting Hamiltonians. Bull. Lond. Math. Soc. 14,1-15. Bloch, A.M., P.S. Krishnaprasad, J.E. Marsden, and G. Sanchez de Alvarez [1992] Stabilization of rigid body dynamics by internal and external torques, Automatka 28, 745-756. Bobenko, A.I, A.G. Reyman, and M.A. Semenov-Tian-Shansky [1989] The Kowalewski Top 99 years later A Lax pair, generalizations and explicit solutions. Commun. Math. Phys. 122, 321-354. Channell, P , and C. Scovel [1990] Symplectic integration of Hamiltonian systems. Nonlinearity 3, 231-259. Cushman, R., and D. Rod [1982] Reduction of the semi-simple 1:1 resonance. Physica 0 6,105-112. David, D, D.D. Holm, and M. Tratnik [1990] Hamiltonian chaos in nonlinear opti­ cal polarization dynamics. Phys. Rept. 187, 281-370. Dellnitz, M., I. Melbourne and J.E. Marsden [1992] Generic bifurcation of Hamil­ tonian vector fields with symmetry, Nonlinearity 5,979-996. Deprit, A. [1983] Elimination of the nodes in problems of N bodies. Celestial Mech. 30,181-195. Ebin, D.G., and J.E. Marsden [1970] Groups of diffeomorphisms and the motion of an incompressible fluid. Ann. Math. 92,102-163. Emmrich, C, and H. Romer [1990] Orbifolds as configuration spaces of systems with gauge symmetries. Commun. Math. Phys. 129,69-94. Fischer, A.E, and J.E. Marsden [1972] The Einstein equations of evolution—a geo­ metric approach. J. Math. Phys. 13, 546-68. Fischer, A.E, J.E. Marsden, and V. Moncricf [1980] The structure of the space of solutions of Einstein's equations, I: One Killing field. Ann. Inst. H. Poincari 33, 147-194. Ge, Zhong, and J.E. Marsden [1988] Lie-Poisson integrators and Lie Poisson Hamiltonian-Jacobi theory. Phys. Lett. A 133,134-139. Golubitsky, M, and I. Stewart [1987] Generic bifurcation of Hamiltonian systems with symmetry. Physica 24D, 391-405. Guillemin, V, and S. Sternberg [1984] Symplectic Techniques in Physics, Cambridge University Press, Cambridge. Hamad, J, J. Hurtubise, and J. Marsden [1991] Reduction of Hamiltonian systems with discrete symmetry, preprint. Holm, D.D, J.E. Marsden, T. Ratiu, and A. Weinstein [1985] Nonlinear stability of fluid and plasma equilibria. Phys. Rept. 123,1-116. Holmes, P J , and J.E. Marsden [1981] A partial differential equation with infinitely many periodic orbits: chaotic oscillations of a forced beam. Arch. Rat. Mech. Anal. 76,135-166. Holmes, P J , and J.E. Marsden [1983] Horseshoes and Arnold diffusion for Hamil­ tonian systems on Lie groups. Indiana Unit. Math. J. 32,273-310.

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S15

Kazhdan, D., B. Kostant, and S. Stembcrg [1978] Hamittonian group actions and dynamical systems of Calogero type. Commun. Pure Appl. Math. 31,481-508. Kummer, M. [1981] On the construction of the reduced phase space of a Hamiltonian system with symmetry. Indiana Univ. Math. J. 30, 281-291. Lewis, D., [1989] Nonlinear stability of a rotating planar liquid drop. Arch. Rat. Mech. Anal. 106, 287-333. Lewis, D., J.E. Marsden, and T.S. Ratiu [1987] Stability and bifurcation of a rotating liquid drop. J. Math. Phys. 28, 2508-2515. Lewis, D., J.E. Marsden, T. Ratiu, and J.C. Simo [1990] Normalizing connections and the energy-momentum method. Proceedings of the CRM Conference on Hamiltonian systems, Transformation Groups, and Spectral Transform Methods, edited by Hamad and Marsden. CRM Press, Montreal, pp. 207-227. Marie, CM. [1976] Symplectic manifolds, dynamical groups and Hamiltonian me­ chanics. Differential Geometry and Relativity, edited by M. Cahen and M. Flato. D. Reidel, Boston. Marsden, J.E. [1981] Lectures on Geometric Methods in Mathematical Physics. SIAM, Philadelphia, PA. Marsden, J.E. [1992] Lectures on Mechanics, London Math. Soc. Lecture Notes 174, Cambridge Univ. Press. Marsden, J.E., and T.J.R. Hughes [1983] Mathematical Foundations of Elasticity. Prentice-Hall, Redwood City, CA. Marsden, J.E., R. Montgomery, and T. Ratiu [1990] Reduction, symmetry, and phases in mechanics. Mem. AMS 436. Marsden, J.E., and T.S. Ratiu [1986] Reduction of Poisson manifolds. Lett. Math. Phys. 11, 161-170. Marsden, J.E., T. Ratiu, and A. Weinstein [1984] Semi-direct products and reduction in mechanics. Trans. Amer. Math. Soc. 281, 147-177. Marsden, J.E., and A. Weinstein [1974] Reduction of symplectic manifolds with sym­ metry. Rep. Math. Phys. 5, 121-130. Marsden, J.E., and A. Weinstein [1982] The Hamiltonian structure of the MaxwellVlasov equations. Physica D 4, 394-406. Marsden, J.E., and A. Weinstein [1983] Coadjoint orbits, vortices and Clebsch vari­ ables for incompressible fluids. Physica D 7, 305-323. Meyer, K.R. [1973] Symmetries and integrals in mechanics. Dynamical Systems, edited by M. Peixoto Academic Press, New York, pp. 259-273. Meyer, K.R., and G. Hall [1991] Hamiltonian Mechanics and the n-body Problem. Springer-Verlag, New York. Montaldi, J.A., R.M. Roberts, and I.N. Stewart [1988] Periodic solutions near equi­ libria of symmetric Hamiltonian systems. Phil. Trans. R. Soc. Load. A 325, 237293. Montgomery, R. [1984] Canonical formulations of a particle in a Yang-Mills field. Lett. Math. Phys. 8, 59-67. Montgomery, R. [1990] Isoholonomic problems and some applications. Commun. Math Phys. 128, 565-592. Montgomery, R., J. Marsden, and T. Ratiu [1984] Gauged Lie-Poisson structures. Contrib. Math. AMS 28,101-114. Morrison, PJ. [1987] Variational principle and stability of nonmonotone VlasovPoisson equilibria. Z. Naturforsch. 42a, 1115-1123. Oh, Y.G., N. Sreenath, P.S. Krishnaprasad and J.E. Marsden [1989] The dynamics of

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coupled planar rigid bodies. Part 2: Bifurcations, periodic solutions, and chaos. Dynam. Diff. Eqs. 1,269-298. Patrick, C , [1989] The dynamics of two coupled rigid bodies in three space. Contrib. Math. AMS 97, 315-336. Routh, EJ., [1877] Stability of a given state of motion. Reprinted in Stability of Motion, edited by A.T. Fuller. Halsted Press, New York, 1975. Shapere, A., and F. Wilczeck [1989] Geometry of self-propulsion at low Reynolds number. J. Fluid Mech. 198,557-585. Simo, J.C, D. Lewis, and J.E. Marsden [1991] Stability of relative equilibria I: The reduced energy momentum method. Arch. Rat. Mech. Anal, 115,15-59. Simo, J.C, T.A. Posbergh, and J.E. Marsden [1990] Stability of coupled rigid body and geometrically exact rods: block diagonalization and the energy-momentum method. Physics Rept, 193,280-360. Sjamaar, R. [1990] Singular orbit spaces in Riemannian and symplectic geometry. Thesis, Utrecht Sjamaar, R., and E. Lerman [1991] Stratified symplectic spaces and Reduction, Ann. of Math. 34,375-422. Smale, S [1967] Differentiable dynamical systems. Bull Amer. Math. Soc. 73, 747817. Smale, S [1970] Topology and mechanics. Invent. Math. 10,305-331; 11,45-64. Smale, S [1980] The Mathematics of Time. Springer-Verlag, New York. Wan, Y.H., and M. Pulvirente [1984] Nonlinear stability of circular vortex patches. Commun. Math. Phys. 99,435-450. Wiggins, S. [1988] Global Bifurcations and Chaos. Springer-Verlag, and Applied Mathematical Sciences, New York.

889 Inventiones math. 10, 305-331 (1970) © by Springer-Verlag 1970

Topology and Mechanics. I S. SMALE* (Pisa)

Introduction This paper is devoted to the subject, topology of mechanical systems with symmetry. The general purpose is to study the topology of a certain map /: T-» Wfrom the phase space of the mechanical system to a linear space W. This map / is the product map of the total energy and a second map which generalizes the notion of angular momentum. Since the "factors" of / are integrals (constant on orbits of the dynamics of the mechanical system), so is / itself. The integral manifolds /„=/ _ 1 (w) (which are usually manifolds) have thus the property, if ue/w,then the orbit through v is contained in Jw. In our estimation, the crudest, but very important, invariant of this orbit is the topological type of Iw, at least from the global point of view. Thus, for example, we can attach as numbers associated to the orbit of v, the Betti numbers of Iw when velw. An important aspect of the problem of studying the topology of the map /: T-» W is then to find the topological type of the /„. Actually it turns out that the Iw are invariant under a certain subgroup G„ of the symmetry group G and indeed the original dynamical system on T in­ duces a dynamical system on each of the quotients IJGW=1W. In fact the study of the dynamics on T can be reduced to the two problems, the first, the study of the map / and the second, the study of the dynamical systems on the spaces Iw. We also use the term integral manifolds to describe the lw, although this is a slight abuse of language. In this paper we do not study the dynamical systems on 7„. However a certain amount of information on thefirstpart of the above reduction is obtained. At least when the symmetry group is not too complicated some picture of the map /: T-* demerges. Central to this is the question, for what we W, does the topological type of lw change? To this end we define the bifurcation set Zc W to be the set of points in W over which / fails to be locally trivial (in the differentiable sense). Thus critical values of / make a contribution to Z * We received partial support in the research from NSF contract GP 8007. We would also like to thank C. Zceman and L. Motchane for their hospitality at the University of Warwick, Maths. Inst, and the Institut des Hautes Etudes Sdentifiques, Bures-sur-Yvette respectively.

890 S. Smale:

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and there may be others due to the fact that T is not compact. Thus we have a main problem, in finding the topology of 1: T-* W, to find the structure of the bifurcation set E. The paper takes the following form. The first three sections are devoted to the simple situation of mechan­ ical systems with two degrees of freedom and with a one dimensional symmetry group. According to ones taste, one might skip these three sections or stop once having finished them. We ourselves have found this low dimensional case very helpful when studying the general picture. The rest of the paper developes some general ideas on the previously mentioned reduction and applys them to a slightly idealized mechanical problem. A subsequent paper applies the work here to find the topology of the integral manifolds in the Newtonian n-body problem in the plane and to find the bifurcation set in that problem. Some good background references are [1,9-11, 13, and 4]. After this paper was written two references have appeared which relate to it. First J.-M. Souriau, Structure des systemes dynamique, Dunod, Paris 1970, part of which is related to our Section 4 and second S. Sternberg, Celestial Mechanics, part II, Benjamin, New York, 1969, part of which is related to our Section 5. Section 1. The Kepler Problem Atfirst,we spend some time going over the topology of some mechan­ ical systems with 2 degrees of freedom and possessing a 1-dimensional symmetry group. Most of this material is not new, but it is very helpful to understand this simple situation before reading the general theory or the more complicated example after that. In this section in particular we study the central force problem with Newtonian potential arising from the planar 2-body problem of celestial mechanics. We call this the Kepler problem. A good classical reference is Goldstein [10] while Kaplan [12] discusses some of the topology. Here as always in this paper we try to emphasize the global point of view. The position of the particle is represented as a point x=(x,,x 2 ) in the plane R2 and the velocity at a point x by veTx(R2)=R2, P = ( P , , V2). The kinetic energy is a function K: T(R2)-»R given by K(v)=±M1 where ||e|| is the Euclidean norm of e, and T(R2)= [) TX(R2). The mass we take to be one. The potential energy is the function V: M2 -► R given

byV(x)=--~,M2=R2-(0,0). Because of the singularity of V at the origin we will frequently use for the manifold of positions M2 instead of R2 and for the space of states T(Af2).

891 307

Topology and Mechanics. I

From this data, the motion of the states is given by a differential equation, 2nd order on M 2 ,1st order on T(M2) or a vectorfieldon T(M2) as follows (Newton's equations): X

X

These equations yield the orbits of states on T(M2). An integral is a real valued function on T{M2) which is constant on orbits. Then classically there are two integrals, energy £, angular momentum J, defined on T(M2)=M2xR2 as follows. E(x,v)=K(v)+V(x) and J(x, v)=xlv2—x2vl or using polar coordinates the last expression can be rewritten J(r,6,f,d)=jr(rd). See for example Goldstein [10] or Abraham [1] for the proof that these functions are integrate. The problem we consider now, which generalizes to the basic problem of this paper, is to make a global, topological, study of the map E x J: T(Af 2 )-RxR defined by ExJ(vx)=(E(vx), J(vx)), vxeT(M2). This mainly includes the study of the inverse images or "fibers" Icp= (E x J)~1 (c, p), (c, p)eR 2 and finding the bifurcation set I in R x R = R2, the energy-angular momentum plane, where the topological type of Ie changes. Strictly speaking I is defined as the set of points over which the map £ x J fails to be locally trivial, but here this happens precisely where the topological type of Ie p changes. (1.1) Proposition. Fixing the values of E at c, J at p, we have f = ]/2(c-K,(r)) where Vp(r)=

V(r)+^r.

Here r= ||x||, r is the derivative of the function r with respect to time along an orbit of our dynamical system. Sometimes we use the words dynamical system to describe the solutions of a differential equation even when these solutions are not defined for all time. Proof of (1.1) (cf. also Goldstein). On the orbit in question we have c=E=$ (f2+r2 ()2) + V and (p/r) 2 =(r 6)2. Putting these together we have (1.1). The function V : M 2 -»R, and its generalizations later, play a vital role in this paper. It can be thought of as a potential function amended by adding a term corresponding to "centrifugal force". It is helpful to keep the graphs of Figs. 1 and 2 in mind. Note the following as a consequence of (1.1). (12) Proposition. For an orbit with energy c and angular momentum p, we have c-K,(r)^0. That is to say if vxeT[M) has EJpJ^c, J(vx)=p, thenc-V,(M)Z0.

892 308

S. Smile: V* p+0

Fig. 2

Fig. I

For p#0, let g(p)=min V (r) and let Ii be the curve in the (c,p) 0
'

plane defined by g(p)=c. Thus It is at least qualitatively described by Fig. 3 and from (1.2) it follows that the image of £ x J lies to the right of r,. For p - 0, the differential equations are not complete since the particle runs into (0,0). This point we hope to pursue in a later paper. In any case a well-known regularization procedure removes the singularity and provides a partly compactified picture.

^

S

Fig. 3

At least if we exclude p=0, the real line R acts on ICf as the dynamical group, as the restriction of the dynamics on T(M2\ since £, J are integrals. Furthermore the group of rotations of the plane S l =SO(2) acts on M2 by rotations and induces an action on T(M2\ Clearly this action of SO(2) leaves the functions V, E, J all invariant as well as the subsets /c<, of T(M2\ Furthermore the action of 50(2) commutes with the action of the dynam-

893 Topology and Mechanics. I

309

ical group R so that together they describe an action of the product group RxS0(2) on T(M2) and on lcf for each (c,p) with p*0. (This situation is described quite generally later where proofs of the preceeding statements will be provided.) ( U ) Proposition. Given (c,p)eR2, p*0, 50(2) xR as above acts transitively on Ic pc T{M2) (of course !e,p-9 if c
Fig. 4

Fig. 6

Fig. 5

Fig. 7

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Then (1.1) completes the picture, for example implying in case (b) that the path of the particle in M1 oscillates between the circles of radius r2 and r3. The point is that r gives the rate at which the distance of the particle approaches or leaves the origin and this doesn't change except for c=Vr{r). Note also the path of the particle in the plane is always bounded away from the origin by some circle (recalling p *0) and except in case (a) from oo by another circle. If p=0 for some trajectory, then that trajectory remains on the same ray from the origin. If c^O, then the trajectory goes from 0 to oo along the ray for - b < t < c o and from oo to 0 for -co0, xeR2. Then the amended function Vp is defined as before by

and again r=/2(c-K,) as in (1.1). Suppose now that V and p are such that Vp is smooth and has nondegenerate critical points on an interval [a,b] with b>a>Q, V{a)= V(b)=c0. Suppose also on [a,b] there exist 3 critical points rltr2,r3 with corresponding values cy
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Fig. 8

(2.1) Proposition. In the mechanical problem described above, the topobgical type of lc p changes precisely at the critical values ofc = c-, .'=1,2,3, ofVp. In fact for c < c,, Icp is empty. For c=Cj, Ic p is a circle in T(M2) which projects onto the circle in M of radius r: about 0. For c,
Fig. 9

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For c 0 >c>c 3 , lCtP will again be a single torus, say corresponding to the rotated outer circle of Fig. 9. Note that the 2 smaller circles rotated form 2 tori which could form lep for some c with c1 for various (c,p). We say a few words about this. Suppose p is fixed and citc2 are two critical values of Vf. Let lef be a toral component of lCf varying smoothly in c,cl,. (Actually p{c,p) is defined on some open set of the (c, p)-plane.) It characterizes completely the topological type of the flow on U Ic>p (or if we vary p, p(c, p) C|
characterizes the flow on the corresponding open set in the 4-dimensions phase space). We would like to raise some questions about the nature of p. But first to clarify matters we have: (12) Proposition. p(c, p) is a C*-function in (c, p\ For the proposition, wefindan exact expression for p(c, p). For this we start with a formula that can be found in Goldstein [10, p. 73]. dr ,0

—]/2mc-2mK(r)--^-

This means the following: A trajectory in time moves in polar coordi­ nates {r{t), 0(t)) in the plane; if the angular momentum p#0, then given the initial condition (r0,6Q) at t=0, we obtain 6 as a function of r, until r reaches its minimum value, say r,. Simplifying by taking m=l,

e-e.-!-,

dr

"> L-y2(c-V,(r))

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. .. 1 and writing a=—, du

e-e0= J *o

iff^))

p Now the rotation number is found by taking a constant times the change in 9 as r varies from r2 to r, and back to r2 or with a corresponding change in u, i.e. ■? du

fhW) The difficulty in differentiating this expression as a function of c comes from the fact that the limits of integration depend on c. In fact it is here that we need that c is not a critical value for Vp. It was Pugh who first showed me how to differentiate the integral as a function of c and in fact that it was C°°. Eventually Markus gave me a reference to these things, namely Loud [14]. In this way, we obtain a proof of (2.2). (2.3) Problem. For most V and p, do the functions p(c, p) above have the property that p(c, p) has non-degenerate critical points as a function ofc? Here most would be described in terms of a Baire set in some function space of the Vs with the C-topology times the real line (for p). This is very likely to be true and would imply quite immediately that in general both the periodic orbits and the non-periodic orbits are dense among the bound orbits of central force problems. See [2] as a reference for this type of problem. (2.4) Problem. Same as above but consider p as a function of 2 vari­ ables (c, p) and look for p with non-degenerate critical points on an open set in R2. One also has the further structure of a product naturally on R1. Section 3. Surface of Revolution In the previous sections we have been discussing a particular case of the following problem. Suppose the group S1 acts on a 2-dimensional manifold M, the action leaving invariant a function V: Af-*R and a Riemannian metric K on M. Then S l also acts on the tangent bundle T(M) and leaves invariant the function still denoted by K, K: T(M)-* R defined by the norm squared. That is on each fiber Tx we have an inner product Kx and for veTx, K(v)=Kx(v, v). Then the energy E=K+ F»jr,

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n: T(M) -»M the projection, defines aflow(up to a scalar factor) on T(M) by Hamilton's equations for example. Then the problem is to obtain a description of this flow. One might call this," the problem of 2 degrees of freedom with symmetry". In this situation we remark that the topology of M is limited if the action of S1 is non-trivial since in this case the Euler characteristic of M is non-negative. In fact it is easy to see that if M is orientable then Af is S2, R x S \ S1 xS 1 or D2 and the action on M is "standard". (In this paper S" is the n-sphere, D" the n-disk, R the real numbers.) In the first sections we considered situations where K was uninter­ esting and the potential energy V gave a non-trivial character to the analysis. Here we look briefly at a problem of this type where V is zero but K has non-trivial curvature. This problem is to understand the flow described by geodesies on a surface of revolution. Here one sees a different aspect of the geometry which is helpful in giving an intuitive picture of the previous sections as well as the more general case to follow. For a classical reference see e.g. [19]. Thus suppose we are given a positive function / on the interval [u,, u2] whose graph we suppose rotated about the u-axis, to give us a surface of revolution Atf. The geodesies define a flow, the geodesic flow on the unit tangent bundle T^(M). We will ignore problems concerned with the boundary of M for the moment. Now we have a function C: 7j(Af)-»R which is constant on orbits, noted by Clairaut, defined by <£(&,)= f(u) • cos 6 where xeM is the point (u,f(u), \l>0) of the surface and 0 is the angle between vx and the circle {(u,f{u), ^)|0g \ji g 2n} in T x (M)cR 3 . See e.g. [19] for this. Later we will see the general method (of Noether type) that gives the Clairaut function and angular momentum. Since S is a real valued function on a 3 dimensional manifold, one can expect in general the level surfaces to be 2 dimensional manifolds.. Furthermore if we keep to the orientable compact case, the existence of a nowhere trivial vector field shows that these level surfaces are tori. Finally since Sl acts on each such surface preserving the flow, the geod­ esic flow is essentially a translation on each torus and the topological type of such a flow is characterized by its rotation number. We consider in more detail the following special case. Suppose our smooth / has only one critical point on [HUBJ], a non-degenerate maximum at um,ui
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I don't know if one can find an answer to the following in the literature (I can't): (3.1) Problem. Does there exist a function / as above with the cor­ responding p constant and irrational ? Note that the sphere gives an example of such an / with p constant and rational. An affirmative answer to (3.1) would give a counter-example to a question raised by Klingenberg. Klingenberg asked if always the periodic orbits were dense in geodesic flows on compact manifolds1. One can prove by the methods of (2.2), that the function p of (3.1) is C°° on [a, b). One replaces the integral formulae there involving Vp to different integral formulae from the differential geometry of geodesies on surfaces of revolution. Struik [19] gives the starting point for this. (3.2) Problem. For most /'s as above do the corresponding p's have non-degenerate critical points? Compare (2.3). Section 4. Symmetry in Mechanics Here we give a general setting for mechanical systems with symmetry. Noether is largely responsible for clarifying the relationship between symmetry and integrals. For recent treatments of this relationship as well as a modern setting (in terms of symplectic manifolds etc.) of Hamiltonian mechanics see Abraham, [1], Hermann [11] and Loomis and Sternberg [13]. What we add to these here, (4.3) below, is an explicit description of the integral as a map onto a Lie Algebra (the dual Lie Algebra of the group) rather than just considering the coordinate functions. In general we will be considering the configuration space of a classical mechanical system as a smooth (C°°) manifold M with tangent bundle T= T(M) as the space of states. The kinetic energy of the system can be thought of as defined by a Riemannian metric on M; i.e., for each xeM, Kx will be an inner product in the Tangent space TX{M), smooth in x. Then the kinetic energy itself will be the function K: T-»R defined by K(v)^Kx{v,v) where veTx(M). The potential energy is given by a function V: Af-*R, smooth with occasional exceptions being certain singularities. The energy or total energy is the function E: T-»R defined by £ - * £ + V- 7i, where JI: T-*M is the projection. From the above data, via Lagrange's equations (or Newton's or Hamilton's, see for example, [1]), there is defined an ordinary differential 1 Added in Proof. Klingenberg has since answerd his question negatively via the route, of showing that (3.1) has a positive answer.

22 Inwntionei math, VoL 10

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equation or smooth tangent vector field on the state space T{M). One can also think of this equation as a 2nd order differential equation on Af. This is called the equation of motion and its trajectories describe the passage of the state in time. We will consider primarily the case with symmetry. That is, there exists a Lie Group G which acts as a Transformation Group (cf. [IS]) on Af as a group of isometries with respect to K and leaves V invariant (i.e. V is constant on orbits of G). We now define in fact the above data to give a simple mechanical system with symmetry or for short a mechanical system with symmetry. (4.1) Definition. A mechanical system with symmetry consists of a 4-tuple (Af, K, V, G) where Af is a manifold, K a Riemannian metric on Af, V a function on Af and G a Lie group acting on Af preserving K and V with all data smooth. Note also G acts on T(Af) as a group of bundle automorphisms (via the derivative). Denote by © the Lie algebra of G. In the situation of (4.1) there is a canonical map <xx: ©-» r,(Af) for each xeAf, defined as follows: Let a.(X) be the vector field on Af gen­ erated by the 1-parameter transformation group corresponding to X e®. Then a.x(X) is the value of a (A") at xeAf. Let Jlx: T*-* ffi* be the dual map of ax: © -► Tx where the * refers to the vector space duals. Then define Jl: T*(Af)-> ©* as the map which restricted to each fiber is Jlx. Let K*: T(Af)-» T*(Af) be the bundle isomorphism defined by the Riemannian metric corresponding to K on Af and J: T-»©* be the composition Jl • K*. (4.2) Definition. The composition J: T-* ©* just mentioned is called the angular momentum of the corresponding mechanical system with symmetry (Af, K, V, G). A justification for this term will be given in the next section. The following proposition is basic for this paper. (43) Proposition. Given a mechanical system with symmetry (Af, K, V,G\ the angular momentum J: T->©* is equivariant with respect to the adjoint action of G on ©*, and an integral. That J is an integral means just that J is constant on orbits of the dynamical system on T. We explain now what "equivariant with respect to the adjoint action" means. Equivariant means that the map commutes with the action of G on the source and target. We have an action of G on T(Af); thus it remains to describe an action of G on ©*. Any connected Lie group G acts on its Lie algebra © via "the adjoint representation". That is for each geG let a f : G-*G be the inner auto­ morphism ot f (fi)=gng -1 . Then af(e)=e and the derivative at e of such

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an inner automorphisms give us a linear automorphism of T,(G)=<5 which together define the adjoint representation of G on ©. Then there is a canonical induced action of G on the dual space ©*, still called the adjoint action. We give the proof of (4.3) in an expanded context. Suppose G acts on M, with induced action on the cotangent bundle T* and that some function H: T*->R, called the Hamiltonian, is invariant under G. Via the symplectic structure on T*, H defines a G invariant vector field XH on 7* (see e.g. [1] for details). In the earlier case H specializes to the sum of the kinetic energy (equally well defined on 7*) and the potential energy. The following will be used to obtain (4.3). (4.4) Proposition. Under the above hypotheses of a G invariant Hamiltonian H on T*(M\ the map J,: 7"*(M)-*<5* defined above is equivariant and an integral. For the proof of equivariance, recall the Lie Group Formula from Nomizu [15] which reads RaA*=(ad(a~1)A)*, where a-»R, is the action of G on M and A*(x)=oix(A% for Ae<5. By dualizing we get the equivariance statement. We have to check now that Jx is an integral. Choose an element n of (5 and define fn: T* -»R by fn(v>)=J(w) (n). Since n was arbitrary, it is sufficient to show that / , is an integral. But for this one can compare with Abraham [1] where each of a class of functions including such /„ is proved to be an integral. This proves (4.4). For (4.3), we only need to note that under the conditions of (4.3), the isomorphism K* is the Legendre transformation which will be G invariant and preserve orbits of the dynamical system. We have the following corollary to (4.3). Of course there is a similar corollary to (4.4). (4.5) Corollary. Let peffi*, Gt be the isotropy group at p of the adjoint action of G on ®*. Then Gp (as a subgroup ofG) leaves J~ i{p) invariant. We are able now to pose with precision one of the basic problem of mechanical systems with symmetry and the basic problem considered in this paper. (4.6) Problem. For a mechanical system with symmetry (M, K, V, G) find the global topological structure of the map ExJ: T-»Rx©* defined by (£ x J)(p)=(E(v), J(v)). We refer to both the general case of mechanical systems with sym­ metry and certain special ones such as the Newtonian 3-body problem, which we will pursue later. Finding the structure of E x J involves at least knowing the following (compare to the Introduction). 22*

902

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(4.6)A. Define the bifurcation set Z of (M, K, V, G) to be the set of points of R x ©* over which E x J fails to be locally trivial (of course in the differentiable sense). For example a critical value of E x J will make a contribution to Z. Over each component of Rx (5* — Z, ExJ is a fibration. Finding the structure of Z is an important part of the basic problem (4.6). (4.6) B. Find the topological type of the integral manifolds. This means the following. Let 7CiP={Ex J)~l(c,p), and lCf=lcJG? jAs in (4.5), Gr also acts on Ief, as well as /,). We call the spaces ICf, Ier the integral manifolds of (M,K, V,G). For exceptional values of c,p, Iep, Ie p may not actually be a manifold. If vele p then the orbit of v is con­ tained in Ic p, and in fact one even has an induced flow on the lcp. In a strong sense there is a reduction of the flow on T(Af), the original mechanical problem, to the flows on the spaces 7C>,, together with the structure of the map ExJ. Finding the topological type of the lCf is an integral part of the problem of studying ExJ. Note that this topological type is likely to change as (c, p) pass through Z. We call the space R x 05* sometimes, the energy-angular momentum spaces. Note that the following formula is a consequence of the definitions. (4.7) Proposition. For XxeTx, Ye®, J(jy(Y)=Kx(Xx,ax(Y)). It might be helpful for the reader to return to the previous sections to see how the simple examples there fit into the framework of this section. Section 5. The Newtonian 3-Body Problem; the Elimination of the Node We see what the preceeding section means in terms of the classical 3-body problem (with Newtonian potential). The first goal is to represent the 3-body problem as a mechanical system with symmetry in the sense of (4.1). As a first step consider the configuration space M' to be the set of x«(x 1 , x 2 , x3) with each x,e£ 3 , £ J being Euclidean 3-space (i. e. a real 3-dimensional vector space equipped with an inner product). One thinks of x, as being the position of the fb particle or body with mass say m(>0. Thus M' is a 9-dimensional linear space and so its tangent bundle has the natural structure of a product T(M')=M' x M'. One can think of a point of this space as a pair (x,p) where xeM' and u—(vltv2,v3), each v,eE3. Here p, is to be thought of as the velocity of the Ph particle. Then Kx(v,v)^K{v,v)^ £ i m i !"iH2 defines the kinetic energy where M is the norm of p, in the inner product structure on E3.

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The potential energy is the function V: M' - A -+ R V(x)=£ -

LJ

r-

II x ( — XjH

where A = \)Aij is the set of singularities of V, AiJ={xeM'\xt=xj}. Summations, unions are taken over the set of pairs (i,j)» lgi',7'^3, i * j . The symmetry group of this mechanical system can be taken to be G=SO(3), the rotation group, or the group of linear transformations of E 3 leaving the inner product invariant (and say orientation preserving). But also see the remark at the end of this section. The natural action of G on £ 3 induces an action, the diagonal action, on M', and then of course a further action on T(M'). Note that this action leaves A invariant. Thus we have at hand a mechanical system with symmetry in the sense of (4.1), (M' - A, K, V, SO(3)) with M' a linear space of dimension 9, K a flat metric and SO (3) represented linearly on M' and T(M'). Of course now, Newton's equations are defined, say as second order equations on M' — A. They take the simple form x = — grad K(x). By writing x=i;, one obtains Newton's equations simply as a vector Field on T(M'-A) which defines the dynamics of this mechanical system. The following proposition gives some justification for the use of the words angular momentum as used in (4.2). (5.1) Proposition. The angular momentum J of (4.2) for the mechanical system with symmetry (M' — A, K, V, SO (3)) (the 3-body problem) takes the form./»(»)=£mi(xi x t>,) where x, x y, is the vector product ofx, and vt in E3. For the proof of (5.1), one need consider simply a mechanical system with symmetry (E3, V, K, SO (3)) with V irrelevant, K the inner product on E3, SO (3) the usual representation preserving K. Then, with (4.7), a well known interpretation of vector product (see e.g. [3]) gives us what we wish. On M'xM', Jx(v)=B(x,v) is a bilinear, non-degenerate, anti-sym­ metric form with values in a 3-dimensional vector space. (5.2) Definition. The center of mass is a function from M' to £ 3 whose value at xis£m,Xj. See Goldstein for a reference for this next very classical fact (53) Proposition. The center of mass of a point in M' moves in time with uniform velocity in a straight line in E3, or this velocity is zero and the center of mass is fixed in time. (5.4) Corollary. Let M = {xe M' \ £ m, x,=0}. Then T{M-A)cT(M'-A) is invariant under the flow.

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Here r(M)={(x,»)6M'xM'|£ffl 1 x i =0, Zmi°t=°}is easily checked

The n

« x t fact

(5.5) Proposition. Suppose (Af, K, V) is a mechanical system (we can ignore the symmetry for this) and M is a submanifold ofM' with the property that T(M) is invariant under the flow on T(M'). Then the restriction of this flow to T(M') is equivalent to the flow on the mechanical system (Af, K/M, V/M) where K/M, V/M are the restrictions of the kinetic energy, potential energy to M. Putting these last two propositions together, we thus obtain a mechanical system with symmetry (M — A, K, V, SO(3)) in the sense of (4.1). This is the problem of 3-bodies with center of mass at the origin. Here Af is a linear space of dimension 6, A a union of 3 linear subspaces of Af of dimension 3 each, K is flat, and SO (3) is represented linearly on Af, T(Af) leaving invariant A (as subspaces, not pointwise). In this system the Ie p can be expected to be 8 dimensional manifolds in general since they are inverse images of ExJ: T-*RxR 3 where T has dimension 12 and the target space 4 dimensions. The isotropy group Gp in this case for p a non-zero angular momentum is the group S1, the group of rotations of R3 keeping p fixed. Then the quotients lctfleJSl can in general be expected to be 7-dimensional manifolds. (It is not difficult to show that Sl acts freely here.) Similar reasoning leads one to expect le0 to be a 5-dimensional space, "the exceptional case". Thus the reduction spoken of in a general context at the end of the last section in the case of the 3-body problem, center of mass at the origin, gives rise to flows on 7-dimensional manifolds. This last step, the step from Ic p to lep is referred to in Wintner's book [21] as "Jacobi's celebrated elimination of the node", with origin in the work of Lagrange. A classical treatment can be found here or in Birkhoff [5]. See also Cartan [7]. Our treatment above, we believe is more conceptual, making it clear that what is happening is a quotient construction. Of course it has the advantage also of working very generally. In any mechanical system with symmetry in the sense of (4.1) (or more generally for that matter), to "eliminate the node" one just divides / c p by the isotropy group of the adjoint action. An earlier special case of our problem (4.6) B is in Birkhoff [5]. He asks in the 3-body problem, what is the connectivity of the Je p. Wintner in [21, p. 334] also raises this question. These authors connect such problems to regularization questions. We remark that one could make a further analysis of the 3-body problem, using the 6-dimensional group of rigid motions of £ 3 as symmetry group before the center of mass is fixed. Such an analysis

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uses the adjoint action of this group and it seems to me would show that one misses something significant by taking the center of mass fixed. Section 6. Amended Potential and Bifurcation We return to the general mechanical system with symmetry of (4.1) and in this section attempt to obtain information on the nature of the bifurcation set Z of this system. Recall associated to the mechanical system with symmetry (M, K, V, G) is the map ExJ: T— R x (5* and I is the subset of R x (5* over which E x J fails to be locally trivial in the differentiable sense. Let E' be the subset of R x ©* of critical values of £ x j . That is to say I' is the set of y=(ExJ){v) where the derivative D{ExJ)(v): T„(T)-»Rx(5* fails to be surjective. Then clearly I' is in fact a subset of T. We are concerned in this section only with i'. First we observe the proposition, essentially from calculus. (6.1) Proposition, veT is a critical point ofExJ: T-*R x ©* if and only if DJ(v): T„{T)-*<5* is not surjective or veJ~l(p) is a critical point of E:J~l(p)-*R for some p. In the last map we mean of course the restriction of £ to J~l{p). The next proposition follows from the definitions. We are supposed given a mechanical system with symmetry as usual. (6.2) Proposition. The following is an equivalent set of conditions on a point xeM. (a) <xx: (5 -*Tx is not injective. (b) Jx: T*-* ©* is not surjective. (c) The isotropy group Gx of the action of G on M has positive di­ mension. We introduce A <=■ M as the set of x in M which satisfy (a), (b) or (c) equivalently of (6.2). Note that A is closed in M and in general, one can expect A to be small. In most mechanical problems the dimension of M is bigger, often much bigger, than the dimension of G. In the 3-body problem of Section 5, A consists of (x lt x2, x3) which are co-linear. For various reasons, it is of interest to consider the planar three-body problem. This is the mechanical system of Section 5 where £ 3 is replaced by the plane E2 and G=SO(3) is replaced by S1, rotations of £ 2 . The other data isn't changed in any essential way. Then the map £ x J : T(Af)-»RxR goes from a 8-dimensional space (fixing center of mass at the origin) to the plane. In general here Ic p will be a 5-dimensional manifold. We mention this simpler case here because A is almost trivial for this system. In fact A consists of the single point (0,0,0) in M as is easily seen. For this reason, what follows is especially directly applicable to this and similar systems.

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Returning to the case of a general mechanical system with symmetry, we have: (63) Proposition. Let pe(5*. Then for each xeM-A, the following two conditions characterize an element ap(x) of Tx. (a) a,(x)ej;lb). (b) ctp(x)U~1(0) in the Kx inner product structure on TX. Furthermore this <xp: M — A-*T{M-A) field onM-A.

is a smooth tangent vector

This is evident. Here ap(x) is the vector in Tx from 0 to J~ '(p) which has minimum length in the Kx norm. (6.4) Definition. Suppose (M, K, V, G) is a mechanical system with symmetry and A, <tp are defined as above for some fixed pe©*. Then the amended potential is the function Vp: M—A -* R defined as the composi­ tion Vp=E*ap where E: T(M-A)-*R is the total energy. (6.5) Proposition. The amended potential has the following properties: (a) V = V+K°amandVJx)=

min E{v).

(b) Over M — A, the critical set of E: J _ 1 (p)-»R is precisely ap(T) where T is the critical set of Vp:M-A-*VLandI' = {(y, p)|ye K,(/*)}(c) Vp as well as T is Gp invariant where Gp is the isotropy group of the adjoint action of G on (6*. (d) In the planar 3-body problem (or in fact for the planar n-body problem, any V\ Vp(x)=V(x)+

4Jg

^

(e) *(/,.,)= K J r 1 (-oo,e]. Remark. Similar to (d), one can check that this amended potential specializes to the Vp of Section 1 and 2, the potential with a term added on corresponding to centrifugal force including the factor of 2. Remark. By (c), the critical set f will consist of Gp orbits. One can expect generically these will be isolated and in fact non-degenerate critical submanifolds in the sense of Bott [6]. Compare this to Wasserman [20]. Perhaps the main importance of (6.5) is that under some circumstances it gives a way of finding Z\ in fact a fairly practical method. We turn to the proof of (6.5). Parts (a) and (e) follows from the definitions, while (c) follows from the definition of Vp as a composition and (4.5). In (d) we take M=(Ra)"; the center of mass at 0 case is taken care of via (5.5). Let K* be the dual inner product to K defined on the dual space M* and M~* M* be defined by x-* Jx where JxeM*, Jx: M -»R defined

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as the planar angular momentum, Jx{v)—£ m,(Xj x v). Let Q be the inner product on M defined by K* via the isomorphism x-*Jx. Wefirstshow P2 that K(a„(x))=—f—- and then that g = 4K. ' Q(x,x) Let || H be the K-norm on M as well as the K* norm on Af*. Then it needs to be shown that ||aJ,(x)||2()(x,x)=p2 or ||ap(x)| ||x|| Q =|p| or \\JX\\ lla,(x)|| = \Jx{af(x))\. But the last is clear. Thus for (d) it only remains to show that Q=4K, K(t>)=£$m, \\v,\\2. Explicitly Q(x,y)=K*(Jx,Jy) by definition of Q. The form of K and Q gives us a reduction immediately to the case n-l, or R2. So we just have to show for xeR2, Q(x,x)=2m ||x||2, the norm now being the ordinary norm on R2. Let x=(x', x 2 )eR 2 be fixed, but we then choose coordinates so that 2 x =0. Recall Jx(v)~m(xiv2-x2vl) or in this case Jx(v)=mxl v2. Then ||/C*JJJC= Sup \Jx(v)\=m ll-IUSi

sup

|xV|

|/fMsi

= |/2m|x1|.

Soe(x,x)=||X*yj| 2 =4X(x,x). We finally give the proof of (6.5)(b). Since the last fact follows from the first, it only needs to be shown that under a, the critical sets of Vf: M-A-+R and £: J _ 1 (p)/M-/1-*R correspond precisely. Here J " l (p)/M - A=J ~i (p) r\ n " l (M - A). Keep in mindI the diagram: T(M)=>J-l(p)-*->R

Since this proof is only concerned with M -A, we are replacing M - A by M hoping no confusion will result (e.g. we suppose A = f>). First suppose v is a critical point of E: 7 _1 (p)-»R. So veJ~l(p) for some xeM. Then v is a critical point of E restricted to the affine space J~ l(p) and £ restricted to T x o Jx" l(p) is a positive definite inner product. From the definition of af(x) it then follows that p=a,(x). Then by the chain rule and the above diagram, DVr{x) must be zero, so x is a critical point of Vf. There remains to show that if x is a critical point of Vf: Af-»R, then o=a,(x) is a critical point of £: J~l(p)-*R. For this we can write TV(J-'to)= T,{J;' to)+Image D a,(x)(T,(M)). Now supposing that x is a critical point of Vp, by the chain rule, DE(v) maps into zero the second term of the summand of the previous

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equation. On the other hand DE(v) maps the first term into zero, since v=ap(x) minimizes £ on J~1 (p). Thus v is a critical point of E: J~ l(p)-*R and the proof of (6.5) is finished. Section 7. An Example with Bifurcation Set and Circular Orbits In this section we consider an example of a mechanical problem which has some relation to various situations that occur in physics, e.g. at a minimum of Sl invariant potential energy, or to the n-body problem. We don't make any attempt to firm up these relations here. We present it as a mechanical problem which illustrates the phenomena discussed in the previous sections in a simple way, but where the topology is non-trivial. In fact the topology permits the use of Morse theory to obtain the existence of certain periodic orbits (the "circular orbits"). Besides that some aspects of the Newtonian 3-body problem are present here in a technically simpler way. Take m,,..., m„ to be positive numbers with n>2 and let configura­ tion space M be the set {(x,, ....x^efR^I^/njX.—O} so that M has the structure of a linear space with a given action of S l (i.e. Sl acts on (R2)" by the diagonal action and restricts to M). Let the kinetic energy K be the function on T(M) defined by K (v)=£ \ mi \\ v, ||2 where t;=(i>, t>„), 2 ^WJOI-XO, and H || is induced from the norm on R . Then of course K defines a Riemannian metric on M and 5' acts on T(M) leaving K in­ variant. We suppose also that a potential function V is given invariant under S1 acting on Af, is smooth on Af, and K(x)-»oo as ||x|| ->oo on M. (7.1) Theorem. The mechanical problem defined above has the following properties: Let m = n — 2. (a) There are at least (m+1) closed orbits of this problem, circular in the sense that they are invariant under S1, for each p^O. (b) In a C function space of the V described above, there is a Baire set 3t described in the proof with the following property: if Ke3 then in R2 = {(c, p)}, each horizontal line of the form Rxp, p4=0, intersects the bifurcation set I at least (m +1) times. Proof. According to Section 6, supposing here p*0, there is defined the amended potential V- M-*R by K,(x)= V{x)+—-£—- which will ' ' C(x,x) now be smooth except at 0 where Kf(x)->oo as ||x||-*0, and Q—4K. Also Vf is Sl(=G} invariant and V,(x)-+oo as Hx|| —»co. Excluding 0 from Af, and letting S1'~i = SQ be the set of xeM with C(x,x)=l, we may write M—0=R+ x S^* 1 . Here wi=/i—2, R + is the positive reals, and this coordinate is given by the Q-norm of xeAf—0. This product structure is S1 invariant.

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Then it is easily seen that the representation of S1 on Af is such that, the quotient S2"*+ '/S1 has the natural structure of /""(C), m-dimensional complex projective space. Thus we have an induced map Vp: R + x P"(C)-»R under our hypotheses above. For consideration of Morse theory, we say that V is non-degenerate if Vp has non-degenerate critical points in the sense of Morse. It is an easy check to see that almost all V of our type are non-degenerate. This check amounts mainly to the approximation theorem that any V can be approximated by a non-degenerate one. If V satisfies (7.1), then Vp. R+ xF"(C)-»R has also the property V~ '(-oo, c] is compact for any c. One can now apply Morse theory or Lusternik-Schnirelman theory to this function Vp: R + x P" (C) -»R. In this case, even though the manifold is not compact, one can use the fact that condition C) of Palais and the author [16,18] is satisfied. Schwartz [17] shows that Lusternik-Schnirel­ man theory is valid under these conditions and thus the number of critical points of Vp is at least the category of R+ x P"(C). This category is m+1, so 9 has at least m+1 =n— 1 critical points. Note that this is under the assumption of (7.1) (a) where no non-degeneracy hypothesis is made. With (6.5), this implies (71)(a). To obtain (7.1)(b), take 3 to be the set of those V: Af-»R, smooth, S1 invariant and K(x)-»oo as ||x||-»oo, with the additional condition that Vp has non-degenerate critical points with distinct values. According to the above discussion, Morse theory with (6.5) yields (7.1)(b). If V were given by an inner product on Af, we could make more precise the nature of Z in R2. This is done in the following theorem in fact (12) Theorem. Suppose that in (Af, K, V, Sl) of (7.1), V is given by an inner product. Then in the energy angular momentum plane R2, Z is given as a union of rays in the positive energy plane. In fact Z is the set {(|p|a i ; p)eR7i=l,...,m-l-l,peR,p*0}

where a,>0.

Generically, the a, can be expected to be distinct. For the proof of (7.2), we note by differentiation that xe M is a critical point of Vp for p * 0 if and only if: VM

-^xY-=0

fora,1 6iV/

"

-

Here we write V(x)= V(x, x), thinking of Fas a function of 2 variables when it is considered as an inner product. Define the induced map V*: M -»M * of Af onto its dual space Af * by V* (x)(y) = V(x, y) and the same for Q, Q*.

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Then our equation for critical points xe Af reads

K w

P2 re (x)

' -et^ * "°

1

or letting K=e*- K*. _

2

Since V is symmetric, V is a self-adjoint linear operator on M, con­ sidering Af as a Finite dimensional Hilbert space with inner product Q. Thus V has real positive eigenvalues. Now all of this picture is invariant under the action of Sl on Af, and so there will exist a sequence of 2-dimensional S1 invariant eigenspaces of V, say El,...,Em+lc:M, with corresponding eigenvalues 0<Xlg, Going back to our previous equation, we obtain the following structure for the critical set T of V': Af-»R. Let /]=/](p) be the set of m+l

xe£, which satisfy p2=X,Q(x,x)2. Then r = U /;. <-i m+l

Now we apply (6.5) (b) tofindthe equation for I — I'. Thus E = (J Z, where Z,={(K,(x),p)|x6^(p)}. Let zeE,, Q(z,z)=l. Then we can write with/?>0,

Z,-{{V,(Pz),p)\p*-llQ{fiz,fizrt}

J(V{fiz)+X,Q{fiz,Pz),p)\p2=^-\

= {(\p\a„p)}. Viz) Here al=—j=r+'\/Tl. This proves (7.2). Section 8. The Integral Manifolds The goal of this section is to give some picture of the topology of the integral manifolds by stating a number of easy propositions. To illustrate the ideas consider first a simple mechanical system (Af, K, V) without symmetry, where M is a compact manifold. Thus K is a Riemannian metric on M and V: M-+R a smooth function. Suppose c is a regular value of Vand define Me= K - I (-oo, c], £ c =£ -, (—oo,c]. Then it is a classical and physically important fact that no orbit with energy c can cross dMc. In fact it follows easily from £ = X + V»n that n{Ee)<=Mc. (Note that K is a positive function.) Toward the problem of determining the topological type of £c, given Mc we define the notion of reduced tangent bundle T(W) of a manifold W

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with boundary. One takes f{W) to be the tangent cell bundle of W with each fiber over 8 W collapsed to a point. Then one can prove that t(W) has a natural structure of a smooth manifold with smooth bound­ ary. The diffeomorphism type of f (W) is determined by that of W. We leave to the reader the check of these facts as well as the following pro­ position. (8.1) Proposition. With notation as above: (a) E c =f(AQ. (b) If one passes through a single non-degenerate critical point of index k of V in going from Me to Mc+a then one also passes through a single non-degenerate critical point of index kofE in going from EetoEc+t. In the more complicated situation of a mechanical problem with symmetry, there is something similar to the above, but more interesting and important. It turns out that Vp will play the role of V. Thus suppose we have a mechanical system with symmetry (M, K, V, G) with p€<6*, J, A, Vf, Gp defined as in Sections 4 and 6. By subtracting A from M we obtain a new system with A empty. Thus we suppose A=$ from the beginning or thus Jx: Tx -» ©* is surjective for each xeM. (8.2) Definition. Let ceR be a regular value of Vp: M-*R. Then let Afe,,= {xeM|K,(x)gc} and £,,,= {i;e.r'(pJIEOO^c}. The following proposition is a consequence of the definitions, (6.5) and is left to the reader. (S3) Proposition, (a) dEcp=Iep, n(Eep)<=MttP. (b) ECiP={vxeJ-l{p)\K(vx)-K(*,{xUc-Vf(x)}. (c) / / Vp: M -»R is proper, so is Eon J~l{p). (d) Similarly if Vp: Me p-*R is proper, so is E: Eep-*R. (e) Let Et be the bundle J - I ( 0 ) restricted to Mep. Then Eep=Ev We make some remarks about (8.3). The second part of (a) has the physical interpretation that any orbit with angular momentum p and energy less than or equal c must stay for all time in the region Mc p of M. The space El used in (e) is constructed as the reduced tangent bundle at the beginning of this section, just replacing T(W) by £j in the definition. The following proposition collects two final usefull but elementary properties about the spaces we have just defined. (&4) Proposition, (a) There is a Gp equivariant diffeomorphism from £,,, to {v.eJ-HO^WvJl^c-V^x)} defined by the map vx^vx-«p(x). (b) ap: Me p-*Ecp preserves a non-degenerate critical manifold in the sense of Bott, and preserves the index as well.

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An important fact about all of the data in (8.3) and (8.4) is that it is G invariant. Thus one can take quotients of all of these objects and all of the properties carry over to the quotient situation. We will use to denote the quotient by Gp. For example i.e P=EC JGf, Mcp=MclJGp and ft: Ecp-*Mcp is the map on the quotients induced by n so that the following diagram commutes:

£<,„-^£

•p i

M

CP

Here Gp represents the "fiber". In a similar manner we have defined £: £ c p ->R, Vp: Afc p -»R and V M , t , - E c , „ . Note 5£ r > „=/,,,. The following proposition gives some properties of the quotient objects and its easy proof will be omitted. (8.5) Proposition. Suppose Gp acts freely on Mcp and Ec p so that the quotients are manifolds. Then (a) Generically Vp\ A?c p -»R will have non-degenerate critical points. (b) IfxeMep is a non-degenerate critical point of Vp, then
913 329

Topology and Mechanics. I

In R 2 -27 we are thinking of R 2 as the energy, angular momentum plane with of course negative energy on the left. We are using the con­ clusion here of (7.2) on the nature of 27. Regions I and III are connected while Region II has a component each in the upper half and lower half planes. The conclusion of (9.1) will be valid for somewhat more general V, depending in principle only on the critical point structure of the Vf. We now give the proof of (9.1). Take p4=0 and let c, =cl{p) be the minimum value of E on J~ l{p) and Cj the only other critical value (see (7.2)). Then for c < c , we clearly have ICfP=9, and for c, c2, Ic , Ic p are as in the Region III case described by (9.1). Fix p as above with c>c2. It will be shown next that Mt,,=S3 x / (we follow the terminology of Section 8). Define SK = {ze M\K(z)= l},and toTzeSKAetR^={PzeMcp\P>0}soRI.c Then clearly Rz e = {f}z\f}e[at
=

UzeM\p2V{z)+-^rZcl.

where 0
C

if c is

bigger than — + m a x Viz). But M. . is the union of the Rz, over zeSK and we may take c as large as we want. This shows that Mt is a normal 1-cell bundle over SK and thus is diffeomorphic to S 3 x /. We have S1 acting on R2 by rotations on (R2)3 by the diagonal action and on M, SK{=S3) by restriction. Call this action of S1 on S c the Hopf action. In fact Sg-^SJS1 = S2 is the well-known Hopf map. Then it follows from the above that Mep is a trivial 1-cell bundle over By (8.4)(a) we can replace £,,, by { i > x e . T ' ( O l H u J ^ c - K,(x)} = Fc. (9.2) Lemma. There is an equivariant map ij/: J~l(0)->M *SK where S acts on the target diagonally. Furthermore the image \jf(Fc) is a cell bundle over Sk. l

Proof of (9.2). Since T{M)=MxM, J~l(0) can be considered naturally as a 3-dimensional linear subspace of M for each x e A f - 0 . Let jx: J^iO)-»M be this inclusion. Denote by Nx the line othogonal to this subspace using the metric K on M and let yx be a unit vector in Nx for each x, continuous in x. On the other hand J - , ( 0 ) is a 3-dimensional vector space bundle over M-0=R+xSK = {(t,z)=tz\OteR+,zeSK}. Then let : J _ I ( 0 ) MxSt be defined by ^(vx)={jx(vx)+{\ogt)yx,z) where x = tz={t, z) and vxeJ~l{Q)cJ~l(0). It is easily seen that ^ is a diffeomorphism. Let geSK Then g acts on M such that g{t,z)=(t,gz), g(Jxl(0))= •/,"'(()), g(yx)=ytx and in fact 4> >s equivariant. It remains to check the last part of (9.2) which is a consequence of the following assertion. There

914 330

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is an equivariant fiber preserving diffeomorphism h: \I/(FC) to D*xSK, (93) Proposition. The group Sl acting on S3 x S3 by the diagonal of the Hopf action has a quotient, S3 x S2. This is a consequence of the following more general proposition which was given me by Bill Browder. (9.4) Proposition. Let S1 act onC(= the complex number) by rotations and diagonally on C, C«+1, C"xS 2 « + l with S 2 « +1 a sphere in C + 1 . Then {C'xS2'*1)^ has the structure of a C bundle over Pf(C) which is the Whitney sum n times y9 where yf is the principal S' bundle over Pf(C). In fact this bundle is associated to y, via the diagonal homomorphism ofS1-»S,x-xS1cl/(n). In (9.3) as a real bundle, (S*x S3)/Sl is S3 x S2 since 2y, is trivial as an SO (4) bundle overS 2 . We obtain (9.1) as follows. From (9.2) together with the statement preceeding it, we have that £ c jis S^quivariantly diffeomorphic to D* x S^ where S1 acts diagonally on the product. Therefore Iep=S3xS3 and Te. is the quotient of the diagonal Hopf action of S1 on S3 xS3. Application of (9.3) thus finishes the proof of (9.1). The methods used for obtaining (9.1) can also be used for the case of general n. We state the results in the following theorem. (9.5) Theorem. In the mechanical system with symmetry of (7.2) with distinct a„ number the regions ofR2 — I, going from left to right by Regions Kj, J ? j , . . . , J?„.

Then (with m = n — 2) for (c,p)eRlt / t > ,=0, 7C>,=0 {c,p)eR2, J ^ - t f x S 4 " - 1 , 7,.,-S 4 "* 1 far 3Zk£n (c,p)eRk, /c>, = S 2 *- 3 xS 4 — " + 5 , and 7C>, is the SAm~2k+3 bundle over P*"2(C) associated to the vector space bundles described in (9.4). References 1. Abraham, R.: Foundations of mechanics. New York: Benjamin 1967. 2. - Robbin, J.: Transversal mappings and flows. New York: Benjamin 1967. 3. Arnold, V. I.: Sur la geometric diflerentielle des groupes de Lie de dimension infmie et ses applications a Phydrodynamique de fluides parfaits. Ann. de Tlnst Fourier, Grenoble XVI, 319-361 (1966). 4. — Avez, A.: Probiemes ergodiques de la mecanique classique. Paris: Hermann 1967. 5. BirkhofT.G.: Dynamical systems. Providence: Amer. Math. Soc 1966 (originally published 1927). 6. Bott, R.: Non-degenerate critical manifolds. Ann. of Math. 60,248-261. (1954). 7. Cartan, E.: Lecons sur les invariants integraux. Paris 1938 (originally written 1921).

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8. Coddington, E., Levinson, N.: Theory of ordinary differential equations. New York: McGraw-Hill 1955. 9. GodbiIlion,C: Geometric differentielle et mecanique analytique. Paris: Hermann 1969. 10. Goldstein, H.: Classical mechanics. Reading, Mass.: Addison-Wesley 1950. 11. Hermann, R.: Differential geometry and the calculus of variations. New York: Academic Press 1968. 12. Kaplan, W.: Topology of the two-body problem. Amer. Math. Monthly 49, 316-323 (1942). 13. Loomis. L, Steinberg. S.: Advanced calculus. Reading, Mass: Addison-Wesley 1968. 14. Loud.W.: Periodic solutions of perturbed second-order autonomous equations. Mem. Amer. Math. Soc. (publisher) and Providence, R.I. 47 (1964). 15. Nomizu, K.: Lie groups and differential geometry. Math. Soc. Japan 1956. 16. Palais, R.: Morse theory on Hilbert manifolds. Topology 2 299T340(1963). 17. Schwartz, J.: Generalizmg the Lusternik-Schnirelman theory of critical points. Comm. Pure AppL Math. 17, 307-315 (1964). 18. Smale, S.: Morse Theory and a non-linear generalization of the Dirichlet problem. Ann. of Math. SO, 382-396 (1964). 19. Struik,D.: Lectures on classical differential geometry. Cambridge, Mass.: AddisonWesley 1950. 20. Wassennan, A.: Equivariant differential topology. Topology, 8,127-150 (1969). 21. Wintner, A.: The analytical foundations of celestical mechanics. Princeton, N.J.: University Press 1941. S. Smale Universita di Pisa Istituto Matematico „Leonida Tonelli" Pisa/Italia (Received July 6, 1970)

23

ln«mcioacsm
916 Inventions math. 11,45-64 (1970) © by Springer-Verlag 1970

Topology and Mechanics. II The Planar n-Body Problem S. SMALE* (Pisa) Introduction Recall the setting of the planar n-body problem of celestial mechanics. One is given n positive numbers, the masses mu ...,m m . Then the con­ figuration space (with center of mass fixed at the origin, once and for all) is the linear space M = {x=(x 1 ,...,x.)€(£ 2 )-|Xm j x i =0}. The phase space is the tangent bundle T(M)=M x M. We have the can­ onical product structure on T(M) because M is a linear space. The kinetic energy K: T(M)-*R is the function K(x,v)=K(v)=^Jdm,\\vi\\2 where (x,v)eMxAf = T(M), D=(P 1 ,...,U B ), £m,t>,=0, a n d II^H i s t h e norm of v{ in£ 2 . Let J u ={(x 1 ,...,x H )eM|x,=x J }, A = \J J u and define the potential energy V: M- J->R by V(xl, ...,x,,)= - £ - j j — '

i

\\xi~

x

■

Here tne

" n i on »

j\\

sum are taken over all pairs («', j) with i+j. Via Newton's equations there is defined the dynamics, given by a vector field XL on the tangent bundle T=T(M—A). Properly speaking the configuration space of the planar n-body problem is M — A and the phase space is T. We let n: T(M)-*M denote the projection (and its various restrictions as well). The total energy E: T-»R, given by E=K+Von or E{x,v)= K(v)+ V(x), is an integral of XL. That is to say E is constant on each orbit. A second integral, angular momentum J: T-»R is defined by J{x,v)—Jx(v)=Ydmixixvl where for example x,xv; can be defined by xj vf—xf v] in terms of Cartesian coordinates on E2. The group of rotations of E2, G = Si acts on M and thus T by the restriction of the diagonal action, leaving invariant A, V, K, E, J, and XL. Let /: T-> R x R be given by I(u)=(E(u), J(u)) for ueT. Note that I is constant on orbits. * We received partial support in the research from NSF contract GP 8007. We would also like to thank L. Motchane for his hospitality at the Institut des Hautes Etudes Scientifiques, Bures-sur-Yvette.

917 S. Smale:

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For (c,p)eR x R let Ic p=I~l(c,p); that is to say ic p consists of all states with angular momentum p and energy c. This space Ic p is invariant under both the dynamics and the symmetry group S1. In fact on the quotient, ICiP = Ic,p/Sl there is an induced vector field. We call these spaces Ic p, fr p, by a slight abuse of notation, the integral manifolds of the planar n-body problem. The goal of this paper is to make a global or topological study of the map I: T-* R x R. This will have two main aspects; first, the investigation of the topological type of the integral manifolds and second, finding the structure of the bifurcation set of /, to be described shortly. Our first theorem on the integral manifolds is the following. Theorem A. In the planar n-body problem for any given choice of masses, suppose p=0. Then (a) for each c^O, there are diffeomorphisms
Ic^S(E(SK-A))xR

and
le,p^E(SK-A)

and q>c: lc_p^E(PH_2{C)-A). All of these maps are smooth in c. Some explanations and remarks are in order. Here SK is the unit sphere in M with the K inner product on M. That is to say SK = {zeM\K(z)=l). Next SK — A is SK with AnSK removed. E(SK) is a Riemannian vector space bundle over SK and E(SK- A) is the part over SK-A. This bundle will be described in Section 3. If £ is a Riemannian vector space bundle, then S(E) denotes the associated unit sphere bundle. £(JJ_2(C)) is a Riemannian vector space bundle over complex projective space JJ_2(C)=SK|Sl and will also be described in Section 3. £{P„_2(C)-A) is the part of this bundle over iJ_2(C) with certain projective subspaces corresponding to A removed. Finally since the Ie , Ic are naturally sub-manifolds of T for the various values of c, one can define smoothness in c simply by saying that the induced map on the subset of R x T is smooth. In (a) of Theorem A, the bundles are clearly trivializable, thus we have the following corollary. Corollary. In the planar n-body problem for zero angular momentum, if the energy c^O, then the integral manifold Ic is diffeomorphic to

918 Topology and Mechanics. II

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S 2 "" 4 x(S K —J)xR. / / the energy c is negative, and p=0, then Ic p is diffeomorphic toR2*-*x(SK-A). In this paper S* is the ^-sphere and R* Cartesian q-space. Our next theorem on the integral manifolds is: Theorem B. In the planar n-body problem for any given choice of the masses, suppose c^O and p+0. Then there exist diffeomorphisms smooth and e>f.

1C,P^E(P._2(Q-A).

with these spaces defined just as in part (b) of Theorem A. To understand our results on the integral manifolds and bifurcation in the negative energy half plane, p+0, we define relative equilibria. Given (mlt..., mj, a position, x=(x t ,..., x j e M - A of the n-bodies is said to be a relative equilibrium and xeRe if there is a 1-parameter group of rotations of E2 which induces a motion of the xt, 3, the nature of relative equilibrium is not very well understood, and there are many unanswered questions in this area.

919 48

S. Smale:

As far as the history of these things, Wintner [5] gives the background to the relative equilibria, although in the slightly more general context of central configurations. Also in general this is a good reference to the n-body problem from a very classical point of view. The following theorem could be considered as implicit in the classical literature (see [5]). In our developement, it comes out as a corollary to a theorem on "rotating coordinate systems" (see (1.1) below). We use this Theorem C in the proof of Theorem D below. Theorem C. For any given choice of the masses in the (planar) n-body problem let SK = {zeM|K(z) = 1} and Vs: SK-A-*R be the restriction of the potential energy to SK with A removed. Let zeM — A be normalized so that K(z) = 1. Then z is a relative equilibrium if and only if z is a critical point of Vs. We proceed now to the bifurcation question. If / : P-*Q is a smooth map from a manifold P to a manifold Q, then the bifurcation set of f, Z(f) is the set of points yeQ over which / fails to be locally trivial. That is to say y$Z{f) if and only there is an open set U containing y and a map g: f~i{U)-*f~1(y) such that h: f~i{U)->f~l(y)xU is a diffeomorphism where h{x)=(g(x),f(x)). The set of critical values off (i.e. the points y, where f(x)=y for some x with the derivative Df(x): Px-*Qy not surjective) will be denoted by Z'(f). Clearly Z'(f) is a subset of Z(f). Write 1=I (I) and Z' = Z'(I). Theorem D. Given the masses of the planar n-body problem (a) The set Z' of critical values of I: T-» R x R is given by Z' = {J^R. Za where Zs={(-V(pz)2,P)eRxR\p±0,ze<x,K(z)=l}. (b) The bifurcation set Z of I is precisely the union of Z' with the 2 coordinate axes o/RxR. Note that V(p z) = — V(z) so that for each a, Z, is a cubic curve in P R x R. Note also that Re may be a finite set (see above) for every choice of the masses. Finally we remark that V(pz) is constant on ze<x with K(z)=l. I would expect that genetically (i.e. for almost all values of (m,,..., m„), the Zt corresponding to the collinear relative equilibria to be distinct from each other and from the other relative equilibria. We pass now to the integral manifolds for negative energy and non­ zero angular momentum. If Q is a manifold with boundary dQ, a new smooth manifold pk(Q) is obtained by identifying for each qedQ the

920 Topology and Mechanics. II

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subset S* ~i x q to a point in Sk ~' x Q. More generally if £ is a Riemannian vector space bundle over a manifold with boundary, /?(£) will denote the smooth manifold obtained by taking the associated unit sphere bundle and identifying to a point each fiber over the boundary. Details of these things are given in Section 5. Let lp: SK-A -*■ R be the function defined by lp(z)= - V(pz)2, V being the potential energy. Let Qcp=lp ' ( - o o . c]. If c is a regular value of lp then Qc p is a manifold with boundary and ffk{Qei p) is a smooth manifold. But for every c, Pk(Qe_ p) can be defined as a topological space in the same way, l~l(c) playing the role of the boundary. One can also define & , . to be the subspace of Pn_2(C)—A defined by Qc p=?"'(—oo, c] where Tp Pn_ 2 (C)—A -* R is induced on the quotient by /,.' With the above and using the notation of Theorem A we have Theorem E. For any given choice of the masses in the planar n-body problem, with c<0 and p+0, lc p has the structure of Pk(QCiP) and Ic p is f}(£0(Qe p)) where k = 2n — 2, and Ea is the bundle described in (9.4) of [4]. Furthermore there is a function y(p)<0/or p#0 (defined by the cubic It farthest to the right in R x RJ such that for y(p)
921 S. Smale:

50

{(c,p)eRxR\cp2 distinct.

= di}di<0,

i = l , 2 , 3 , 4 with the dt not necessarily

(b) The bifurcation set I of 1 consists of I' together with the coordinate axes. (c) For p=0, c < 0 or for p4=0, c^O, Ic p = {i.e. is diffeomorphic to) R3 x (S 3 - A) and Icp=R3 x (S2 - 3 pts.). Note that S3-A = S1x(S2-3 pts.). (d) For p=0, c^O, Ic p is diffeomorphic to S 2 x R x ( S 3 - 4 ) and /c„ = S2xRx(S2-3pts.). ' (e) For each p#=0, there is c,(p)<0 (in fact a cubic) such that for cR has 3 non-degenerate saddle points (the Euler points) and 2 non-degenerate maximums (the Lagrange points). With this hypothesis and our above theorems it is an easy exercise to write down the topology of the integral manifolds for the remaining cases. We leave this to the reader. Birkhoff at the end of his book "Dynamical Systems" [2] discusses these questions here for the problem in 3-space. He restricts his discussion to n = 3, negative energy, non-zero angular momentum and the result he states (in our language) is that Ic p changes just at the Lagrange and Euler relative equilibria (without studying the structure of Z or T themselves). He gives some of the argument for this without attempting to make a complete proof. Birkhoff also raises the question as to what is the topology of the integral manifolds (for the 3-body problem in £ 3 where he relates his question to regularization). Wintner [5] raises a number of similar questions. For background material of this paper, [4] is recommended. Also motivation is given in [4] for the study of the bifurcation set. Smooth means C°°, and if \pc: X-*Y depends on cett, then smooth in c means that the induced map C x X -> Y is smooth.

922 Topology and Mechanics. II

51

Section 1 The goal of this section is to give a proof of the following theorem, as well as background generalities on mechanical systems. (1.1) Theorem. Suppose (M, K, V, G) is a mechanical system with sym­ metry in the sense of (4.1) of [4] so that M is configuration space, K kinetic energy, V potential energy and the Lie group G acts on M leaving K and V invariant. Let Xe (5, the Lie algebra of G. Then X can also be considered as a vector field on M generating local solution curves, say tl/,(x),xeM. Denote by
923 52

S.Smale:

development in a more general context, or say [5] from a classical point of view. One has a similar situation from the Hamiltonian point of view. In this case configuration space M is given together with a function H: T*->R on the cotangent space T* = T*{M) which satisfies the analogous condition to (1.2). (1.2') Condition. H on each fiber of T* is a polynomial of degree two and in the 2nd order terms is positive definite. Again as a consequence of (1.2'), H can be written as a sum of its homogeneous parts H = K1+fx+Vloit where Kt is given by a Riemannian metric on T* -»M, X is a tangent vector field on M, fx(wx)={wx, X(x)), wxeT* and K, is a function on M. Via the canonical symplectic form on T*, H defines a vector field XH on T* which gives the dynamics of the system. In this case H itself is an integral. Note the following aspect which can lead to much confusion if one is not careful. In the Lagrangian point of view, the integral E is independent of the linear terms in L, while in the Hamiltonian point of view, the integral depends directly on the linear terms of H. Again one can see [1] for a developement in a more general context. Now one can pass from the Lagrangian point of view to the Hamil­ tonian (and back again) by a map * L : T<-> T* called the Legendre transformation. One can compute directly the 4>L in this case using [1], [5] for background, obtaining
824 Topology and Mechanics. II

53

Proof. Apply the above formula for L(vx)(vx)=2K (vx) + F(vx).

this context is defined simply by

For the proof of (1.1) we will also need (1.4) Proposition. Given L, H as in (1.3), the singular points of the equations of motion XL on T or XH on T* are precisely of the form on T, the 0-vector at x, 0x where the derivative, DV(x)=0; and on T*, R x r * be defined by \l/(t,w) = (t, \l/,(w)). Strictly speaking since ip, may be only defined for small t, \f/ has as domain some open set of R x T* containing 0 x T*. Following the notation of [1] let ^ be the map on vector fields of R x T* defined by Or1)*(1.5) Proposition. Let X be a vector field on M, fx: RxT*-»R defined by fx(t, rix)=(t]x, X(x)) and ^ defined using X as in the previous paragraph. Suppose H: Rx T"*->R is some function. Then
925 54

S. Smale:

Proof of (1.6). We use (1.5), with K = H' and the function H,K of (1.5) defined from the H, H' of (1.6) by making them constant in t. So from (1.5) one obtains ^ XH = XH,, or XH = \ji* XH.. Let q>, be the local 1-parameter group generated by XH, , (w)=iff, (w) and we obtain the conclusion of (1.6). Now we prove(l.l). Let // = £<>tf>->where L = K-VonE = K + V°n. Thus H = ^R + Von by (1.3). Let H' = H+fx, X the vector field on M as in (1.1). By (1.6) we are looking for the zeros o{XH.. Using (1.3) again, the L corresponding to H' is given by L = K-2X-(V-K(X,X)) and the corresponding integral of Jacobi, E1 = K+V-K(X,X). Then (1.1) follows from (1.4). Section 2

The goal of this section is to give the proofs of Theorem C and part (a) of Theorem D. We use the function Vp of Section 6 of [4] and we refer the reader to this reference for a background in what follows. Recall from this reference that given a mechanical system with sym­ metry (M0,K, V, G), there is associated a function Vp: M0 - A -> R where in our case, A = , M0 = M — A, angular momentum p takes values in R p2 and simply Vpv(x)= V(x)+ v ' ' 4K(x) Write SK = {zeM\K(z)=l} so that M - 0 = R + x S K , the diffeomorphism Af-0-»R + xSK being given by x-> I l|x||K, V

1. IWIIL '

Under this same map M — A is carried diffeomorphically onto R + x (SK- A). Let Vs: SK-A ->R be the restriction of V: M-A -»R. Let a(f) denote the set of critical points of a map / (2.1) Proposition. (a) <x(Kp)= |(t, z)eM - A \zea(Vs), t =

^~^\-

(b) a(V- \\X\\2K)=|(t, z)eM - A \ze
2\\x(z)\\2Ky

The function in (b) is that from M — A -> R of (1.1) where G = S' and X can be characterized by a real number. Note that in (a) the case p=0 is included and so there are no critical points of Vp in this case. Proof of (2.1). By calculus (t, z) is a critical point of Vp when the two partial derivatives are zero or D, Vp(t, z)=0 and Dz Vp(t, z)=0. Now nv,t

\ n lV^zK

P2 \

-V&

P2

926 Topology and Mechanics. II

Setting this to 0 gives us t =

55

. On the other hand

DzVp(t,z)=

2 V(z)

j

— Dz V(z) and the first half of (2.1) foliows. For the second half first note that D,{V(t, z))-K(X(t, z), X(u z)) = ^^—2t

K(X(z),X(z))=0

-Viz) and since t>0 one obtains that t = 1 / ^ „ ^,/~K2'u2 • The rest of (2.1) follows 2\\X(z)\\ K from the fact that the map M-*M defined by x -»X{x) is an isometry for the K metric, times a constant factor. From the second half of (2.1), using (1.1) we obtain immediately this corollary. (2.2) Corollary. Let zeM be normalized so that K(z)= 1. Then z is a relqtive equilibrium if and only if z is a critical point of the potential energy Vs: SK — A -* R. Here as elsewhere K becomes naturally a function on M itself, since M is a linear space, and K is flat (see Section 4 of [4]). This (2.2) is Theo­ rem C of the Introduction. Using the first part of (2.1) we obtain (23) Proposition. Z'(V)= {-

V(pzf\zea(Vs)\.

Pr 0 0 /r(K p )=|K p (tz)|z€(7(K s ),t = p2 At2K(z)

T

^ - j

zeo(Vs),t=-

P

2V(z)

V(zf zecj(V )}. s 2 P

The following proposition is easily checked noting that J: T-+R has no critical points. (2.4) Proposition. ueT is a critical point of I:T-*RxRif and only if J(u)=p and DEp{u)=0 where Ep: J -1 (j>)-» R is the restriction of E: T-* R. Furthermore (c, p)el'(I) if and only if ceI'(Ep). Now it was shown in Section 6 of [4] that I'{Ep)=Z'(Vr). this together with (2.2), (2.3) and (2.4) we obtain:

Putting

{IS) Proposition. In the planar n-body problem, Z'(I) is the set of (c, p ) e R x R with c= -V(pz)2, where z is a relative equilibrium with K(z)=l. This is part (a) of Theorem D of the Introduction.

927 S. Smale:

56

Section 3 The goal of this section is to give a proof of Theorem A. Wefirstgive a description of the bundles involved. Here E=E(SK) is the vector space bundle over SK with fiber Ez for zeSK defined by Ez=J~ '(0). The union of the Ez over zeSK has the struc­ ture of a vector space bundle. For each z, the inner product K on M induces an inner product on £ z c M and thus E has the structure of a Riemannian vector space bundle. The group S' acts naturally on £ as a group of bundle automorphisms preserving K, and the quotient is a Riemannian vector space bundle over Pn_2(C) which we denote by £ or E(P„_2(Cj). As in the introduction we denote the restrictions of these bundles to SK-A, PK_2(C)-A respectively by E(SK-A), £(Pn_2(C)-A). Now let c ^ 0, p=0. Using the identification M - A with R + x (SK - A), we are able to write Ic0={(t,z,v)e(M-A)xM\v6Jz-i(0),K(v)=c-V(tz)}. Note here that J,;1(0)=./Z"l(0), \\vf2 = K(v\ and c-V(tz)>0.

Define a

map q>c: / r 0 -► S(E(SK - A)) x R + by q>c(t, z, v)= /—— e£ z , f 1. It is seen immediately that q>c is a diffeomorphism depending smoothly on c. The group S' acts on /c 0 naturally, and on S(E(SK—A)) x R + by the natural action on thefirstfactor, trivial action on the second. The diffeomorphism R be defined by t,(c, z)= V(cz) so that r,(c, z) is characterized by V(tt(c, z) z) = c. Then we may write 7 c0 ={(f,z,t')|0t(t,z,v) = vsE,. The map
Section 4 The goal of this section is to prove Theorem B of the Introduction. Thus we consider c^O, p + 0. Define r0: R + x(S*-d)x(R-0)->R so that Vp{t0(c,z,p),z)=c; i.e. let t0(c, z,p) be the unique positive solution of the quadratic equation to

4(g

4

2c

928 Topology and Mechanics. II

57

if c>0 and fn =

if c=0. Note that this defines a smooth function 4 V(z) of (c,z,p) and that here R + = [0,oo) contrary to our previous usage R + = (0, oo). We hope this last causes no confusion. It is easily seen that we may write Icp={(t,z,v)e(M-A)xM\t0(c,z,p)^t
by


(4.1) Lemma. K(k(v-txp(tz)) = k2(c- Vp(tz)). Proof. Since K is homogeneous in k of degree 2, we only note that K(v-a,{tz)) = K(v)-K(a,{tz))=K(v)-V,(t2)

= c-V,(tz).

(Compare Section 6 of [4].) Using (4.1) and the properties of k one can now check that
929

58

S. Smale:

The following proposition gives an alternate description of a(£) and shows that as a differentiable manifold a(£) is well-defined. (5.1) Proposition. Let E be a vector space bundle equipped with a Riemannian metric over the manifold M, dM = 0, and f:M-*Ra smooth function with regular value ceR. Let g: £->R be the function g(vx) = \\vx\\2+f(x) for vxeEx, the fiber over x. Then c is a regular value of g and so g~'(— oo, c] = X{E,f c) is a differentiable manifold with boundary. Furthermore if E, £' are two bundles over M as above with c a regular value for two functions ff with / ~ ' ( - o o , c] =f1~1(-oo.c] and £, £' are isomorphic as bundles over Mc (the isomorphism not necessarily pre­ serving the metrics), then X(E,f c) is diffeomorphic to X(E',f,c). Here Mc = {xeM\f(x)^c} and Ec is £ restricted to Mc. One sees that X(E,fc) is an alternate description of a (£,.). The proof of (5.1) is straightforward. Remark. One may give a little more general proposition by considering in addition to g: £ -»R of the form g(vx) = || vx \\2 +f(x), a smooth function say h: £->R with this property: The restriction of h to each fiber Ex is proper and has only a single critical point, a non-degenerate minimum at the origin and k: M -* R defined by k(x)=min h(x) is smooth. Then i>e£ x

if c is a regular value of h, h~'(— oo, c] = a(£ r ). If M is a manifold perhaps with boundary, <xk(M) will denote the differentiable manifold a(£) where £ is the trivial bundle Rk x M over M. This is well-defined by proposition (5.1). If £ is a vector space bundle let da (£)=/?(£) and if M is a manifold with boundary let d<xk(M)=Pk(M). It is easily seen that a(£) has as defor­ mation retract M and we have the following proposition. (5.2) Proposition. For any manifold with boundary M and any k > 0,j ^ 0, Hi{)k(M)) = Hi(M) + Hi+1-k(M,dM). This proposition was given me by Browder but since we don't use it, we omit the (easy) proof. (5.3) Examples. (a) IfdM = 0 , & ( M ) = S * - ' x M . (b) 0,(M) = the double of M. (c) pk(D')=Sk+l-\ (d) IfSA# 1 =p,j8 t (M 1 xM 2 )=M,x^(M 2 ). (e) Example from mechanics. Let (M, K, V) be a simple mechanical system so that K is a Riemannian metric on M, dM = 0, and V: M -* R is a function. Let c be a regular value of V, Mc = {xeM\ V(x)^c) and Tc the tangent bundle of M restricted to Mc. Then the energy level E~x(c) isf}(Tc).

930 Topology and Mechanics. II

59

Section 6 The goal of this section is to prove Theorem E. Thus we suppose always here that c < 0, p+0. Recalling that T=(M-A)x M = R + x(SK-A)xM, one can write ICiP={(t,z,v)eT\veJ-1(p),K(v)

= c-V(tz)}.

Now let t~(c,p, z), t+(c,p,z) be the two real valued positive smooth functions defined on the domain in (— oc,0)x(R—0)x(SK-/4) of (c, p, z) with zeQcp, characterized by Vp(r{c,p,z\z) = c,

Vp(t+(c,p,z),z)=c,

t-(c,p,z)^t+(c,p,z).

One can take these functions as solutions of i^ t

+

- £ - = c or 4t 2

cr'-K(z)t--£=0 4

or _ -V(z)±\/V(z)2 + cpz *~ 2c + Note that t~, t coincide on V(z)= - ] / — cp2 which is the boundary of Qc p and for V{z)> -\/~cp2, t~, t+ fail to be defined. In fact one sees - V(z)2 that the domain of definition for t , t+ is just as stated, i.e., 5 —^c p or / p (z)gcor zeQ c p . Recall from Section 4 the smooth tangent vector field ap defined on M-A, characterized by txp(x)eJ~l(p), and ap(x) is normal to ^"'(O) in Tx in the K inner product. We may now alternately use the expression for lc given by the following: (6.1) Lemma. Icp={(t,z,v)eT\veJ-i(piK(v)=c-V(tz), zeQcpt-(c,p,z)^t^t

+

(c,p,z)}.

Proof. If K (v)=c - V{t z) then K{v-zp(tz))=K(v)-K(<xpUz))

= c-V(tz)-K(<xp(tz)) = c-Vp(tz)2:0

so that Vp(tz)^c. From the discussion preceeding the lemma it follows that zeQ c p and t~ ^t^t+. This proves the lemma. A look at the graph of Vp(t z) for fixed z, p as a function of t is helpful in understanding the previous part of this section and in fact the whole paper. For example lp(z) is the lowest point on this graph.

931 S. Smale:

60

Noting that J~l (0)=J~' (0), let I^p={{t,z,u)eT\t-^t^t+,ueJ-1(0),K(u)=c-Vp(tz)}. Then from the previous considerations it follows that the map fie p: Icp->I'cp defined f}cp(t,z,v) = (t,z,v-txp(tz)=u) is a diffeomorphism between / and I'e which is smooth in c, p. Our goal is to represent I'c p as ftk(Q.cp). Let E be the bundle over SK we have seen in the Introduction and Section 3, and let f be the trivial line bundle over SK. By thinking of the fiber £2 at zeSK as the normal to Ez in M, one sees that the Whitney sum E + £ is a trivalizable vector space bundle over SK. Using the notation of Section 5 and counting dimensions, it is then sufficient to represent l'c p as /?((£+Z){QCtP)) where of course (E + £)(QC p) is the restriction of the bundle £ +
value of Vp(tz) as t ranges over R + . We construct a diffeomorphism gp: R + x(S K -0)->^ which is a diffeomorphism "on each fiber" gp 2: R + x z -»<J2 taking xp(z) into 0. Then let fp: (E + £)-► R be the function defined on the fiber over z b y /,.,(«,8*.W) = K(u)+K,(tz). We note that /,-'(c) = /; p is a conse­ quence of the construction of I'c and the map /?c p . Furthermore lp(z)= min/ pz (u,g pz (:)) is a smooth function on SK — A and fp satisfies the remarks after (5.1). Therefore / c p = /; , = & ( / ; ' ( - oo, c]) = ^k(Gc,„). The results about Ic p follow as well. It is easily seen that lp is proper and has a maximum value y(p)<0 onSK — A. Then for y (p) < c < 0, Qc p = SK - A and so Ic p is Sk'l x (SK - A). By Theorem D, y(p) is a cubic. This finishes the proof of Theorem E. Section 7 We give the proof of part (b) of Theorem D. From the definitions, this is a direct consequence of the following. (7.1) Proposition. For any given choice of masses in the planar n-body problem: (a) The bifurcation set I of I contains the coordinate axes of the energy-angular momentum plane.

932 Topology and Mechanics. II

61

(b) Over the set {(c,p)sRxR|c>0, p*0} the map I: T-»RxR has a global differentiable product structure. (c) Let (c 0 ,p 0 )eRxR be such that c 0 <0, p 0 +0, and {c0,p0)$Z'. Then there exists a neighborhood U of (c0, p0) in R x R and a diffeomorphism lnPo for each (c,p)eU, smooth in c,p. Proof. By Theorem E, there is a region Q in R x R, c < 0, with boundary including the set c=0 and the set c<0, p=0 such that for {c,p)eQ, Icp=S2"-ix(SK-A). By Theorem A, Corollary, if c<0, p=0, le p = Sin-* x (SK-A)x R. By Theorem B, if c^O, p*0, / c p = K a " - 3 x(SK-A). Putting these together we see that even the betti numbers of the lc p change at the coordinate axes, so that surely these coordinate axes are in I. This proves (7.1) (a). Furthermore (7.1)(b) follows directly from Theorem B. Thus it only remains to prove (c). For this we proceed as follows. Recall that Qc-P = lp '(-oo,c] where lp: SK-A-*R is defined by - Viz)2 a n c t n a t ls z lp( )= 2—» * lp proper with i ts critical values associated to I as in Section 2. Thus Qcp={zeSK - A \ K ( z ) g / - c p 2 } . It follows easily that one may choose a neighborhood U of (c0, p0) and construct diffeomorphisms acp: QCiP-*Qeo,po smooth in {c,p)eU. We will need such a family of diffeomorphisms with an additional growth property near dQc p. Denote by Y the bundle E + <J over SK - A withfiberover zeSK — A, Yz and the restriction of Y to Qc p by Ycp. Recall the function fp: Y-*R defined in Section 6 with min f(y)=I Jz) and satisfying the remark after (5.1). Thus on each Yz,fp has the topological structure of a norm in a finite dimensional Hilbert space. One may furthermore write rcJz)=Y2nf-l(c)

and

/ ; „ = U /;,,(z).

We will define diffeomorphisms yc p: Yc p-+ YCo po coveringCTCp : QeiP-* QCOiPO which will have the property that besides being smooth in (c,p), they induce by restriction diffeomorphisms 'C.P^^CO.PO- This will be sufficient to finish the proof of (7.1). One can immediately obtain such a family of homeomorphisms ycp by simply asking that on each Yz, yc p: Yz-> Y„cp(z) maps the level surface (a sphere) f~' (c) into f^' (c0). In fact this construction can be easily done so that it meets all smoothness conditions except for dif­ ferentiability at 0 in Y2 when zedQc p. Furthermore if one modifys the diffeomorphisms
933 S. Smale:

62

Appendix on Moulton's Theorem Here we give a proof of the following theorem of Moulton [3], a proof using critical point theory which fits into the methods of this paper. Theorem. For any given choice of masses in the n-body problem, there are exactly n !/2 classes of collinear relative equilibria; that is to say, there are n!/2 classes of relative equilibria x = (x,, ...,x„) where the x ; all belong to the same straight line through the origin. For the proof we use the notations, definitions, and Theorems of this paper, especially from the Introduction. Choose some line £ 1 c£ 2 .ThisdefinesasubsetM r cMofx = (x,,...,x II ) such that each xteEl. Let Sr = SKr\M„ Sr-A = S,-(Srr\A). Now while S' acts on SK, only the rotation by n radians leaves S r invariant. Thus Z 2 the group of order 2 acts on Sr and on the quotient we have (P,_ 2 (R)- A)<=P„_2(C)-A -^R where P„_ 2 (R)=S r /Z 2 is real projective space, contained naturally in P n _ 2 (C). Here V is induced by the potential energy. From Theorem C it follows immediately that the number of classes of relative equilibria is the number of critical points of V: P„_ 2(C)—A -» R (the normalized relative equilibria themselves corresponding to critical points of V: SK — R). Now if a relative equilibrium (x1,...,x„) is collinear, then one may bring all the xt into El by a single rotation of E2. From these considerations we obtain: Proposition 1. The classes of collinear relative equilibria are in a bijective correspondence with the critical points of V: P B _ 2 (C)—A-*R which lie in Pn_2(R)~ AcP„_2(C)- A. Next we reduce the problem to simply the study of the function V:PH_2(R)-A^Rby: Proposition 2.1f xePn_2(R)- A isacritical point of V: P„_2(R)- A->R then x is also a critical point of V: P„_ 2 (C) — A -»R. For the proof of this and also later we will need to know the derivatives of the potential function. Discussions with Shub were very helpful with this next proposition. Proposition 3. For given masses m=(m,,..., v

( X )= - Y

mn),

mm

' t

(a) The first derivative of Vm: M — A-*R is given by

934 Topology and Mechanics. II

63

for veM, and the 2nd derivative D2VJx)(v,w) = Qx,m(v,w)= _ £ _ " V " i

x,-x,.|| 3

• ( | | X _ X | | 2 (*i ~ Xj, V, ~ V}){xi -

Xj,

Wt - Wj) - (», - VjHWi - Wj)j

with, v,weM. (b) The second derivative of the restriction Vm: SK — A -* R is: D2 VJ{SK-A)(x)(v,

W)=Qx,Jv,

w)+ VJx) KJv, w).

Here ( , ) denotes the inner product on E2, \\ || the norm on E2, Km the kinetic energy as an inner product on M for the given masses m = (TOJ ,..., mm). The same formulae are valid for £ l or E3 instead of E2. Proof of Proposition 3. For (a) one can compute on (E2)" (and restrict). Then DVm{x)(v)= t DkVJx)(vk)= - X I A ( " ' " ' . ) (vk) and for each i,j y

M.M; Uxt-xJl){V,)-

D

k

£i

(

\(n.(xi-Xj,V,-Vj)

\\xt-xjf

■

These together yield the first part of (a). For the second part of (b), one proceeds in the same way using D2Vm(x)(v, W ) = I D 4 D , K 1 1 ( X ) ( P „ w,). For (b) one uses the formula D2VJSK-A(x)(v,

w) = D2VJx)(i{v), i(w)) + DVJx)(D2 i(v, w))

where i: SK-* M is the inclusion. A standard calculation finishes then the proof of Proposition 3. We now give the proof of Proposition 2. For vteE2, let t?,=(t?J, v") where rJeE1 and uj'efE1)1 in E2. Then we can write v=(v',v") with v'( = v[,...,v'K) etc. for each veM. Now if xeSrczSK, x$A, TX(SK)= {veM\v±x} and Tx{Sr)={v'eMr\v'±x} where on M as always we use the K inner product. If veTx{SK) and v=(v',v") then v'eTx(Sr) since K(v,x)=K(v',x). By Proposition 3, if xeSr-A, veTx(SK), one sees that DVm(x)(v) = DVJx)(v'). Then DVJx)(v') = 0 implies DV(x)(v)=0. This yields Pro­ position 2.

935

64

S. Smaie: Topology and Mechanics. II

Proposition 4. PB _ 2 (R) — A has n !/2 components. Proof. Let x = (x,,..., xn)e S„ — A, and associated to x, let (i,, i2,..., i„) be an ordering of (1,2,3,..., n) defined by i, < ik if and only if X; < xik (note that the x, are all distinct). With this correspondence, one sees that Sr-A has n! components and that the quotient P„_2{R)-A has n!/2 components. Proposition 5. IfxeP„_2{R)-A is a critical point of Vn:P„_2(R)- A ->R, then x is a non-degenerate maximum. Proof. Apply part (b) of Proposition 3 to see that for V: Sr—A -* R, D2VJ(Sr — A){x) is a negative definite form in this case. The proposition follows. We now obtain the proof of Moulton's Theorem itself. Toward A the function V on Pn _ 2 (R) — A tends to — oo and so has a maximum on each component of P„_2(R) — A. But elementary Morse theory excludes the possibility of 2 critical points on a single component, since each is a non-degenerate maximum. Thus one counts nl/2 critical points and hence n !/2 collinear relative equilibria. References 1. Abraham, R.: Foundations of mechanics. New York: Benjamin 1967. 2. BirkhofT, G.: Dynamical systems. Providence: 1966 (originally published 1927). 3. Moulton, F. R.: The straight line solutions of the problem of N bodies. Annals of Math. Vol.12, 1-17(1910). 4. Smale, S.: Topology and mechanics I. To appear. 5. Wintner, A.: The analytical foundations of celestial mechanics. Princeton, N. J.: Prince­ ton University Press 1941. S. Smale University di Pisa Istituto Matematico " Leonida Tonelli" Pisa/Italia (Received July 6,1970)

936

PROBLEMS ON TEE NATURE OF RELATIVE EQUILIBRIA IN CELESTIAL MECHANICS S. Smale

Hara va taka an essentially very old problem and discuss it in the framework of modern topology ; in this way some new questions on the subject of ralatira equilibria can be explicated. For a classical background, aee the sections on central configurations in Wintner [2]. For a modern background one can sea [1] and in fact this reference details much of what is said here. Some of these questions were worked out with M. Shub. Since general questions of relatire equilibria reduce to planar questions, we start by recalling the planar n - body problem.

Here, glren the masses, 9 1*

B m., 1' ...,m_ *••' *n > 0, one has a "configuration spaoa", the subset of defined by 11 - { x - ( x 1 , . . . , x n ) € (I**)11 j 1 . ^ - 0 }

(E )

2 Here E i s the Euclidean plane and we suppose that the center of mass i s fixed at the o r i g i n . Then the tangent bundle T(M) - M x « - { ( x , r ) I l i j i j - 0 , I m ^ - OJ. And the kinetic energy K I T ( H ) - R is giren by K(x,r) - K(T) - iZm± H T J I * where || || is the norm on E . Thus K can also be considered as an inner product on M itself. The potential energy is the function T > II-A -» R glren by

v(x) - - z - xa =x L ll i- jll

and the diagonal A

is the union

where A. . » \x € M x, - x A . i v i<j J 13 i d ' Then ria Newton's equations, K and T define the dynamics of this mechanic­ al system as for example a rector field on T - T(M - &) . The group of rotations S0(2) - S acting on E , Induces an action also on (E 2 ) n , M and T(M) , learing inrariant K , V and A .

937 195

A point is a

x € M-A, is said to be a relative equilibrium or 2

1 - parameter group

if. of rotations of

E

variant by

S

equilibria.

acting on

M-A

H - R /_1 _+ e e/ o ,«

The quotient

if there "

such that

(*+(*.,)» •••> *+(*• )) satisfies Xewton'a equations. 1

x 6 R

Hote that

R

is in-

and also by non - sero scalar Multiplication.

will be called the set of classes of relative ——™—

We are interested in the nature of

R

for a given choice of

naBses. For the case

n - 2 , it is not difficult to see that

For the case

n ■ 3

fact that

R

R

has one element.

and any given choice of the masses it is a clasaioal

has 5 elements.

Three of these, found by Euler, are collinear,

that is to say that the corresponding

x.

lie on a straight line.

The other two, found by Lagrange, have the

x,

at the corners of an equi­

lateral triangle and differ from each other by a permutation of two of the vertices. Also, for any n and choice of the m. , there are precisely n' -r-

classes of relative equilibria which are collinear.

of Moulton's Theorem. the structure of Is

R

This is the content

Beyond these things there are no general theorems on

R . Perhaps the outstanding question on this subject is i

always a finite set ? It seems to me that the basic structure of

R

is essentially unresolved and to understand this structure is a beautiful problem. Besides the fact that the study of relative equilibria is an old and classical question, there are direct physical implications. For example there are bits of matter (the "Trojans") in the sky that seem to correspond to the Lagrange relative equilibria.

Furthermore, there is a

direot relationship between the relative equilibria and the bifurcation of the integral manifolds in the planar precisely

let

I i T - RxR

is the total energy and the trajectories in to know the set

Z'

T

J

l(w) - (E(w),J(w)) where

is angular momentum.

Then

I

E

is constant on

defined by Newton's equation and it is of interest

of critical values of

the union of cubic curves in an element of

n - body problem (cf. [l]). More

be defined by

RxR

I.

It is a fact that

Z'

is

each of which corresponds precisely to

R . e

Our starting point for considering the relative equilibria is the following theorem. Theorem =

Let

S R - {x e M | K(x) - 1} .

equilibrium if and only if potential energy

x

Then

x € SR-A

is a relative

is a critical point of the restriction of the

V t S^ -A — • R .

938 196

The symmetry group

S

acts on

Sv

leaving A

and

T

invariant.

quotient S „ / S can be seen to be complex projective space /^ & & - A / S 1 the union of complex projective subspaces. Let

V : P -(C) - 6 -» R n—e.

be the map induced from

Conjecture » For almost all choices of

P (l) n—^

and

V.

(m , ..., m ) , the corresponding *\, t

Just defined is a Morse function (i.e.

The

T

*\* 7

n

has non - degenerate critical

points). It is even conceivable that for every choice of

(m,, . . . , « ) , i

V

is a

n

Morse function although sometimes I feel that the classical literature may contain a counter-example. For the implications of the conjecture, the following two propositions are very relevant. Proposition 1 t

7 t Pn2(l) -A

-• R

is a proper map.

This is a direct consequence of the definition of translating the picture to Proposition 2

(M. Shub) :

Pj^gW

(supposing given

in

(See Appendix.)

N.

T,

its behaviour near A ,

7. There exists a neighborhood m^, ..., m

) such that

7

N

of A

in

has no critical points

From propositions 1 and 2 and the fact that a non - degenerate point is iso­ lated, we have t First implication of conjecture : almost all values of

For

(«., ..., m ) , there are only a finite number of

classes of relative equilibria. Second Implication of conjecture » For almost all values of one may proceed as follows. equilibrium

x

Fixing such an

one may attach a non - negative integer

of the corresponding critical point of

7.

(m., ..., m )

(m., .... m ) , to each relative i n lnd (x) ,

the index

Then the numbers of relative

equilibria of given index are related by the corresponding Morse inequalities. In particular the rich homology of

P

(S) - A

gives a strong existence

theorem for the relative equilibria. Although the homology of

P _ 2 (l) -Z.

is rather complicated, I believe that

finding this homology is not a very deep problem. to me that

F n _ 2 (E) - £

A. Andreotti pointed out

is a Stein manifold and thus

^ ( P ^ J t ) -A ) - 0

939

197

for

i < n-2.

index

< n-2.

One night ask if also there are no critical points of In looking at the homo logy of

P

.(it) - A ,

V

of

I encountered the

partition function, which seems to be related to the Betti numbers. By choosing a line P

? (B)

E

in

E , a real protective subspace

is determined by asking that the

x

€ E .

P__.(R)

colllnear relative equilibria) then can be thought of as points of (cf. [ 1]).

These by Moulton's theorem are

reflection across

E

point set

P „(R) a11"2

points of

V

and leaving A

precisely the Houlton points. number of critical points.

of

F-oC) ~"

• ^ *

exist

- R

T

R

) -

A

Furthermore

-(E)

having as fixed

This involution maps critical

and the fixed critical points are

is the minimal number forced by the homology

First for

must look like.

the 3 points of A

P

n_2(

This gives a further homology condition on the

existence of \f

We wind up with 2 examples. V i P.(B) - A

T

of

p

I wonder if (at least part of the time) the

number of critical points of he

if

invariant. ~

into critical points of

n

"r- in number.

induces an Involution

of

The Houlton classes (of

, where

On 7

with the above properties. n - 3 » one knowB what P,(R) - S 1

goes to

-<*> .

in

P.(B) - S 2 , there

Also

P.(R) - A t~

'

contains the 3 Euler (or Houlton) points which are saddle points of V . The Lagrange points are off P.(R) and are interchanged by the involution if . They are maximum points for For

n - 4

and

have index 3. with the

x.

7.

m. ■ 1 , 1 - 1 , . . . , 4

I checked that the 12 Houlton points

Also there exist 6 relative equilibria classes in this case at vertices of a square.

These are local maximum of

7

(of

index 4 ) . Finally there are 8 relative equilibria classes which consist of th the vertices of an equilateral triangle with the 4 position at the center. These have index 2.

I presume there are no others in this case, especially

since these 26 are consistent with the Horse inequalities and the homology of

P 2 (a) - A.

(H 4 - Z , Hj - 5Z , H 2 - 62) .

940 198

8SP8M3ICES (1) Smale, S., Topology and Mechanics I, Inventiones Mathematicae 10, 305-331, 1970. , Topology and Mechanics II, to appear in Inventiones Math. (2) Vintner, A.,

The Analytical Foundationaof Celeatial Mechanics, Princeton UniTeraity Press, Prinoeton, H.J. 1941. • • « Added in proof t Emery Thomas pointed out to me tbe following paper, related to the abore i R.V. Baaa, "A characterization of central configuration solu­ tions of the n-body problem, leading to a numerioal technique for finding them", Adraneea in Engineering Science, 1969, pp. 323-330.

UHIYERSITY OP CALIPOHMIA BERKELEY, CALIPORHIA

941 PERSONAL PERSPECTIVES O N MATHEMATICS AND MECHANICS Steve Smale

1. These days especially, provocative questions confront a socially conscious scientist when he begins to contemplate where applications of his work might lead. As one whose main work has been in pure mathematics, and who is beginning to concern himself with areas of applied mathe­ matics such as electrical circuit theory, I wonder to what extent I should explicitly direct my work toward socially positive goals. Many issues on the relations of the profession of a scientist to the social crisis of this time deserve a format, for example, at various professional meetings and conferences, we well as in the departments of universities. There are strong pressures on the scientist, from the sources of funding research, scientific tradition, and a basic conservatism of maintaining the status quo, which act to prevent discussion of these issues. Although I am not going to pursue such a discussion here, I feel that mathematicians and scientists in general must face these questions in a much more serious way than they have done to date (myself included).

2. A fairly general procedure for the mathematical study of a physical system starts with explica­ tion of rlie space of stotes of that system. Now this space of states could reasonably be one of a number of mathematical objects. However, in my mind, a principal candidate for the stare space should be a differentiate manifold; and in case the system has a finite number of degrees of freedom, then this will be a finite dimensional manifold. Usually associated with the physical system is the notion of how a state progresses in time. The corresponding mathematical object is a dynamical system or a first order ordinary differential equation on the manifold of states. Too often in the physical sciences, the space of states is postulated to be a linear space, when the basic problem is essentially nonlinear; this confuses the mathematical development. For the International Union of Pure and Applied Physics Conference on Statistical Mechanics, Chicago, March 1971.

942 On the other hand, recent decades have seen big developments in parts of mathematics related to differentiable manifolds. Thus there should be no reason for reticence on the part of applied mathematicians to use these developments. We have tried to provide a little background for what follows. I will describe breifly aspects of three areas in which I have been working. These are mechanics, differentiable dynamical systems, and electrical circuits.

3. Recent years have seen several textbook accounts of the development of classical mechanics on manifolds. Two of these are Abraham and Marsden, Foundations of Mechanics [1 ] , and Loomis and Sternberg, Advanced Calculus [ 2 ] , A number of journal articles also have appeared on aspects of this subject, including my "Topology and Mechanics" [ 3 ] . A basic motivation of this literature is to make the discipline of classical mechanics more elegant, more geometric (and so perhaps less analytic), more global. Furthermore, this develop­ ment has done a lot to unify mechanics with large parts of geometry and part of topology. The globalization has removed the need for a linear space framework, such as one finds in the traditional classical mechanics text. I will give a quick indication how this goes. Configuration space is a smooth manifold M. For the space of states one takes the tangent bundle T of M or the cotangent bundle T* of M. The manifold T* has a natural sympletic structure, i . e . , a canonical nondegenerate 2-form O defined on it. This form defines an isomorphism between vector fields and 1-forms. Thus if H:T* -♦ R is a function on T* (a "Hamilton!an"), dH is a 1-form which corresponds to a unique vector field X,, via O. A vector field is the same thing as a first order ordinary differential H equation; in fact, X,, is one way of representing Hamilton's equations. It is immediate then n that X U (H) = 0 or that differentiation of H along X u is zero. This is conservation of energy. H n Symmetry and other conservation laws are related similarly, simply and conceptually. Further­ more, X u preserves O and hence the volume element O , where n = dim M . This is Liouville's H theorem. Suppose one starts with a kinetic and potential energy as well as M.

How does that relate

to this context ? Potential energy is a smooth function V:M -* R, and kinetic energy can be thought of as a Riemannian metric on M with K:T -• R given by norm squared of a tangent vector with

943 respect to this Riemannian metric. is the projection.

Total energy E:T -• R is the function K + V o l l , where IT:T -• M

The Riemannian metric induces a bundle isomorphism (the Legendre transforma­

tion T -• T* and thus pulls fJ back to T to give a symplectic structure on T. field X r naturally associated to E via this symplectic structure. Newton's equations, and of course the energy E is conserved.

Thus there is a vector

The vector field X_ generalizes If one has a Lie group 9

of

symmetries acting on M leaving E invariant, one has a natural induced map a of the Lie algebra V of 9 into vector fields on M . Let a : V "*T vector field a ( X ) at X € M where X 6 V-

be the linear map obtained by evaluating the

One then lias a map J :T* - V * obtained by letting

J . restricted to a fiber T * be the dual linear map of a . The composition J , T - T * -• V * generalizes angular momentum.

This map J:T -• V * is constant on orbits, and furthermore it can

be shown that J is equivariant with respect to the adjoint action on G on "j/*.

This description

of angular momentum helps clarify Jacobi's "elimination of the node" in celestial mechanics. In the above if V ^ 0 , one has simply the case of Riemannian geometry, and the projections into M of the integral curves of X . are geodesies.

More generally the projections of these integral

curves are the trajectories in configuration space. W e have given here the flavor of how this modernization of mechanics goes. The assertions above can usually be proved with very little effort, if one has accepted the basic elements of differential topology and geometry.

The above-mentioned three references do cover and expand

on this material. For many modern mathematicians, the elegance of this treatment of mechanics would be justification for its development.

Beyond this, some new insights have been obtained already by

this globalization w i t h , I believe, much more to come.

4. A lot of research has been done in recent years in the area of mathematics now called differentiable dynamical systems. that give an assessment.

Let me describe btiefly, roughly, what it is about and after

For more substantial and precise information on this subject one can see

my "survey, " "Differentiable Dynamical Systems" [ 4 ] , and its references, as well as the more recent Global Analysis [ 5 ] . The discipline of differentiable dynamical systems attempts to study systematically dynamical systems on a manifold.

So one forms the topological space D y n ( M ) (with a topology respecting

differentiable properties) of all dynamical systems on a manifold M .

944 One has a reasonable notion of "almost all" in Dyn(M) meaning a Baire set (a countable intersection of open dense sets), and when convenient one restricts attention to such a set. A theorem is regarded as useful if, for example, it gives a property valid for all systems of some Baire set of Dyn(M). Related is the idea of structural stability and certain variations. This kind of stability is a property of a dynamical system itself (not of a state or orbit) and asserts that nearby dynamical systems have the same structure. The "same structure" can be defined in several interesting ways, but the basic idea is that two dynamical systems have "the same structure" if they have the some gross behavior, or the same qualitative behavior. For example, the original definition of "same structure" of two dynamical systems was that there was an orbit preserving continuous transforma­ tion between them. This yields the definition of structural stability proper. It is a recent theorem that every compact manifold admits structurally stable systems, and almost all gradient dynamical systems are structurally stable. But while there exists a rich set of structurally stable systems, there are also important examples which are not structurally stable, and have good but weaker stability properties. One series of theorems has been in the direction of characterizing various stability properties of dynamical systems in terms of more tangible properties of the system. One cannot expect this development per se to help in the study of mechanical systems as in section 3 above, because small perturbations will destroy the Hamiltonian behavior of a mechanical system: in particular, dissipation accomplishes this. On the other hand, engineering, and problems from the biological sciences, could profit from this approach through stability questions. What happens near an attractor or sink would be of special interest in applications. Most simple is the steady-state case (fixed point of the dynamical system), then the oscillation (or periodic behavior); but recent studies have shown how easy it is to have far more complicated attractors appear. In particular, even the structurally stable attractors in rather simply given examples contain a vast mixture of periodic, almost periodic, homoclinic, and other kinds of phenomena. So differentiable dynamical systems have developed as a purely mathematical subject: the attempt to study all ordinary differential equations on a manifold. As such, a wealth of theorems and knowledge has been accumulated. On the other hand, the subject has not achieved any definitive state. One of the main recent achievements has been to remove the fear of making a global study of differential equations on manifolds of dimension greater than 2. Many examples, and in fact

945 large classes of systems, with good stability properties are now known and well understood on manifolds of all dimensions. On the other hand, we have concrete examples not understood as to their stability properties; in fact, there remain good questions to be found. The subject is in a state of activity and flux at this time. As I have said, up to this point the study of differentiable dynamical systems has developed as a purely mathematical discipline with only minimal direct relations to physics and engineering. I suspect there is a good potential for interaction. For example, the above-described work might be used to extend the horizons of applied mathematics by expanding the realm of possibilities for mathematical models.

5. I would like to give here the flavor of some recent work of mine on electric circuits—from "Mathematical Foundations of Electrical Circuits, " preprints in existence. Our development proceeds in a natural manner, working with what is given by the circuit and making no arbitrary choices. We consider circuits—for example, as in C. DesocrandE. Kuh, Basic Circuit Theory [6]—with nonlinear elements, either resistors, inductors, or capacitors. Associated with a circuit is a linear graph, say connected and oriented, consisting of nodes or 0-cells and elements, branches, or 1-cells. An unrestricted state is the set of currents and voltages in each branch and is thus a 2b-tuple of real numbers, where b is the number of branches. The space 6 of all states then is real Cartesian space of 2b dimensions. Kirchoff's and Ohm's laws will impose conditions so that a physical state must lie in a certain subset £ c 6 which we proceed to define. First, Kirchoff's current and voltage laws define linear relations on the currents and voltages in the branches, and thus restrict states to lie in a linear subspace K of 6 . Let iDxV)_: 6 -• R x R' be the projection defined by taking the resistance currents and K K resistance voltages. Thus R consists of currents through the resistors and R' is the linear space of voltages across the resistors. We take Ohm's law to say that for each resistor p , the characteristic is a closed one-dimensional submanifold A these A

of the current voltage plane of that resistor. Then

define a closed submanifold (their product) A in R x R'. Let I I ' : K - R x R' be the

restriction of i . x u „ . Then consider. Hypothesis: I I ' i s transversal to A . Although this is a key hypothesis, rather than recalling its rather technical definition we note the main consequences that are of concern to us. Under this hypothesis, IT 4 (A) = L Is a smooth manifold of S and dim E = the number of inductors plus the number of capacitors.

946 Furthermore, this hypothesis is generic in that it will be satisfied with almost all choices of resistor characteristics. The space of physical states is naturally defined as £ (those satisfying both Kirchoff 's and Ohm's laws). We see the above hypothesis as key, because in giving a manifold structure to S , it permits analysis of the space of states. In fact, via Faraday's laws, we can obtain an ordinary differential equation, perhaps singular, on E. In brief, this goes as follows (but see the above reference for details). The inductances and capacitances define a symmetric form I over S; this form cannot, in general, be expected to be positive definite at each point of S.

In fact, besides being indefinite, I may also be degenerate

on part of E. The resistances define a closed Worm w on 3D. Then the differential equations are the gradient of w with respect to I. These are going to be much more complicated than the usual gradients because of the general nature of I. These equations contain in increasing generality those of Van der Pol, Liertord, and Brayton-Moser. On the other hand, they contain examples which are of a different type corresponding to "relaxation oscillations" and thus con­ tribute to achieving a certain unity in the differential equations of circuit theory.

6. There is a remaining aspect of this global analysis approach to applied mathematics that I would like to mention. The approach taken in this paper may indeed be a little hard to accept for the applied mathematician trained in traditional methods. However, the approach here tends to make applied mathematics and also ordinary differential equations accessible and attractive to the modern mathematician, one who has been brought up in the purist, Bourbaki style of education. For several decades, the most important mathematical tendencies in the United States and Western Europe have been very separated from applications and there has even been a certain disdain for applied mathematics by many leading mathematicians. I believe that there have been some healthy sides to this separation. Many fields in pure mathematics have flourished in this time, and perhaps because of the division a certain clarity has been achieved. On the other hand, at least since before the time of Newton and Leibniz, never have the main schools of mathematics, the main mathematicians, been so sharply separated from other disciplines. I believe it worthwhile to try to change this course of events. The modernizations

947 I have described in mechanics, ordinary differential equations, and circuits will hopefully help to break this isolation by bringing modern mathematics more substantially into communication with other fields.

REFERENCES 1.

Abraham and Marsden, Foundations of Mechanics ( N e w York, 1967).

2.

Loom is and Sternberg, Advanced Calculus (Reading, Mass., 1968), chap. 13.

3.

Steve Smale, "Topology and Mechanics, " I and I I , Inventiones M a t h . 1970.

4.

Steve SmaI.e, "Differenfiable Dynamical Systems," Bull. Amer. M a t h . Soc. 1967.

5.

S. S. Chern and Steve Smale, Global Analysis, vol. 1 (Providence, Ft.I.: American

Mathematical Society, 1970). 6.

C . D e s o e r a n d E . Kuh, Basic Circuit Theory ( N e w York, 1969).

951 J . DIFFERENTIAL GEOMETRY 7 (1972) 193-210

ON THE MATHEMATICAL FOUNDATIONS OF ELECTRICAL CIRCUIT THEORY S. SMALE

The goal of this note is to derive the differential equations for simple (non­ linear) electrical circuits with resistors, inductors and capacitors. I would like to express my deep indebtedness in these matters to George Oster. Oster sup­ plied me with the basic references to the literature, and conversations on this subject with him were very helpful. The reference Brayton & Moser [4] was also very helpful. 1. A simple electrical circuit provides us first of all with an oriented graph G which will be assumed to be connected but not necessarily planar. This is a one-dimensional cell complex with branches or elements (1-cells) and nodes (0-cells or vertices). The states of the circuit have two components, the currents through the branches and the voltages across the branches. Let C} be the vector space of real /-chains of G, Cs the /-cochains of G, j = 0,1. Thus O can be thought of as the dual vector space of C}. As is well-known, the currents in the circuit can be represented as an element i of C,. Thus i = £ i.c. where a ranges over 0

all the branches, c. is the
which is 1 on c„ and 0 on the others. Let y = Q X C . Then s = (i, v) e £f is a state (unrestricted) of the circuit. Physical laws (Kirchhoff and Generalized Ohm) will constrain the physical states to lie in a submanifold 2 ot £f which we proceed to define. Denote the boundary map by 3: Cl-+C0 and coboundary by 3*: O —*Cl. Thus 3 is a linear transformation of vector spaces, and 3* is its adjoint on the dual spaces. In this context the Kirchhoff laws KCL, KVL can be expressed as follows (see Branin [2] and the references therein): KCL: i 6 Ker3 , KVL: v 6 Image3* . The first condition just expresses the fact that the currents entering a node Communicated December 9, 1971.

952 194

S. SMALE

sum to zero while the second asserts the existence of a voltage potential. Note that we do not use "meshes", and also that C, has the natural structure of real Cartesian space R" where b is the number of branches of G; the same holds for O. Let Bt: C< x C* —* R be the bilinear pairing of a vector space with its dual, i = 0,1. Then Bt(i, v) = £ '.*.■ The Kirchhoff laws yield that any physical .-i

state (i, D) e C, x O is restricted to lie in the linear subspace K = Ker 3 x Image 3* of Ct X C1. The first part of the following well-known proposition is essentially what is called Tellegen's theorem. Cf. Desoer [5], Brayton & Moser [4]. (1.1) Proposition. Bl vanishes on K and dim K = dim C,. Proof. This is standard linear algebra, i.e., if (/, v)eK, then di = 0, v = d*u and therefore B,(i, v) = B,(i, 3*K) = B0(3i, M) = 0. Furthermore dimK = dimKer3 + dimImage3* = (dimC, - dim C0 + 1) + (dimC, - 1) = dimC,. (1.2) Corollary. The l-form 2 v.di, on Cx x Cl vanishes when restricted toK. Since each /, is a linear function on if, this corollary is just a restatement of (1.1). The next step is to introduce the branch elements which a simple electrical circuit gives us. The branches of G in our framework can be classified into exactly three categories, resistance branches, inductor branches and capacitor branches (batteries are included in our definition of resistor). We let 9t denote the real vector space of currents through the resistance branches, so that if i e 91, then i = 2 ifcp, summation over the resistance branches. Similarly we define & as the space of currents through the inductor branches, and # for the currents through the capacitor branches. Analogously we then have 91', ££', <€', the dual spaces for the voltages across the respective branches. We suppose that the />-th resistance is given by its characteristic, a closed 1-dimensional manifold Af c R x R = {(«',, v,)} for each resistance branch p. Let A C 5? x 9? be the product of the A, so that A = {(.i, v)e9l X9i'\ (/,, v,) e Af}. Then A is a closed submanifold of ^ x 91' with dim A = dim St. Let iB x vB: Sf —► 9t X 9t' be the natural projection, and JT' its restriction to K. (In the future, we will frequently use the symbols iL: Sf —* 2£ etc. for the various projections.) (1.3) Hypothesis, vf is transverse regular to A in the sense of Thorn; see, e.g., [1]. This means that if x e K, y = J(x) e A, then the composition TX(K) ► T„{9t X 9tT) * Ty(9l X 9ir)/Tv(A) is surjective. Note that (1.3) is a generic property (i.e., (1.3) will be true for almost all choices of the Ar), and we will always assume it to be satisfied.

953

195

ELECTRICAL CIRCUIT THEORY

Assuming (1.3) then, let 2 = n'-\A). (1.4) Proposition. ^ is a submanifold of K with dim 2 = dim (if X «")■ Proof. That 2 is a submanifold is a standard fact of differential topology (the implicit function theorem) as well as the fact that codim* 2 = codim,,*, A; see [1]. Then we have from (1.1) that dim 2 = dim C, — dim & = dim (if X # 0 since codim* 2 = dim 0t. We call 2 the manifold of states (physical) of the network. Next we wish to define a symmetric bilinear form / over 2 which will come from the inductance and capacitance. Thus for each x € 2» h wiH be a s v m " metric bilinear form on the tangent space T x (2)- For this we first define a form / over if x «" by /„,„ = - 2 L^di* + 2 C^dv). Here the first sum is over the inductor branches and the second over the capacitor branches. L, is a smooth positive function of ix e R, the inductance in the i-th branch and given in advance. Similarly, Cr is the capacitance in the f-th branch and is also a smooth positive function on R given in advance. The above / on if x # ' is a smooth nondegenerate symmetric form, i.e., an indefinite metric. Let n: 2 -* & X # ' be the restriction of the natural projection iL x vc: Sf -► if X #', and let / = n*J so that / is a symmetric form defined over 2 fr°m J via *. If at * e 2> the derivative Dic(x): T x (2) -* r, (X) (i? X V) is an isomorphism then (and only then) the form Ix on T x ( 2 ) will be nondegenerate. Recall from (1.4) that dim r x ( 2 ) = dim r. (x) (if x «")• The next step in our development is to define a certain closed 1-form w on 2> and using this define the equations of motions for the states. To this end, let h':Sf-+R

duality pairing

—

be the composition ffX <*'

=► R, and h be the restriction of h! to 2 - Thus

h

„ .

0'»*>) = 2 W T

summation over the capacitance branches. Next let ft be the 1-form on 9t X 3t' defined by jy, = 2 v,di,, ^ ^ " * ^ symbol TJ for the 1-form on 2 induced by the map it": 2 - » & X &' previously defined. (1.5) Proposition, jy is a closed \-form on 2 Proof. It is sufficient to show that d^ vanishes when restricted to A since n" factors through A. But d(vfdif) = 0 for each p; it is a 2-form on a onedimensional manifold A,. Define w = -q + dh on 2 • Thus w is a closed 1-form on 2 • (1.6) Main theorem. The equations of motion for the network are Hdx/dt, Y) = w(Y) for all Y <= 7X2). This is an equation for a curve of states in time, t —► x{t), which must be satisfied. So x: [a, b] -> 2 , and at x(t) e 2 we have 7X(O(^(0, Y) = w,it)(.Y) for all y 6 r x ( t ,(2). (1.7) Remark. Let U be the open subset of 2 where DM.x): T x (2) -*■ Tmlx)(y X # 0 is an isomorphism. Then on U, I defines an isomorphism

954 196

S. SMALE

between vector fields and 1-forms. Thus on U, there is a vector field X which corresponds to w under this correspondence. This X when integrated gives the passage of a state in time on U. That is, X on U can be thought of as the ordinary differential equation (nonsingular) for the circuit. However, for the full picture, one must keep all of 2 . Of course if # ' ( £ ) = 0 or even if w is cohomologous to zero, we can write w = dP for some smooth function P: 2 —► R and the right hand side of our equation simplifies accordingly. For the usual electric circuits H*(AJ = 0 for each p, so Hl(A) = 0 and w = dP; the equation becomes I(dx/dl, Y) = dP(Y) for all Y. Our starting point for the proof of (1.6) is (1.2) which we can write

E v,di„ + 2 M
i

r

as a 1-form on 2 C K. Here according to our usual convention the first sum is over resistance branches, the second over inductor branches and the third over capacitor branches. Each of these sums can be interpreted as 1-forms on 2 as induced by maps:

y 9ty.0t' 2 — > K — > y — * s e x2"

\?x«" By the Leibniz rule we have d(Z irvr~> = S VA

+ S 'Ar •

Putting this into the first equation, we obtain - 2 M ' l + 2 'rdvr = Z V,dif + d Ti KVr =

W

•

We also have the following basic relations of circuit theory (Faraday, etc.): v, = Lt(Odit/dt,

iT = Cr(.vr)dvt/dt.

Making another substitution we obtain -XLAOidiJdOdi,

+ 2 Cr(vrHdvT/dt)dvr = yv ,

which is just another way of writing the equation of (1.6). The final part of this section is devoted to showing how energy and power fit naturally into our framework. Compare, e.g., Valkenburg [12, p. 23]. First we define a real valued function W on 2» the total energy stored in all the inductive and capacitive elements, or energy for short.

955 197

ELECTRICAL CIRCUIT THEORY

Let WL: SC -> R be the function WL(i) = f £ LAQi/ii, where F is any o,r

path from 0 to / in if. This function is well-defined since clearly the integral is independent of r. Similarly define Wc:<#'-+R by Wc(v) = C* £ Cr(vr)vrdvT. Then let o,r

r

W: 2 -* R, W = WLoiL + Wco7>c be the energy of the network where iL: J]—* if is the restriction of the projection iL: £f —* SC, etc. The power, or the power in all the resistance elements, can be represented by another function PB: £ —► J? as the composition

The last map is just the pairing of a vector space with its dual. (1.8) Theorem. On the part of £ where X is defined via (1.7), XW =

-PH-

Here X- W is differentiation of the function W with respect to X or in other words with respect to the natural action of X on W. The theorem expresses mathematically that the energy decreases along the trajectories according to the power dissipated in the resistors. For example, if there are no resistors one has X- W = 0 or conservation of energy (but these are not Hamiltonian systems, however). Proof of (1.8). By Tellegen's theorem, we can write PR + PL + Pc = 0 on K or on 2] where PL, Pc are defined via <£ X 2", & X W respectively, similarly to PR. Then it is sufficient to prove that X- WL = PL, X- Wc = Pc on 2 . where WL, Wc can be thought of as functions on £ • We prove the first, the second being similar. For this we project a solution curve <j>t(x) of X into & x if' via the restric­ tion p of iL X vL: Sf -► Se X &' to £ • It is sufficient to check

A-WL(pMx)) dt

PL(PMX))

= t-a

PL(PX)

Using the definition of WL and noting for each X that L(h)hdit

= Hi^ii-^-dt dt

= ijV^t ,

this last is easily checked. 2. We give some simple examples here of electrical circuits, largely to show how the framework of the preceding section can act as a unifying force. Example 1. This is a simple RLC series circuit with the resistance charac­ teristic current controlled. That is to say, we have the situation of Fig. 1 with

956 198

S. SMALE

I—MAMMA——1(c

—r00000"000"0
Fig. 1

the orientation as depicted and A, is described by {(»',,/(/,))} C 01 x 91 where /: R —» R is a real smooth function. Here our 2 is diffeomorphic to if x «" = {(i„ vr)} by *: 2 -♦ <£ X «", with ff the restriction of iL X vc: 5* —» if X 3". In fact it can be easily seen using the definition of £ and Kirchhoff's laws that an inverse to K is given by

(h,vt) -* Ox,h, -ii,KQ,

-/(«*) +

vT,vr).

Thus our basic equations get transferred to if x V. Here the forms / and / are essentially the same and letting (x, y) = (/„ vr), where / = —L(x)dx* + C(y)d?. Then w = 9 +
Now one deduces for the basic equations (see Remark (3) below): Ux)^- = -^(x,y)=ydt dx

f(x) ,

C(y)^- = ^-(x,y) dt dy

= -x .

Thus in the case L = 1 and C = 1, the equations are dx/dt = y — f(x), dy/dt = — x. This is exactly Lienard's equation as a first order equation (cf. Hartman [6, p. 179], Lefschetz [8, p. 250]). Remarks. (1) For f(x) = x1 — x, this becomes Van der Pol's equation. (2) Of course a battery can be included, by incorporating it into the resist­ ance term. (3) Formally the equations are derived as follows from the main theorem of § 1. Suppose on R1 = {(*„ *,)}, / = -Uxt)dx\ + C{x^dx\, 1{X, Y) = dP(Y) for all y. Let Y, = (1,0), Y, = (0,1). Then -L(.x1)X1 = I(X, Y,) = dP{Y,) = (dP/dx^x^xj, and similarly C(x2)X1 = I(X,YJ = {dPjdx^x^x^ where

* = (*„*,).

(4) The power dissipated in the resistor is — PB{x,y) = —xj{x), and the energy of § 1 is given by W(x, y) = x* + y1 assuming L = C = 1. Let us look at the phase portrait of the above system of differential equations in the (x, y)-plane, under the assumptions there exist positive constants c, k with

957 199

ELECTRICAL CIRCUIT THEORY

(a)

xf(x) > c\x\ for JJC] > k

(b)

f(0) < 0

(natural assumption),

(nonlinearity assumption).

Proposition. Under the above assumptions {including L= 1,C = 1), orbits starting with large energy tend towards a periodic orbit, there is one zero of the vector field, a source, and there is a cross-section to the flow in the strongest sense. First it is clear that x = 0, y = — /(0) is the unique zero, and the vector field has I — j ' ~Q) as its matrix of first partial derivatives at (0, — /(0)). The eigenvalues are X = $(—f(0) ± Vf (0)2 — 4) and will both have positive real part since f(O) < 0. Thus the singularity is a source. Using the theorem of § 1 that X-W — —Pr, remark (4) of this example and our assumption (a), one can check the following lemma. Lemma. There is some disk D in R2 defined by W(x, y) < K such that every orbit 0(0 has an associated t0 so that {0(/) | / ;> t„} c D. This gives us our proposition except for the cross-section Q which we define by Q = {(x,y) e R2\x = 0, y ^ -/(0)}. Then dQ = the zero, and the vector field at every other point of Q is perpendicular to Q (i.e., horizontal). Further­ more, fairly direct, and well-known arguments yield that every non-trivial orbit meets Q and after leaving Q returns again to Q. Thus we define T: Q —► Q by taking this point of first return to obtain a diffeomorphism which is the identity on dQ, which contains all of the qualitative information of the system. Further­ more T expands away from oo and dQ. Thisfinishesthe proof of the proposi­ tion and our discussion of Example 1. Example 2. A simple RLC circuit in parallel with voltage controlled characteristic as in Fig. 2 with if = /(v,).

t

J *1r i -J _

ig

1

p^ /

W

Fig. 2

We have here a diffeomorphism .Sf x # ' —» £ Example 1) (/„ vr) -* (f(vr), i„Kvr)-it, Then w = -vrf(v7) ti(x,y) = (i^v,). So

defined by (compare

-vT, vr> vr) .

+ d(vr(f(vr) - /,)), and w = d((J(y) - x)y) - y df(y)

958 200

S. SMALE

P(x, y)= -xy+

jVf(t)dt

,

I = -L(x)dx* + C(y)df ,

and the equations are L(x)dx/dt = -dP/dx = y ,

C(y)dy/dt = dP/dy = -x + /(y)

These equations are essentially the same as in Example 1. Behind this is a duality, see for example Desoer & Kuh [5]. Example 3. We will see in detail how an example from Brayton [3, p. 14] fits into our framework. This is the circuit in the following Fig. 3. ■—A/WW

««)

1

— • — i »Fig. 3

Here L, the resistance R in branch 2, and C are all linear. E is the constant voltage source and can be thought of as a resistance with characteristic (i,, v,) = (i'„ E) and the resistance in branch 3 is voltage controlled with characteristic defined by the function / on R. In this case the map it: £ —* if x # ' is a diffeomorphism, and in fact an inverse is given by (h, Vs) — 0«, h, f(vt), i„ it - /(v,), -E, Ri„ vt, E - Ri, - vt, v„) ,

se x «" — s

c

(^ x if x *f) x (#' x if' x «"),

where we have used the natural .R3 structure on 9t ond 3?'. Also we have used Kirchhoff laws which in this case read: it = i, = it = i, + j , and v3 = viy Vi + vt + v3 + vt = 0; it is easy to see that we have exhibited an inverse to K. Let {x, y) = (it, vj, and we get for / or / even, / = —L dx2 + C dy*. We obtain the "mixed potential" P as follows: dP(x,y) = w = dh + rj = d((x -f(y))-y)-Edx + Rxdx + ydf(y), P(x,y) = xy-Ex

+ R^-

- J*/(«)du . 0

The differential equations are L dx/dt = - y - Rx + E ,

Cdy/dt = x - f(y) .

959

201

ELECTRICAL CIRCUIT THEORY

Furthermore we have expressions as follows for the work and power: W(x,y) = Lx* + Cf ,

PR = Rx> - Ex + yf(y) .

For the function / considered by Brayton, one has the fact that there is a disk in the (x, v)-plane such that every orbit stays in the disk for large enough time. This is similar to the case of Example 1 and uses the fact that

X.W=-PB. Furthermore, if in addition there is just one zero, one can easily check that this zero is an attractor and that there is a 1-dimensional cross-section. Note that up to now all the examples satisfy the condition that JT: 2 —♦ J*? x *€' is a (global) diffeomorphism. Compare this to the Brayton-Moser hypothesis [4] that the currents through the inductors and the voltages across the capacitors determine all currents and voltages in the circuit via Kirchhoff's law. In fact it is a reasonable interpretation that the B-M hypothesis means exactly that x: £ —► & X # ' has a well-defined inverse and hence is a diffeomorphism. In this case, the derivation of Example 1 generalizes to give the equations: L&JdiJdt = —dP/dit , CT(vr)dvT/dt = —8P/dvr ,

each X, an inductor branch, each y, a capacitor branch.

These are the B-M equations (up to a sign; Brayton and Moser seem to use an unusual sign convention). Remark 1. These equations contain as special cases, all examples to this point. If there are no inductors, then one has a gradient dynamical system. Remark 2. As with Brayton and Moser, these equations admit an easy extension to the case of mutual inductance, capacitance. Example 4. This circuit has just two elements, a linear capacitor and a nonlinear current controlled resistor as given in Fig. 4.

m

I i

^ tyWWW Fig. 4

Fig. 5

Here i, f(i) are the current and voltage in the resistor where we assume that / has the qualitative properties indicated in Fig. S. Let i be a parameterization of 2 so that

960 202

S. SMALE

£ = {(/, - l , / ( 0 J ( 0 ) 6 St X # X &' X «"} . Then dP(0 = -d(i/(0) + f(i)di,P(i) = -if(i) + ffU)dj,

and the equation

0

can be read off from the main theorem of § 1 as C(dKQy(X, Y) = dP(Y) , or CfiQ'di/dt = dPJdi = - i f ( 0 , or finally Cdi/dt=

-ilfii)

.

Note that this equation does not fit into the framework of the B-M equations, it: 2 —> & X # ' is not a diffeomorphism and in fact the equation is singular where f (0 = 0. So we have obtained a singular first order, ordinary differential equation on the 1-dimensional manifold 2 or a vector field as in Fig. 6 where the singularities (not the zero) are marked with an x.

S

Fig. 6

What happens at the singularity is undetermined by the mathematics, i.e., by the differential equation. However, one can give a prescription for what happens at these singularities which is consistent with experiment and can be justified by circuit theory. This has to do with the theory of "relaxation oscillations" and proceeds as follows. At a singularity, the state jumps instantaneously to the part of 2 described by the dotted arrows jt, ;2 in the following Fig. 7. Thus, at least after a while, the state oscillates, the oscillation including two portions which take no time and two passages along the manifold. To justify this interpretation of what happens at the jumps, we proceed as follows according to a suggestion of C. Desoer and some ideas in the literature [7], [9]. R. E. Kalman first mentioned to me that this study was connected with relaxation oscillations (in connection with Example S). We introduce into the circuit of Fig. 4, an inductor in series with inductance L, to obtain the

961

203

ELECTRICAL CIRCUIT THEORY

h

1

I

h

circuit of Example 1, (see Fig. 1). The differential equations for this example are nonsingular and thus so to speak we have regularized the differential equa­ tions of the original circuit. We may in fact assume that L is as small as we want, so that there is a good physical justification in introducing L. The new differential equations are L di/dt = + vr - f(0 ,

C dvjdt = - i ,

where i is the current through the inductor (or through the resistor, which is the same thing) and vr is the voltage across the capacitor. Thus if L = 0, then vr = f(i),dvr/dt = f(i)di/dt and dvr/dt = —i/C so we recover the equations of Example 4. Now to justify our prescription of the jumps one looks at the phase portrait of the above as L —* 0. For this we refer to the literature [7], [9]. The idea is that £ of Example 4 can be imbedded naturally in the (i, vr)-plane of Example 1: i —► (/, fit) = vT), and one can imbed the cycle of Example 4 including the jumps also (in fact just look at Fig. 7). Then given any neighborhood of the cycle, by taking L small enough, the unique cycle of Example 1 (for the choice of f) falls into this neighborhood, with the time taken along the jump part arbitrarily small. The proof is not difficult. Example 5. The circuit we discuss here has some features of Example 4 but is more complicated. It and Example 4 both fit naturally into the framework of § 1 and the differential equations there apply. In both of these examples, the map JT: £ —* & X # ' has a singular derivative and the Brayton^Moser framework does not fit. The indefinite metric / is degenerate and this leads to singularities of the differential equation as a vector field on 2] (singularity in the sense that the vectorfieldis not defined there). A regularization is developed in Example 6 for Example 5. The circuit then is exactly that of Example 1 except that the resistor charac­ teristic is not assumed to be current-controlled and the orientation is different. See Fig. 8. We do assume that the resistor is voltage controlled so that its

962

204

S. SMALE

c

i-AWWV

1^

^QQfiQib

'

Fig. 8

characteristic is the graph of a real function f(vf) = /,. In this case one has a global coordinate system for 2 given by the restriction of the projection vR x vc: if -+ &.' x «" to 2 - ^ inverse # ' x «" -»• 2 c & i s gi v e n b v (x, y) = (v,, vr) -> (/(v,), /(v,), /(v,), v,, -vt

- vr, vr) .

Again this is checked from the definitions and Kirchhofi's laws. Then w = rj + dh = vfdir + d(iTvT) = xdfix) + d(y f(x)) and w = dP where P(x, y) = (x + y)Kx) — l*f(t)dt. In this case / has the form 0

/ = -WJdZ + C(vr)dv) = -L(f(x))(dKx)y + C(y)d? = CWdy* - L(J(x))(dKx)/dxydx* . Note that this form is degenerate when (df/dx)(x) = 0. This fact prevents one from using the B-M (i2, vr) as coordinates for 2 • Our equations for the circuit now become: L(Kx))(df/dxydx/dt = -{dP/dxXx,y)

,

C(y)dy/dt = (3P/9y)(x,y) .

Thus if C = L = 1, we obtain (df/dx)(x)dx/dt = - ( * + y) ,

dy/dt = /(*) .

One can write down the energy and power as W = ?x + v) = fix)1 + y1 and PR = i,vt = xf(x) respectively. One can expect that df(x)/dx = 0 for certain isolated x if the resistance is sufficiently nonlinear. In this case the equations for the circuit have a 1-dimen­ sional set of singularities. Remark. The same equations are valid for the dual circuit, that of Example 2 where the resistance is current controlled. We now proceed to study the phase portrait in the (jr,y)-plane of these equations when the characteristic has the qualitative behavior described by Fig. 9. There will be three zeros of this vector field given by f(x) = 0, x + y = 0 or (*„ — *,), (x,, —xj, (*,, — *,) where xf are the three zeros of /. The local behavior of the flow in the neighborhood will in general be determined by the eigenvalues of the matrix of partical derivatives

963

205

ELECTRICAL CIRCUIT THEORY

/«,) - v,

Fig. 9

(

a

Vfto

- ^ w ) with a = * r o - *»> 0

/

f(*)«

for x = some xt. This is an easy calculation from the basic equations. Then the eigenvalues have the form X = |(a ± Vo2 — 4). Typically under these conditions if xt < x2 < J:3, one might expect that |a| < 2 and a(xt) < 0, a(*2) > 0, a(x3) < 0, in which case the first and third zeros are sinks while the second is a source. We will make this assumption in what follows. Next let x", x" be the values of x where f (x) = 0. The vector field will be undefined then at the two vertical lines through x1, x".

*

*l

*

*,

Fig. 10

*"

X.

\

964 206

S. SMALE

One knows exactly the qualitative behavior of the vector field along the curves x + y = 0, fix) = 0, as well as f (x) = 0. This follows from the form of the differential equations. Putting all of this information together, we can obtain the phase portrait as partially depicted in Fig. 10. This leaves open the question of what happens as a trajectory runs into a singular line fix) = 0. We give a prescription for this with a justification in the next section. Example 6. We follow a suggestion of C. Desoer to regularize Example 5 by adding a (small) capacitance C as in Fig. 11.

Fig. ll

The goal will be to give a prescription for what happens in Example 5 at the singularities (undefined places of the vector field) and to provide some informa­ tion on the phase portrait of Example 6. We emphasize that the resistor in Example 6 is the same non-linear one as in Example 5. Proposition. As C —» 0 in Example 6, one obtains Example 5 with the following "prescription". When a trajectory hits the line x = x", the state jumps instantaneously to the line x = x keeping the same y value. When a trajectory hits the line x = x', the state jumps instantaneously to the line x = x, again keeping the same y value. In what follows, clarification will be added to this proposition. Thefirststep is to write down the equation for Example 6. As usual, we find a coordinate chart on 2 , this time as a map SC x W —* 2 c & which is an inverse to n: £ — Se X «" and given by: (/„ vr, v'r) —• ijiv',), it, i„ i, - fiv'r), v'r, —vr - v'r, vr, v'T), where it is of course the current through L, and vT, v'T is the voltage across the capacitor C, C respectively. Here we have used the Kirchhoff laws if + ?7 = ir = h, v>i + v, + v, = 0. v, = KThen one checks easily that P = ixivT + v'r) — I Tfiv'r)dv'T. The equations 0

are L dijdt = -ivT + vp , Cdvr/dt = i,,

C dv'r/dt = i, - fiv'r) .

965

207

ELECTRICAL CIRCUIT THEORY

If C = 0,L = C = 1, then it = j(v% dijdt = j'WJdv'Jdt, v'r = v, and dvr/dt = /(v,) ,

f'(v,)dv,/dt

= -(vT + v,) .

This checks; these are the equations of Example 5. We let (x, y, z) = (v'r, vr, ix), and L = C = 1. So we have dx/dt = (z-Kx))/C

, dy/dt = z,

dz/dt =-(x

+ y) .

Energy and power have the form W = z1 4- y2 + C'x2, PR = xf(x). Now we consider the surface S imbedded in JF, (x, y, z) = (x, y, /(*))• This surface we think of as Example 5 imbedded in Example 6, and the question is what happens on S as C —► 0. Now we leave to the reader to carry out this process in detail to prove the previous proposition. One chases along trajectories for C very small, and keeps sharp account of how the signs of components of our vector field change. Now we discuss the more general problem of the phase portrait of Fig. 11, and we do not necessarily assume that C is small. My feeling is that one can expect the qualitative behavior of this differential equation in 3-space to be rather complicated. One property is that for the kind of resistor characteristic we have been considering, there will be three zeros of the vector field. The study of the local behavior of the system about these zeros should be rather straightforward. Furthermore, using (1.8), our expression for W and PB and the form of these differential equations, one can check that the states eventually contract onto some fixed, compact subset of our 3-dimensional phase space; thus one knows the qualitative behavior of this differential equation at oo of J?3. Finally we remark that there is a very well-behaved cross-section Q in this example, defined by Q = {(x,y,z) € fl5!* + v > 0, z = f(x)} with associated diffeomorphism T: Q —> Q. It should not be a problem of too great difficulty to study the behavior of T on the boundary dQ of Q. The general problem then reduces to the study of T. I would think that this should be a relevant, interesting and challenging problem, to study the qualitative properties of this transformation T, say for / of the type we have been con­ sidering. It might be useful to impose other conditions such as /, say of a generic type. Can one use (1.8) to obtain further information? Is the fact that this system is of gradient type for an indefinite metric of some use? Example 7. Consider the circuit of Fig. 12. Note that the map iL X vc: & —* & X *" restricted to K has image in a one-dimensional subspace of if X f', & X ■
966 208

S. SMALE

Fig. 12

ih\ Ker3 -» &. But dimKer 3 = 1. Thus no matter what the characteristic of 01 is, or Li,Lt are, the differential equations of (1.6) are degenerate every­ where on 2 for this Example 7. Let's look briefly at the generalization of this forced degeneracy. Consider generally the map iL X vc: if —► & x *" restricted to K. This is a product Ker3 x l m 3 * — JSP x «" of the restriction of projections iL: Ker 3 -► :SP, vc:Imd* —■ <$'. Thus the condition for forced degeneracy is precisely "either iL: Ker 3 —► .2? or vc: Im 3* —> tf' fails to be surjective". In the case of forced degeneracy, the differential equations are everywhere ill-defined on 2 C K independent of the charac­ teristics. There is one sufficient criterion for the above necessary and sufficient con­ dition for forced degeneracy. Namely, if either dim JSf > dim Ker 3 or dim # ' > dim Im 3* , then iL x vc —»JS? x # ' will fail to be surjective. This last criterion can be easily verified by a counting procedure, e.g., as in the original example of Fig. 11. More generally, dim ££ = the number of inductors, dim Ker 3 = # branches — # nodes + 1, so if # inductors > # branches — # nodes + 1 then one has forced degeneracy. A dual statement can be made for the capacitors, namely, $ capacitors > # nodes — 1 implies forced degeneracy. 3. This section consists essentially of a number of remarks related to the preceding sections. Theorem (1.6) suggests that it might be worthwhile to consider abstractly dynamical systems of that type. The ordinary gradient systems of a function with respect to a (positive definite) Riemannian metric are well understood (see, e.g., [11 §1.2] or [10]). The equations of (1.6) involve extensions of these in several ways. For example, consider this. (3.1) Problem. What can one say about the dynamical systems which are gradient systems of a function with respect to a nondegenerate indefinite metric, say on a compact manifold? That is to say, let M be a compact manifold with

967 ELECTRICAL CIRCUIT THEORY

209

a nondegenerate symmetric form defined over it (e.g., a Lorentz manifold), and /: M — ► R a smooth function. Then grad / is the vector field correspond­ ing under this form to the 1-form df. What special properties do such systems have? Now (1.6) motivates one to consider a further extension of (3.1). This consists of replacing df by a closed 1-form w on M. We will call such a system grad w where grad w is the vector field corresponding to w under the nondegenerate metric. Moe Hirsch pointed out to me the facts in this paragraph. We construct examples of systems of this generalized gradient type. Let a compact manifold M have a vector field X which is a suspension as defined in [11, §11.1]. Then one has a canonical map x: M —► S\Sl the circle. Now take any Riemannian metric on M such that ||AX*)|| = 1 for each x e M and n~\q) is orthogonal to X for each q e Sl, and let w>„ be the canonical 1-form on S1. It can be checked that X = grad (TT*^). One also has a converse construction. Suppose X — grad w, dw = 0 and X is never zero on a compact manifold M. Consider [w] in H\M, R) and take an approximation Wj with [wj e H\M, Q), Q the rationals. Let p be a positive integer so that [p, wj 6 //'(Af. z), Zthe integers. Then there is a smooth function /: M —► S1 generated by p >v, (Bruschlinsky). If the ap­ proximation w, is good enough, f~\q) will be transversal to X for each q € S1. Thus /"'(?) is a cross-section. Is the following true? (3.2). Suppose X = grad (w) is the gradient of a closed 1-form with respect to a Riemannian metric on a compact manifold Af. Suppose further that w is not cohomologous to zero and that X is well-behaved in the sense it satisfies the conditions of (2.2) of [11]. Then X has a closed orbit, not a point, which is asymptotically stable (i.e., a sink). The conclusion would seem to be relevant to electrical circuit problems since a stable periodic solution is of direct physical interest. However the hypothesis on the behavior of A" is strong. The following would seem to be interesting problems. (3.3) Problem. Can one always regularize the equations of (1.6) by adding arbitrarily small inductors and capacitors to the circuit appropriately? How? By regularizing we mean obtaining new equations which have the property it: 2 —► & X # ' is a diffeomorphism (or at least a local diffeomorphism), e.g., as in Example 4 and Examples 5, 6. (3.4) Problem. Suppose the power PR of a circuit can be written in the form PR = nfFf where each P/. Af^R has the property Pf(if, v,) > Cr[?P + *>J) for large (/,, vf) with some positive constant Cr Suppose also that it: 2 —► if X c€' is a diffeomorphism. Then (under possibly additional conditions) does there exist a compact set U C 2 with the property, given x e £ is there a <„ such that the orbit $t(x) c U if t > f„? In other words is there a compact set of attraction for the system? The idea would be to use (1.8).

968 210

S. SMALE

References [ 1 ] R. Abraham & J. Robbin, Transversal mappings and flows, Benjamin, New York, 1967. [ 2 ] F. Branin, The algebraic-topological basis for network analogies and the vector calculus, Generalized Network, Polytechnic Press, New York, 1966, 453-491. [ 3 ] R. Brayton, Nonlinear reciprocal networks, I.B.M. Report RC2606 (#42927), September, 1969. [ 4 ] R. Brayton & J. Moser, A theory of non-linear networks. I, II, Quart. Appl. Math. 22 (1964) 1-33, 81-104. [ 5 ] C. Desoer & E. Kuh, Basic circuit theory, McGraw-Hill, New York, 1969. [ 6 ] P. Hartman, Ordinary differential equations, Wiley, New York, 1964. [ 7 ] J. La Salle, Relaxation oscillations, Quart. Appl. Math. 7 (1949) 1-19. [ 8 ] S. Lefshetz, Differential equations; geometric theory, Interscience, New York, 1957. [ 9 ] N. Minorsky, Dynamics and nonlinear mechanics, Wiley, New York, 1958. [10] J. Palis & S. Smale, Structural stability theorems, Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968) Amer. Math. Soc., 1970, 223-231. [11] S. Smale Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967) 747-317. [12] M. E. Van Valkenburg, Network analysis, Prentice-Hall, Englewood Cliffs, N. J., 1955. UNIVERSITY OF CALIFORNIA. BERKELEY

969 A MATHEMATICAL MODEL OF TWO CELLS VIA TURING'S EQUATION

By S.SMAIE

University of California at Berkeley

1. Here we describe a mathematical model in the field of cellular biology. It is a model for two similar cells which interact via diffusion past a membrane. Each cell by itself is inert or dead in the sense that the concentrations of its enzymes achieve a constant equilibrium. In interaction however, the cellular system pulses (or expressed perhaps over dramatically, becomes alive!) in the sense that the concentrations of the enzymes in each cell will oscillate indefinitely. Of course we are using an extremely simplified picture of actual cells. The model is an example of Turing's equations of cellular biology [5] which are described in the next section. I would like to thank H. Hartman for bringing to my attention the importance of these equations and for showing me Turing's paper. The general idea of our model is first to give abstractly an example of a dynamical system for the chemical kinetics of four chemicals (or enzymes). This dynamics represents the reaction of these chemicals with each other and has the property that every solution tends to one unique stationary point or equilibrium in the space of concentrations as time goes to °°. This is the sense in which the cell is dead, where the cell consists of these four chemicals. After a period of transition, the chemical system stays at equilibrium. We emphasize that our reaction process is an abstract mathematical one and that we have not tried to find four chemicals with this kind of chemical kinetics. The next step is to give four positive diffusion constants for the membrane which could describe the diffusion of the four chemicals past the membrane. The cellular system consisting of the two cells separated by the membrane will be described by differential equations according to Turing. With our choice of the chemical kinetics and diffusion constants this new dynamical system will 17

970

18

8. SHALE

have a nontrivial periodic solution and essentially every solution will tend to this periodic solution. Thus no matter what the initial conditions, the interacting system will tend toward an oscillation (with fixed period). After an interval of transition, it will oscillate. Both the equilibrium of the isolated cell and the oscillating solution of the interacting system described above are stable (or are attractors) and even stable in a global way. But more than this, the equations themselves are stable so that any equations near ours have the same properties. Our dynamical systems are "structurally stable." This gives them at least a physical possi­ bility of occurring. In Turing's original paper some examples of Turing's equations are given with oscillation. However, these are linear and it is impossible to have an oscillation in any structurally stable linear dynamical system. Linear analysis can be used primarily to under­ stand the neighborhood of an equilibrium solution. Development of linear Turing theory has been carried very far in the very pretty paper of Othmer and Scriven [3]. Our example has reasonable boundary conditions, as one or more of the concentrations goes to 0 or to °°. Also, a complete phase portrait of the differential equation in eight dimensions for the cellular system is obtained. This example and Turing's equations as well go beyond biology. The model here shows how the linear coupling of two different kinds of processes, each process in itself stationary, can produce an oscillation. This is the coupling of transport processes (in this case diffusion across a membrane) and transformation processes (in this case chemical reactions). In ecology, Turing's equations have another interpretation; see, e.g., [2]. Also S. Boorman's Harvard Thesis has a related interpretation and analysis. Wefinallyremark that our results could equally well be interpreted as putting a single cell into an environment which could start it pulsating. 2. We give a brief description of Turing's equations [5]. These are sometimes called Rashevsky-Turing equations because of earlier work of Rashevsky on this subject.

971 MODEL OF TWO CELLS VIA TURING S EQUATION

19

One starts from a cell-complex, in either the biological or mathematical sense of the word, e.g., as given in Figure 1.

FlGUKB 1

From the mathematical point of view this system is a cellcomplex structure on a two- or three-dimensional manifold (e.g., an open set of Rz or R3). Suppose there are N cells and they are numbered 1, •••,iV. It is supposed that the cells contain enzymes (or chemicals, or "morphogens" in the terminology of Turing) which react with each other. Suppose there are m of these chemicals. Then the state space for each cell is the space P = {x6R"|x = (x 1 ,...,x-), x*£0, each i), where x* denotes the concentration of the ttn* chemical. The state space for the system under discussion is the Cartesian product PX ••• X P (N times) or (P)N. Thus a state for this cellular system is a point, x£(P)N, x-=(x t , •••,*,,) with each Xj£P giving all the concentrations for the »th cell, t = l, •••,N. The dynamics for the typical cell by itself is given by an ordinary differential equation on P; this can be described by a map R: P—» R" and dx/dt*=R(x). This R describes how the chemicals react with each other in that cell; the subject of chemical kinetics deals with the nature of R. Most typically, the dynamics of dx/dt — R(x) on P is described by the existence of a single equilibrium x £ P such that every solution tends to x; at least this will be the case if conservation laws have been taken into account one way or another (as in the situation in [5] or [3]).

972 8. SHALE

20

A natural boundary condition on this equation is that if xG P, *■= (*'»•• •,*"). with x*—0, then the *th component Rk(x) of R(x) is positive. So far we have discussed each cell in some kind of hypothetical isolation. The cells are separated from each other by a membrane which allows for diffusion from one cell to adjoining cells. In the simplest case of diffusion, if a certain chemical has a bigger con­ centration in the rth cell than an adjoining cell, then the concentra­ tion of that chemical decreases in the rth cell, at a rate proportional to the difference. This gives some motivation to Turing's equations which add this diffusion term to give an interaction between the cells.

(T)

^ -/?(*) + «*

£

*»(*-*>.

* - l , ••-.*.

i € wt of tafc •4oUn«ith oB

Let us explain (T) in detail. The first term above, R(xk), gives the chemical kinetics in the Ath cell. The 2nd term above describes the diffusion processes between cells. Thus Xi — xk E R" represents the difference of the concentrations of all the chemicals between the ith and Jkth cells. Here Mi* is a linear transformation from R" to R" or an m X m matrix. In the most natural simple case, and the case we develop here, M,-» is a positive diagonal matrix. Also the chemical kinetics for each cell is considered the same. (T) is a 1st order system of ordinary differential equations on the state space (P)N of the biological system, and will tell how a state' moves in time. We specialize to a case of 2 cells adjoined along a membrane which is the example pursued in the rest of the paper.

1st cell

}

FIOUBX2

2nd cell

973

MODEL OF TWO CELLS VIA TURING S EQUATION

21

dkb/A-.*<«,)+«(*,-*), dz2/dt^R(z1)+lt(zl-Z2).

( T ) 2

This is an equation on PxP will be of the form

with ( z i , % ) G P x P . Here M

with each Mi > 0. This is the most simple case. We may also write (T2) as given by a vector field X on P X P where X(ZUZ2) = ( « ( Z , ) + M & ~Zl),R(Zt)

+ M(*! - * 2 » .

3. Here we state our results. Let P = \z^fL\z = (z\zl,zi,z*),zi ^Q\. There exists a smooth (C~) map R:P—>R\ and M I , - - , , M 4 > 0 u>»t« fA« following properties (1), (2), (3) Wou>: (1) The differential equation dz/dt = R(z) on P is globally asymptotically stable and is structurally stable. In other words there is a unique equilibrium z£P of the differ­ ential equation and every solution tends to z as t-+ a>. That R is structurally stable means that the equation dz/dt = Ro(z) has the same structural properties as dz/dt = R(z) if RQ is a Cl perturbation of R. (See [l] for details on these kinds of stability and background on ordinary differential equations.) (2) On PxP with z = (zi.Zj) G ^ X P the differential system, MAIN THEOREM.

dzJdt^R(zl)+n(zl-zl), dz2/dt = R(zt)+„(zl-z2),

/Mi 0\ " ° \ 0 ' ' -M4/'

is a "global oscillator" and is structurally stable. More precisely, a global oscillator on Px P is a dynamical system which has a nontrivial attracting periodic solution y and except for a closed set 2 of measure 0, every solution tends to y as t—> <*>. In fact, here 2 is a six-dimensional smoothly imbedded cell and on 2 every solution tends to the unique equilibrium (z,z) of (T). (3) The boundary conditions are reasonable in the following way.

974

22

8. 8MALE

Let Zo-c(«t,z£,z3,a£)£P satisfy z*-=0 for some k between one and four. Then the kth component /?*(*>) of Rfo) is positive. The above theorem gives mathematical precision to the state­ ments made in $1 via the interpretation of (T) as in | 2 . The example is related to the phenomena of Hopf bifurcation; see for example [4]. Our analysis is more global however, and we have a complete description of the phase portrait. 4. In this section we show how to construct the differential equations of the previous section. Towards obtaining the vector field R:P—>R* of the Main Theoremof §3 we will find a C" vector field Q:R4—R4 on R4 and

with these properties: (1) Q has the origin, 0, as a global attractor for the equation dz/dt = Q(t) onR 4 . (2) There is a if > 0 such that if zER 4 , | z | £ K, then Q(z) - - z . (3) On R4 X R4 the vector field Ui,4)-»(2Qta) +M(«2 -*i).2Q(zi) +M(*i -«2» is a global oscillator. Once such a Q has been found we finish as follows: Choose some zEP so that z - z G P for all z with | z | £ K. Let R(z) = 2Q(z — z). Then R will have the properties of §3. This changes the question from P to R4 where we can use the linear structure systematically (even though our equations are not linear). To find a Q as needed above we first ignore property (2), or behaviour of Q at <*> and concentrate on (1) and (3). In fact we first find S:R 4 —R 4 satisfying (1) and (3) with S replacing Q and after that S is modified to satisfy (2). Toward constructing this S, observe that the set A — {(zlf 2») G R 4 XR 4 |zi —Zi) has the property that the vector field

975 MODEL OF TWO CELLS VIA TURING S EQUATION

(*)

(2S(zl) +ti(zi-zi),2S(zi)

23

+ „ ( * -%))

is tangent to A. That is, A is invariant under the flow and on A the flow is contracting to the origin. Now suppose S satisfies S( — z) = — S(z) or that S is odd. Then on A 1 ={U 1 ,z 2 )GR 4 XR 4 |2 1 = - 2 j } the vector field (•) is invariant and has the form z^S(z)

-v(z)

(up to a factor of 2). From these considerations we are motivated to seek an odd map S:R4—»R4 which satisfies the following: (Si) S has 0 as a global attractor and S — M is a global oscillator onR 4 . (52) A1 is an attractor for (*) on R 4 xR 4 . (53) Boundary conditions can be made good. The heart of the matter lies in (1); we consider that next. First consider the matrix a 0 ya 0 ] __ 0 a 0 70 M_ -ya 0 -2a 0 0 -ya 0 -2a] where a < - 1 and y/2 M2. MS> M«. This can be checked easily since M resembles a 2 X 2 matrix. Also there is a linear change of coordinates which changes the matrix ti = ( ' • .

) into H.

The coordinates of chemical concentrations are those in which n has the diagonal form. Thus the y, do not represent concentrations. However the y coordinates are much easier to work with. We give now 5 as the sum of a linear map Sj: R4 —»R4 and a cubic map S3 in terms of the ^-coordinates.

976 8. SHALE

24[

Thus let

S-SJ+SJ

with

/l+o -1

s,=

o\

1

TO

O

0 TO 2o 0 0 2a,

-TO

0

o

-TO

Sa(y)=(-(y1),,o,o,o). One notes now that S is odd and since the inner product (Sy.y) < 0 if y j*0 (an easy check) it follows that the origin of R4 is a global attractor for S. The next step is to check that S —J takes the form

/ / i n

Ui) o

.

\

/4o 0\ \ 0 4o/

+S,.

Thus the (y'.y2) 2-dhnensional subspace is a contracting in­ variant subspace in R4 for S — ji, since a < 0; on this subspace, the equations for S — J take the form dy'/dt = y - ((y')» - y 1 ) ,

dy 2 /* = - y 1 .

This is Van der Pol's equation (see [l]), which we know is a global oscillator. Thus S — n is a global oscillator on R4. The next step is to show that (82) is true. This can be proved along the following lines. The vector field X on R4 X R4 given by X(z) =(2S(2 t ) + M ( 2 2 -2,),2S(22) + H(z1 -22))

can be written in the form Yx(z) + Y2(z) where Yi(z) E A, Y2(z) E A x . Then YiW-WzJ

+S(zt),S(zi)

+S(zt)),

Yt(z) « (S(*,) -StoJ + M (2, - 2 , ) . - (S(*,) -S&) x

+„(**

That X points toward A follows from the Lemma.

-Z,)))-

977

MODEL OF TWO CELLS VIA TURING S EQUATION LEMMA. <S(ZI)

25

+S(%),2, + 2 J ) ^ 0 .

PROOF OF LEMMA.

Write S=Sl+S2.

We already know

(SM) +SAZt),2l +Z2> =<S,(Z, +2t),Zi +Zx) £ 0. But <S3(z,) +§,(2i),«, +22) = [- (y,,)J - (y,1)8] • (y,1 +yl) and that is ^ 0 since, for any real numbered and b, (a3 + 6s) (a + b) ^ 0. Finally one "straightens out" the flow of S outside some large ball. One uses a smooth function R+, 0 ^ ? ^ 1 ,