Smooth Dynamical Systems (Advanced Series in Nonlinear Dynamics)

maps

Proof. By the definition of flow equivalence (a x h)(D) <=• E, and it suffices to show that (a x h)(D) = E. But (a, h) induces a partial flow tp: (a x h)(D) -» Y on Y, and tp = tp on (a x h)(D). Since by Proposition 2.13 (ii) partial flows cannot be extended, the domain of tp equals the domain of ip. • Now let cp be any flow (or, indeed, partial flow) on X and let U be an open subset of X. Then, for all x € X, <S>~X (U) is open in R and hence a countable union of disjoint open intervals. Let Dx be the one containing 0. We call (px(Dx) the orbit component of through x (in U) (see Figure 2.16). We

FIGURE 2.16

denote by D the subset U x e uD* x {*} of R x [/ (see Figure 2.17), and define a map (p\U:D^U by (. By the same abuse, we call c/>\U the restriction of the flow 4> to the subset U. It is very easy to check that cp | U is a partial flow on U. Thus if ip is a flow on a topological space Y and V is an

FIGURE 2.17

V

LIMIT SETS OF FLOWS

41

open subspace of Y, then to say that <£ | U is flow or topologically equivalent to if/\ V means that they are equivalent as partial flows. If p e X and q e Y, we say that <\> is equivalent at p to tp at q if there exist open neighbourhoods U of p and V of q and an equivalence from \U to iff\V takingp to q. Once again, we write that , p):4>isa flow, p £ the phase space of<j>}. Ifcf> \p is flow equivalent to t//\q then \p is topologically equivalent to ip\q. O

V. LIMIT SETS OF FLOWS We now begin an investigation of properties of orbits of flows that are preserved under topological equivalence. An obvious example is the topological types of the orbits; for instance if the sets of fixed points of two flows have different cardinalities then we can say immediately that the flows are not topologically equivalent. Let and 4> be flows on topological spaces X and Y. Suppose that h is a topological equivalence from
FIGURE 2.19

42

EQUIVALENT SYSTEMS

CH. 2

the reverse flow. In order to handle these differences, we must analyse closures of orbits more carefully, and pick out those parts corresponding to "large positive t" and "large negative t". These limit sets will be the main objects of study for the rest of this chapter. Let /, denote the closed half-line [t, oo[. The co-set (or co-limit set) w{x) of a point x e X (with respect to the flow cp) is defined by

(2.20)

oj(x)=njAT).

Intuitively, co{x) is the subset of X that t.x approaches as f->oo. For instance, in the saddle point picture of Example 1.15, if x is on either of the two vertical orbits then t.x approaches the origin as r-»oo, and this is reflected in the fact that a)(x) = {0}. It follows immediately from the definition of flow that, for all teR, co(t.x) = co(x). Thus we may define the co-set by co (T) = co (x) for any xeT. Notice that if T is a fixed point or periodic orbit then x{I,) = F for all x e T and t e R, and so co(T) = T. Thus w ( 0 is not necessarily part of f \ r . (2.21) Exercise. Determine the w-sets for the orbits of the examples of flows in Chapter 1. For instance, show that any orbit of an irrationalflowon T2 has T2 as co-set. Notice that in the definition (2.20) one could equally well have taken the intersection over any subset of R that is unbounded above, for example N = {1, 2, 3 , . . . } . Notice also that a point y is in co(x) if and only if there is a real sequence (f„) such that t„-><x> and tn.x-*Y as n-*<x>. Similarly, we define the a-set a(x) of a point x eX by

«(*)=n«M/,) IER

where /, = ]—oo, t], and the a-set a(X) of an orbit T by a(T) = a(x) for any xeT. The a-set is the subset that t.x approaches as f-»-oo. To avoid continual repetition, we confine attention in the results that follow to w-sets, since the corresponding results for a-sets are always exactly analogous in statement and proof. In fact, we can give a precise formulation for this correspondence in terms of the reverse flow <j>~ of described in Example 1.26, as follows: (2.22) Proposition. Let T by an orbit ofcf> and let a'"(0 andco~(T) denote the a-set and co-set of V as an orbit of <j>~. Then a~(T) = co(T) and co (r) = a ( r ) . We now derive a few simple properties of w-sets (of a flow cj> on a topological space X unless otherwise stated). First we show that any topological equivalence maps w-sets onto o>-sets.

43

LIMIT SETS OF FLOWS

V

(2.23) Theorem. Let h:X^Y be a topological equivalence from to ift. Then, for each orbit Y of 4>,h maps w ( 0 onto co(h(T)), the co-set of the orbit h{Y)ofiP. Proof. Let A : R -* R be an increasing homeomorphism such that hx = t(fhMX where x e T. Then h (
Pi hcf>x(I,)

(since A is injective)

(since A is a homeomorphism)

ieR

= Pi tphuMh) = Pl ^Hxihu)) IER

(since A is increasing)

IeR

= 6,(ft(D).

•

(2.24) Proposition. Lef T Z>e an orfe/f o/ 0. Then co (r) « a closed subset ofX, andw(r)cf. D (2.25) Exercise. Show that if T is an orbit of a flow on X, then f = r u a (O u to (r). Deduce that if a (T) = co (D = 0 , then T is a closed subset of X. (2.26) Proposition. Let T and A be orbits of such that r c w ( A ) . Then a>(D c ' for all t e R). (2.27) Proposition. Any co-set of is an invariant set of cf>. Proof. We have to show that, for all s e R and for all p e co(T), s.peco(T), or, equivalently, that cf>s(co(r)) = co(T). Now, for any x e T,

4>s(o>(D) = s ( n 4Afi) VleR

/

= PI *<£*(/,) reR

«ER

Thus, for instance, if co(T) is a single point q, then a is a fixed point of 4>. As we have seen in the examples of Chapter 1, an orbit may have an empty <w-set. This phenomenon seems to be associated with the orbit "going

44

EQUIVALENT SYSTEMS

CH. 2

to infinity" so it is reasonable to suppose that if we introduce some compactness condition we can ensure non-empty w-sets. What can we say about connectedness of w-sets? At first sight it seems plausible that co(T) inherits connectedness from I\ However, with a little imagination one can visualize a flow on R 2 having orbits with non-connected w-sets. If we wish to ensure connected w-sets, the simplest answer is, once again, a compactness condition. In considering these, and later questions, it is convenient to quote two purely topological lemmas, whose statements and proofs are given in the appendix to this chapter. (2.28) Proposition. Let K be a compact subset ofX, such that, for all n e N, 4M)nK*0. Then co(x)nK # 0 . Proof. For all n s N , F„=4>x(In)nK is a closed subset of the compact subset K and hence is compact. By Lemma 2.44 of the appendix, C\Fn is non-empty. • (2.29) Proposition. Let Kbea compact subset ofX, such that, for some reN, x(Ir)(=K. Also x(In+r). • (2.30) Corollary. If X is compact Hausdorff then, for any orbit F, co(F) is non-empty, compact and connected. D (2.31) Proposition. Let X be compact. Then for any neighbourhood U of w(x), there exists n e N such that 4>x(In)c U. Proof. We may suppose that U is open, whence X\U is closed and compact. The result now follows immediately from Proposition 2.28 with K = X\U. • This is one of several results that we shall improve upon in the appendix when we discuss compactification (see Exercise 2.61). (2.32) Note. Suppose that U and V are open subsets of X and Y respectively and that h : U -> V is a topological equivalence from | U to if/\ V. We should like to know that h takes w-sets of orbit components of onto w-sets of orbit components of ip. There are various awkward points to contend with. For example, an orbit in U may have part of its w-set outside U. Again, an orbit wholly in U may be mapped by h to a component of an orbit that leaves V. However, suppose that X and Y are Hausdorff, that K is a compact (and

LIMIT SETS OF FLOWS

V

45

hence closed) subset of U, and let T be an orbit of cf> such that, for some x e F and t e R, x(It)c K. Thus, by Proposition 2.29, co(F) is a non-empty subset of K. One may prove that if, by abuse of notation, we denote by h (F) the orbit of if/ containing h<j>x(I,), then the orbit component of h(F) containing hcf>x(I,) contains iphw(It') for some f'eR, and a>(h(T)) = h(co(F)). We commented earlier that if F is a fixed point or periodic orbit then co (F) = T. We now investigate this property more thoroughly. (2.33) Lemma. Let X be Hausdorff and let F be a compact orbit of cp. Then

«(r) = r. Proof. The set F is closed in X. Hence, by Proposition 2.29, co(F) is a non-empty subset of F. Since it is an invariant set, it must be the whole of

r.

•

(2.34) Proposition. Let X be compact and Hausdorff. Then an orbit Fofcf) is closed in X if and only if co (F) = F. Proof. Necessity is a special case of Lemma 2.33. If &>(r) = r then, by Proposition 2.24, F is closed. • (2.35) Theorem. Let X be Hausdorff. Then an orbit Fofcf) is compact if and only if it is a fixed point or a periodic orbit. Proof. Sufficiency is immediate. Suppose, then, that F is compact and that F is neither a fixed point nor a periodic orbit. Then, for any x e F, x is injectiye. For each n e N, C„ = x ([-n, n]) is a compact subset of X, and hence closed. Moreover, co(F) = F by Lemma 2.33, and so, for allpeF, each neighbourhood of p contains points in cf>x(In+1)cF\Cn. We may therefore apply Lemma 2.45 of the appendix with Y = A = F, and deduce that F is not locally compact, which is a contradiction. • (2.36) Corollary. Let X be compact and Hausdorff. Then the following three conditions on an orbit Fofcf) are equivalent: (i) T is a closed subset of X, (ii) T is a fixed point or a periodic orbit,

(iii)«(r) = r.

•

As we commented in Chapter 1, this result partially justifies the use of the term "closed orbit" as a synonym for "periodic orbit". See Exercise 2.63 of the appendix for a generalization of Corollary 2.36 to non-compact X. (2.37) Exercise. A minimal set of a dynamical system is a non-empty closed invariant set that does not contain any closed invariant proper subset. Use Zorn's Lemma (see Lang [2]) to prove that if, for any orbit r of , F is compact then it contains a minimal set. Give an example to show that F may have a non-empty co-set that is not minimal.

46

EQUIVALENT SYSTEMS

CH. 2

VI. LIMIT SETS OF HOMEOMORPHISMS The theory of a- and w-limit sets may also be developed in the context of discrete dynamical systems. I f / i s a homeomorphism of a topological space, then the (x) of xsX with respect to / is denned by

neN

The a-set a(x) of x is the w-set of x with respect to / _ 1 . All results of the previous section have analogues, with the one obvious exception that w-sets of homeomorphisms need not be (and seldom are) connected.

VII. NON-WANDERING SETS The fundamental equivalence relations in the theory of dynamical systems are indisputably topological equivalence (for flows) and topological conjugacy (for homeomorphisms). However, when the going gets rough in the classification problem, one tends to cast about for a new (but still natural) equivalence relation with respect to which classification may be easier. We shall describe some attempts in this direction in Chapter 7. One of these, generally accepted to be the most important, is concerned with a certain invariant set known as the non-wandering set, and now is a suitable time to explain this concept. The definition is due to David Birkhoff [1] and the logic behind it is as follows. If one compares phase portraits of dynamical systems, for example the two in Figure 2.38, it seems that certain parts are qualitatively more important than others. If one were asked to pick out the significant features of the left hand picture, one would inevitably begin by mentioning the fixed points and closed orbit. Generally speaking, qualitative features in a phase portrait of a dynamical system can usually be traced • >

•*•

-» > • > • >

-»•

-» -» FIGURE 2.38

•

VII

NON-WANDERING SETS

47

back to sets of points that exhibit some form of recurrence. The strongest form of recurrence is periodicity, where a point resumes its original position arbitrarily often, but there are weaker forms that are also important. One uses the technical term recurrent point for a point that belongs to its own w-limit set. For example, all points of the torus T2 are recurrent with respect to an irrational flow (Example 1.25), although none are periodic. By definition, a point is recurrent if and only if, for all neighbourhoods U of x, t.xeU for arbitrarily large t e G. We define x to be non-wandering if, for all neighbourhoods U of x, (t. U) n U is non-empty for arbitrarily large t e G. Thus we have a form of recurrence that is weaker than technical recurrence. To see that it is strictly weaker, observe that any point of the non-recurrent orbit T in Figure 2.39 is a non-wandering point. Rather more importantly

FIGURE 2.39

(because it is in a stabler configuration) consider any point x ¥= 0 on the stable manifold of the fixed point 0 of the hyperbolic toral automorphism (Example 1.30). It is clearly not recurrent, since u> (x) = {0}, but it is non-wandering, since any neighbourhood of it contains periodic points. The term "non-wandering point" is an unhappy one, since not only may the point wander away from its original position but, as we have seen, it may never come back again. K. Sigmund has put forward the attractive alternative nostalgic point, for although the point itself may wander away, its thoughts (represented by U) keep coming back. Non-wandering points are also called Cl-points, and the set of all non-wandering points of , the non-wandering set of cf>, is denotedft( is discrete a n d / = (p1). (2.40) Exercise. Prove that for any homeomorphism / of X, n ( / _ 1 ) = Cl(f). Note the distinction between w-sets and the ft-set. All limit points are fl-points, but the converse is false, as the following exercise shows: (2.41) Exercise. Prove that if y ea(x) for some x e X, then y eCL((f>). Sketch the phase portrait of a flow on R 2 such that (i) Cl() = R 2 , but (ii) for some x e R , x is neither an a-limit point nor an w-limit point of any orbit of <j>.

48

EQUIVALENT SYSTEMS

CH. 2

The following result sums up some elementary properties of fl-sets: (2.42) Theorem. For any dynamical system on X, Cl(cf>) is a closed invariant subset of X, and is non-empty if X is compact. Topological conjugacies and equivalences preserve il-sets. Proof. The complement of Cl(4>) is open in X, for if x has an open neighbourhood U such that (r. U) n U is empty for all sufficiently large t, then so has every point of U. Thus Q,() is closed in X. Moreover, for all s E G, X has such a neighbourhood U if and only if s.x has a neighbourhood (namely s. U) with a similar property. Thus X\Q,(4>), and hence also Cl(), is an invariant set. If X is compact, then, by Proposition 2.29, any orbit of has a non-empty w-set, and by Exercise 2.41, this is part of fl(). Finally, suppose that h : X -* Y is a topological conjugacy or equivalence from to a dynamical system ip on Y. Let p e fl(). Let V be a neighbourhood of q = h(p) in y and let t0 £ G. We have to prove that, for some t g r0, and some y &V,t.y eV. This is trivially true if V n (f0. V) is non-empty, so suppose that it is empty. Let h~1(t0.q) = t\ .p. By continuity of 4>, there is a neighbourhood U of p in h~l{V) such that ti. U a h~l(t0. V). Since p e fl(), there exists xe U and ( 2 e G with t2>ti, such that t2.xe U. Let & (fi. x) = /o • y. Then y € V. Also, since h is orientation preserving, h (t2 • x) = f. y for some f > t0, and, since ( 2 . x e [ / , ( . y e V (see Figure 2.42). •

FIGURE 2.42

The new equivalence relations mentioned at the beginning of the section, are called ^-equivalence (for flows) and Cl-conjugacy (for homeomorphisms). They are just the old ones, topological equivalence and conjugacy, restricted to fl-sets. Thus if \Cl() denotes the restriction of the flow <j> to n(), defined by (4>\tl())(t, x) = (t, x) for all (t, x)eRxQ,(4>), then <j> is

VII

NON-WANDERING SETS

49

Cl-equivalent to ij/ if and only if \ £l{4>) is topologically equivalent to if/1 fi(
Appendix 2 In this appendix we prove the two topological results quoted in the main body of the chapter, we further discuss some points arising from orientation of orbits, and we extend results about limit sets from compact to locally compact spaces by the process of compactification.

I. TWO TOPOLOGICAL LEMMAS Let X be a topological space. A sequence (F„)„ eN of subsets of X is decreasing if, for all m, n e N, m > n implies Fm <=• Fn. For the definition of increasing, reverse the inclusion. Thus, in the above context of w-sets, (4>x(In))neTi is a decreasing sequence. (2.44) Lemma. Let (Fn) be a decreasing sequence of closed compact, nonempty subsets ofX. Then F = D„G^Fn is non-empty. If further, Xis Hausdorff and each Fn is connected, then F is connected. Proof. Without loss of generality we may assume that X =FU and hence that X is compact. Suppose that F = 0 . Then {X\F„}neN is an open cover of X. Since X is compact, this cover has a finite subcover. But X\Fm => X\Fn for all m^n. Hence X=X\Fno for some n0
II

ORIENTED ORBITS IN HAUSDORFF SPACES

51

dense open subsets of the space is a dense subset. Baire's theorem asserts that locally compact Hausdorff spaces (and complete metric spaces) are Baire spaces. We give a direct proof of an equivalent but slightly more technical statement which is particularly convenient for our applications. (2.45) Lemma. Let the topological space Y be the union of a finite or countably infinite sequence (C„)n>o of closed subsets. Let A be a non-empty subset of Y such that, for all as A and for all « > 0, a e A\C„. Then Y is not locally compact Hausdorff. Proof. Suppose that Y is locally compact Hausdorff, and hence regular. We choose inductively a sequence (an) of points of A and a decreasing sequence (F„) of compact closed neighbourhoods F„ of an, such that, for all n > 0 , C„ nFn is empty. Then, by Lemma 2.44, f~)Fn is non-empty, and yet, by construction, it contains no point of Y = U C , which is a contradiction. We may as well start the induction by taking C 0 = 0 , a0 = any point of A and F 0 = any compact closed neighbourhood of a0. Suppose that an-\ and F„-i have been constructed. Then int F„^i contains a point a„ of A\Cn. Choose Fn as any closed neighbourhood of an in (int F„_i) r\(Y\Cn). • II. ORIENTED ORBITS IN HAUSDORFF SPACES Let and if be flows on topological spaces X and Y respectively, and let h : X -* Y be a homeomorphism mapping orbits of onto orbits of t//. Recall that h preserves the orientation of an orbit r of 4> if there is, for some x e F, an increasing homeomorphism a :R-»R, such that, for all f e R , h(t.x) = a(t).h(x). Similarly, h reverses the orientation of T if there is a decreasing homeomorphism /3 : R-* R such that h(t.x) = fi(t).h{x). According to these definitions, h may both preserve and reverse the orientation of I\ (2.46) Exercise. Prove that h both preserves and reverses the orientation of r if and only if T is a fixed point. A more worrying possibility is that h may neither preserve nor reverse the orientation of I\ (2.47) Example. Let be the flow given by (t, x) = x +1 on the space real numbers with indiscrete topology. Then the topological equivalence h : R^ -* R^ from <j> itself defined by h(x) = x

for

x^±l,

fc(±l)==Fl

neither preserves nor reverses the orientation of the unique orbit of (f>. The situation revealed by this example is, of course, pathological, and we shall show that the phenomenon cannot occur in Hausdorff spaces. The

52

APPENDIX

CH. 2

proof is rather tricky; a much easier result of the same type is: (2.48) Exercise. Prove that if T is a closed orbit of then h either preserves or reserves the orientation of T. (Hint: See the first part of Exercise 2.4.) We begin by proving a slightly off-beat property of real intervals (or, more generally, of connected, locally connected, locally compact, Hausdorff spaces). (2.49) Lemma. Let I be a real interval. If lis the union of a sequence (C„) of two or more disjoint non-empty closed subsets, then the sequence is uncountably infinite. Proof. Suppose that (C„)„ >0 is a finite or countably infinite sequence with the given property. Let An be the frontier of C„ in /. Then An is a non-empty (because / is connected) subset of C„. Let A = yjA„, and let aeA„ for some n. Then, for all m ^ n, a e A\Am. We prove that a e A\An. Let V be any neighbourhood of a in I. Then V meets C„ and also C m for some m^n. We may assume that V is connected, in which case, since V meets both Cm and its complement in /, V meets the frontier Am of Cm. Since Am <= A\A„, we conclude that a e A\A„. Lemma 2.45 now gives a contradiction. • (2.50) Theorem. IfXis a Hausdorff space and the homeomorphism h:X ^Y maps an orbit T of <j> onto an orbit of tj/, then h either preserves or reverses the orientation of F. Proof. The theorem is trivial if T is a fixed point, and is just the result of Exercise 2.48 if T is a closed orbit. It remains to prove the result when x is a continuous injection. In this case, by Theorem 2.35, i//hM is also a continuous injection, and therefore it induces a continuous bijection from R onto h(T). Let (i :/i(r)-»R be the inverse of this bijection. The map n is not necessarily continuous; nevertheless, we assert that A = /u,h<j>x :R-»R is a homeomorphism. This latter statement implies, of course, that A is increasing or that A is decreasing, and the conclusion of the theorem follows immediately. To prove our assertion about A, it is enough to show that A is continuous at each t e R. Let / be a compact interval such that t e int /. Then hx(J) is a compact subset of the Hausdorff space Y, and is therefore closed in Y. Since tjjh(X) is continuous, A (/) = K, say, is closed in R. In fact, we shall prove that K is compact. This implies that 4ihM maps K homeomorphically onto hx(J), hence that A maps / homeomorphically onto K, and in particular that A is continuous at t. Suppose first that k contains an unbounded interval, say [a, oo[. Then, by Proposition 2.29, x(J). Hence, using Proposition 2.27, hx(J) = h(T)-h(f>x(R). But hx is injective and

II

ORIENTED ORBITS IN HAUSDORFF SPACES

53

/ # R, so we have a contradiction. Similarly, K can contain no unbounded interval of the form ]—oo, b[. Thus there exists an increasing sequence (an)nez of points of R\K such that a„ -> oo as n -* oo and an -» -oo as n -» -oo. Thus K is the union of the sequence (K n [a„-u an])neZ of disjoint compact sets. These are mapped homeomorphically by A ~ into compact sets whose union is / . It follows from Lemma 2.49 that K is a single compact set contained in [a„_i, an] for some n. • Note that a similar theorem holds when h : U -» V is a homeomorphism between open subsets U of X and V of F, and both T and / i ( 0 are orbit components. The difficulty in proving the above theorems stems from the fact that the continuous bijection 4>x : R -* X is not necessarily an embedding. To see that it is not, consider any irrational flow on the torus (Example 1.25). One has, in fact, the following simple criterion for x to be an embedding. (2.51) Theorem. Let Xbe Hausdorff, and let c/> have stabilizer {0} at x. Then x:R^>X is an embedding if and only if its image V is locally compact. Proof. We may as well take X = T. Necessity is trivial. For sufficiency, we have to prove that the inverse of the continuous bijection ( ^ : R - » r is continuous. Suppose not. Let F be a closed subset of R such that <j>x (F) is not closed. Let p be a point of x(F)\x(F). For all n > 0, x maps F n \—n, n] homeomorphically onto a closed subset. Hence p is in the closure of either 4>x(Fr\[n, oo[) or x(Fn]-oo, - « ] ) . We deduce that p is in either a(T) or w(r), say the latter. Thus w(r) = T. This is impossible, by the proof of Theorem 2.35. • (2.52) Exercise. Make the suggested generalization of Lemma 2.49. (2.53) Exercise. Let G be a second countable (i.e. having a countable basis for its topology), locally compact topological group acting on a Hausdorff space X. Prove that, for any x e X, the map G/ Gx -» G. x taking gGx to g. x is a homeomorphism if and only if G.x is locally compact. (Hint: either generalize the proof of Theorem 2.51 or prove directly that the map is open, using a version of Baire's theorem, as, for example, in § 3 of Chapter 2 of Helgason [1]). (2.54) Exercise. Let G and H be second countable, locally compact, locally connected topological groups acting freely (i.e. with trivial stabilizers) and transitively (i.e. with only one orbit) on a Hausdorff space X. Prove that, for all x e X, the map tp~ 1x : G -* H is a homeomorphism (where <$> and 4> are the actions). The w-set argument in the proof of Theorem 2.50 does not seem to generalize easily, but it can be replaced, as in Irwin [1].

54

CH. 2

APPENDIX

III. COMPACTIFICATION Let X be a non-compact topological space. There is a standard procedure for associating with X a compact topological space, called the one-point compactification X* of X. Let oo denote some point not in X. We define the set X* to be X u {oo}. To turn X* into a topological space, we define a subset U of X* to be open in X* if and only if either U is an open subset ui X or X*\ U is a closed compact subset of X It is an easy exercise to verify that X* is indeed compact, and that it is HausdorfT if and only if X is locally compact and Hausdorff (see Proposition 8.2 of Chapter 3 of Hu [1]). (2.55) Example. Let X = Rn. Then X* is homeomorphic to 5", the unit sphere in R" + . A particular homeomorphism h may be constructed by mapping R", identified with the hyperplane x„ + 1 = — 1 of R n + 1 , onto Sn\{en+i} by stereographic projection from e„+i, and oo to en+i, where e n + 1 = ( 0 , . . . , 0,1). See Figure 2.55. « „ t l = A(flo)

FIGURE 2.55

Now let 0 be a dynamical system on X. We define the one-point compactification of to be the map (f>* : G x X*^X* defined by (g, x) if x ^ oo, and (g, oo) = oo. For instance the phase portraits of 4>* for the flows of Examples 1.8 and 1.9 are illustrated in Figure 2.56 (identifying R* with S 1 , as above).


{t,x)*x*t

FIGURE 2.56

Ill

COMPACTIFICATION

55

(2.57) Exercise. Sketch the phase portrait of * near oo for the flows of Examples 1.14-1.17. We now prove that * is a dynamical system on X*. This is trivial if G = Z, so we consider the flow case. (2.58) Lemma. Let $ : R x X ^ R x X be defined by <£>{t,x) = (t,4>{t,x)). Then <J> is a homeomorphism. Proof. The inverse _1(f, y) = (t, (-t, y)). Moreover if 77i and 7T2 denote projection of R x X onto its factors, then 771$ = 771 and 7T-24> = (/>. Since both ir\ and are continuous, it follows that <& is continuous. Similarly, «J>_1 is continuous. • (2.59) Theorem. The one-point compactification cf>* of a dynamical system on X is a dynamical system on X*. Proof. It is immediate that * satisfies conditions 1.1. We have to prove that *(t, x) in X*, there exists an open neighbourhood U of (t, x) in R x X * such that <£*([/) c V. There are two cases to consider. (i) Suppose x € X. Then we may assume that V is open in X. Thus U = ((j,*y\v) = 4>~\V) is open in R x X and hence in RxX*. (ii) Suppose that x = 00. The set V is an open neighbourhood of 00 in X*, and so F = X*\ V is a compact closed subset of X. Let if = f 1 ( F ) n ( [ ( - U + l ] x I ) = 4rx([>-l,? + l]xF), where <1> is the homeomorphism of Lemma 2.58. Hence A" is a compact closed subset of R x X, and Tr2(K) is a compact subset of X. Moreover, since TTI(K) is compact, ^Wii^xX is proper, and so ir2(K) is closed. Let W = X*\TT2(K). Then W is an open neighbourhood of 00 in X*. Finally, we take C/ = ] r - l , f + l [ x W, and check that <(>*(U)<=V. • One-point compactifications of spaces and dynamical systems are useful in extending results on compact spaces to locally compact spaces. One applies the known theory to * and deduces properties of . It is immediate that any orbit T or <j> is also an orbit of cf>*. It may happen, however, that w*(T) (the co-set of T regarded as an orbit of *) contains the point 00. For instance, this is always so when w(T) = 0 . (2.60) Exercise. Prove that w(D<=o,*(r)c w(r)u{oo}. Deduce that if *(T) = {00}.

56

APPENDIX

CH. 2

(2.61) Exercise. Let X be locally compact Hausdorff and let *(T) =
(T) has a compact connected component W, then «(r) = W. (2.63) Exercise. Prove that if X is locally compact Hausdorff then the following four statements are equivalent: (i)ft»(n = r,

(ii) w *(0 = r, (Hi) T is compact, (iv) T is a fixed point or a periodic orbit.

CHAPTER 3

Integration of Vector Fields

At the end of the introduction we said a few words about the motion of a liquid whose velocity at any point of space is independent of time. This example was used to motivate the axioms for a flow. Conversely, given a flow 4> on a space X, we can recover a notion of velocity at a point provided that X has the structure of a differentiable manifold and has a certain degree of smoothness. The velocity of cf> is a vector field on X (a fact that is often obscured in Euclidean space, which is commonly identified with its own tangent bundle). In this chapter we discuss existence and uniqueness of a flow whose velocity is a prescribed vector field. The local problem is, using a chart, equivalent to the problem of existence and uniqueness of solutions (integral curves) of a system of ordinary differential equations. In terms of the moving liquid model, an integral curve is the path of a given particle of fluid. Picard's theorem deals with the local existence of such integral curves. The idea behind its proof is as follows. One chooses at random a curve y 0 (a continuous map from time to position in space). Taking the velocity of the liquid at y0(t) for each t gives a vector valued function v of time. Now consider a journey with the same starting point as y 0 but with velocity v(t) at time t. Let yx be the path of this journey (see Figure 3.1). Applying this process again with y 0 replaced by yi we obtain a new curve y2, and so on. The curves form an infinite sequence (y,). At each stage we modify the previous curve to fit in with the velocity of the fluid along it, so it is not surprising that, for large i, y, approaches an integral curve. What we have described is rather like holding onto one end of a light thread while the thread gradually drifts into streamline position, but it is a discrete rather than a continuous process. Convergence of such a sequence of paths may be thought of in terms of convergence of the sequence (y,(f)) of positions, for all times t. A more fruitful approach, however, is to consider convergence in an infinite 57

58

INTEGRATION OF VECTOR FIELDS

CH. 3

Fluid flow

FIGURE 3.1

dimensional Banach space, each of whose points is a continuous map from time to position. We shall make use of map spaces at many crucial stages in this and in later chapters. This is not sophistication for its own sake; it leads to considerable simplifications of proofs. For example, most of the complications in the classical proof that the integral of a smooth vector field is a smooth function of initial position vanish once we have at our disposal a few elementary results on map spaces. We describe the requisite theory in Appendices B and C. The latter contains a version of Banach's contraction mapping theorem. This extremely useful tool is applied to avoid longdrawn-out successive approximation arguments, for example in Picard's theorem and in other parts of the integration theory below.

I. VECTOR FIELDS Let AT be a differentiate manifold, and let / be a real interval. A vector field on X is a map v: X -» TX associating with each point x o f X a vector v{x) in the tangent space TxX to X at x. We think of TXX as the space of all possible velocities at x of a particle moving along paths on the manifold X. Thus we have in mind a picture like Figure 3.2, where X is the submanifold S2 of R 3 and TXX is embedded in R 3 as a plane geometrically tangent to S at x. However we really need an intrinsic definition of TXX, independent of any particular embedding of the manifold X in Euclidean space. This is given in Appendix A, together with further notes on vector fields. Notice that a particle moving along / can have any real velocity at t e /, and so T,I is a copy of the real line. In fact one may identify T,I with {/} x R in / x R, and hence

VECTOR FIELDS

FIGURE 3.2

identify 77 with J x R. Similarly we show in Example A.31 that if U is an open subset of Euclidean space R" then TU may be identified with U x R". In this case, if v is a Cr vector field on U then the first component of v{x) with respect to the product is always x, signifying that v{x) is in the tangent space at x. Thus v(x) = (x, f(x)) for some C map f:U -» R". Conversely, any Cr map / : £ / - » R" corresponds to a C vector field {id, f) on U. The map / that determines and is determined by v in this way is called the principal part of v. It is common, and usually perfectly harmless, to blur the distinction between vector fields and principal parts. We shall resist the temptation to do so, however, at least for the rest of this chapter. Note that all the above remarks hold equally well when R" is replaced by any Banach space E. (3.3) Example. {Gradient vector fields). Let f:X->R be a Cr+ function (/•>0), and suppose that X has a Riemannian structure (see Appendix A). Then we may associate with the linear tangent map Tf: TX -» 7"R a C vector field V/ (or grad /) on X, called the gradient vector field of /. Example A. 57 of Appendix A describes in detail the construction of V/. If X = R", fix) has principal part {df/dxi,..., df/dx„). Thus, for example, if /(JCI, x2) = x\x2, then Vf{xu x2) has principal part {2x\x2, x\). If X = S2, embedded as

FIGURE 3.3

60

INTEGRATION OF VECTOR FIELDS

CH. 3

the unit sphere in R 3 , a n d / is the negative height function f(xu x2, x3) = then Vf(x) is tangent to the path taken by a raindrop on S2, that is to say line of steepest descent through x, which is a meridian of longitude. Figure illustrates this, and also V/ for / the negative height function on embedding of T2 as a tyre inner tube.

-x3, the 3.3 the

(3.4) Example. Two rather trivial vector fields needed later are the zero vector field on X, x >-^0x, where 0X is the zero vector in TXX (0X is identified with the point x itself in Figure 3.2), and the unit vector field on 7, fi-»l ft where 1, is the vector (t, 1) in r / = {r}xR. Now consider the product manifold IxX. At any point (t,x), the velocity of a moving'particle has components in the 7 and X directions, so we may identify T(I x X) with 77 x TX (see Exercise A.33). We denote by u the unit vector field on IxX in the positive t direction. That is to say, u(t, x) = (1„ 0 J (see Figure 3.4).

x

i

1

FIGURE 3.4

II. VELOCITY VECTOR FIELDS AND INTEGRAL FLOWS Let 4>:IxX-+X be any map such that <j>':X^X is a homeomorphism for all 161. Given x e X and t e7, let y = (<£')"^x). Then y(t) = <j>(t, y) = x. If we regard 4>y: I -* X as determining the position of a particle moving on X, then the particle reaches the point x e AT at time /. Now suppose that the function (f>y is differentiable at t. Then the velocity of the particle when it reaches x is the vector \ (t) of TXX (see Example A.37). We call this vector the velocity of at the point x at time t. Thus, for example, if is defined for t > 0 and x e R by with respect to t at (t, y) gives the velocity of (j> at x at time t as the vector (1/t) sinh" 1 x cosh (sinh -1 x) in the tangent space to R at x. Note that if is C 1 then its velocity at x at time t is (T)u(t, y), where « is as above.

II

VELOCITY VECTOR FIELDS AND INTEGRAL FLOWS

61

We shall show below (Theorem 3.9) that of is a flow on X then its velocity at x is the same at all times t. Thus, in this case, cj> gives rise to a vector field v, called the velocity vector field of/, where v(x) is the velocity of (j> at x at any time f. Since 4>x(0) = x, the simplest formula for v is i>(x) = <j>'x (0). For example, if AT = R and <£ is the flow / . x = (x 1 / 3 + tf, we differentiate with respect to t at t - 0, and obtain u(x) = (x, 3x 2 / 3 ). If is C 1 then v(x) = (T(f>)u(0, x), so the diagram T(RxX)

—

RXX commutes. If <£ is C(r>

* TX

*X

1), then t> is C _ 1 .

(3.5) Exercise. Find the velocity vector fields of the flows in Examples 1.14-1.17 and Exercise 1.18. Conversely, if v is a given vector field on X, we call any flow

, and say that v is integrable if such a flow exists. (3.6) Example. Let X = R and let v:X-+TX = R2 be the vector field v(x)~ (x, - x ) . Then : R x X -»X given by ^(t, x) = xe~' is an integral flow of 4>.

(3.7) Exercise. Show that the zero vector field on any manifold X has the trivial flow as its unique integral flow. (3.8) Exercise. Find an integral flow of the unit vector field u on R x X defined above. (3.9) Theorem. Let 4> be a flow on X such that, for all x eX, the map 4>x: R -* X is differentiable. Then the velocity of at any point is independent of time. Thus 4> has a well defined velocity vector field. Proof. We prove that the velocity at x at time t equals the velocity at x at time 0. Let y = 4>~'{x). Then, by the basic property of flows <j>y{u) = x(-t + u) for all u € R. Differentiating with respect to u at u = t, using the chain rule for the right-hand side, we obtain 4>'y(t) = , which is itself

62

INTEGRATION OF VECTOR FIELDS

CH. 3

C. The technique used in the proof of this result can be visualized as gluing is a flow on X and v is the velocity vector field of (f>, then the orbit of through x e X is the image of an integral curve of v at x. A local integral of v is a map :I xU->X, where I is an interval neighbourhood of 0 and U is a non-empty open subset of X, such that, for all x e U, 4>x: I -* U is an integral curve of v at x (see Figure 3.10). For all x e [ / , w e call a local integral atx, and say that v is integrable atx if such a local integral exists. If it does so, and is C 1 , then the diagram T(I x U)

—

IxU

* TX

*X

commutes. If further I = R and U =X then the local integral is, as we shall show later (Corollary 3.38) a flow on X, and is hence an integral flow of v. The term local integral should not be confused with the term local first integral. See the appendix to this chapter for a discussion of first integrals. I*U

/,fjr}

(0,x)

\o}xU

FIGURE 3.10

In investigating the existence of local integrals, we may restrict our attention to vector fields on (open subsets of) Banach spaces. For suppose that we wish to show that a vector field v on a manifold X is integrable at a

HI

ORDINARY DIFFERENTIAL EQUATIONS

63

point x. We take an admissible chart £: V -» V <= E at x, where X is modelled on E. The C°° diffeomorphism £ induces, from u| V, a vector field £*(t>) on V , which is integrable at £(x) if and only if v is integrable at x. This is a consequence of the following result: (3.11) Theorem. Let v be a vector field on X, and let h:X-*Y be a C diffeomorphism. Ifyis an integral curve ofv then hy is an integral curve of the vector field h*(y) = (Th)vh'1 induced on Y from v by h. If <j>:Ix U->X is a local integral of v at x, then h(id x (h\U)~ ) is a local integral of h^iv) at h(x). Proof. The diagram 77

*TX

I

*TY

*X y

*Y h

commutes, where t is the positive unit vector field on /, and T(hy) — ThTy. Thus hy is an integral curve of h*(v). The proof of the second statement is similar. •

III. ORDINARY DIFFERENTIAL EQUATIONS Let V be an open subset of a Banach space E, and let v be a Cr vector field on V(r s= 0). Suppose that :I xU-> V is a local integral of v. Let us make explicit the relation so established between and the principal part / of v. According to the definition of local integral, for all (t,x)eIxU, (3.12)

y'(t) = v{t, x) = ((t, x), f(t, x)).

Hence Di4>(t, x)(l) = fcj>(t, x). We make the usual abuse of notation that identifies a linear map from R to E with its value at 1, and write (3.13)

£>! =f<j>.

Conversely, any map 4>: I x U -* V satisfying (3.13) also satisfies (3.12). We are led to make definitions of local integrals and integral curves of any map / : V->E in the obvious manner. That is to say, 4>:IxU-*V is a local integral of / at x e U if is the inclusion and Di4> = /<;6 on / x [/. An integral curve of / is a C 1 map -y:/-» V such that y' =fy on /. We say that y is an

64

INTEGRATION OF VECTOR FIELDS

CH. 3

integral curve at x if 0 e int / and y(0) = x. Trivially: (3.14) Proposition. A map is a local integral (resp. integral curve) of a vector field v on V if and only if it is a local integral (resp. integral curve) of the principal part of v. • We observe that the relation y' = fy implies that any integral curve of a C° vector field is C 1 . More generally, we obtain, by induction, the following result: (3.15) Proposition.Any integral curve of a C vector field is Cr+1.

D

In classical notation, with E = R", the relations D\4> =/ and y' =fy are both written in the form dx/dt = f(x), which is called an autonomous system of first order ordinary differential equations. Here the term "autonomous" refers to the fact that the variable t is not explicitly present on the right-hand side. It is a "system of equations" since there are n scalar equations of the form dxildt=fi(x\,... ,xn). The function y is called a solution of the equations, and 4> is described as a solution regarded as a function of initial conditions, or general solution. (3.16) Example. Let v be the vector field on R 2 with principal part / : R -»R given by f(x, y) = (—y, x). The corresponding pair of scalar differential equations is dx _ dt

dy dt

The solution x = cos t, y = sin t of these equations is, in our parlance, the integral curve y: R -» R given by y(:) = (cos t, sin t). The general solution x = A cos t-B sin t, y = A sin t + B cos t, where A and B are "arbitrary constants", is in our parlance, the local integral (and, indeed, integral flow) <£:RxR 2 ^R given by (t, (x, y)) = (x cos t - y sin t, x sin t + y cos t). (3.17) Note. Non-autonomous systems. In the present context, we lose no generality by restricting our attention to autonomous differential equations. Any non-autonomous system dx/dt = g(t, x), where g maps an open subset V of R x E to E, gives rise to an autonomous system dy/dt=f(y), where / : V - > R x E is defined by /(«, JC) = ( 1 , g(u, x)). In other words, we are replacing the given differential equation by du/dt = 1, dx/dt = g(u, x). Any solution of dy/dt=f(y) yields a solution of dx/dt = g(t,x). Specifically, suppose that we want a solution 8 of dx/dt = g(t, x) satisfying the boundary condition 8{t0) = p. Suppose that we have found an integral curve y: / -» V of / at (to, p). Then we assert that 5(f) = y2(t -10) defines a suitable solution 8,

Ill

ORDINARY DIFFERENTIAL EQUATIONS

65

where the index denotes the component in the product R x E . For observe that5(f 0 ) = y2(0) = p and S'(t) = y'2(t-t0)=f2y(t-t0)

=

g(yi(t-t0),S(t)).

But, since y[(t)=fiy(t)=l and yi(0) = t0, yi(t) = t + t0, and so S'(t) = g(t, 8(t)). We should also point out that an wth order ordinary differential equation in "normal form" dmy

,/

dy

dm~ly\

may be reduced by the substitutions _dm'1y d t „ ^

dy xi-y,x2-dt,...,xm=

to the form dx/dt = g(t, x), where x = ( x i , . . . , xm) e E m , and g(t, x) = {x2, x3,..-,

xm, hit, xi,...,

xm)).

Moreover, the implicit mapping theorem (Exercise C.14) can often be used locally to reduce more general mth order equations to the above normal form. (3.18) Example.The simple pendulum equation 6" = -g sin 6 of the introduction was reduced to 6' = CJ, CJ' = -g sin 6 by the substitution o> = 6'. (3.19) Example. In conservative mechanics, Lagrange's motion are usually written

equations of

d_(dL\_ f^_ n dt \dqi' dq( where q, (i = 1,2,..., n) are "generalized coordinates", the Lagrangian L is T — V, T is the kinetic energy and V is the potential energy. If we choose qt such that the positive definite quadratic form

T=t kmte'if, ;= i

where the m, are "generalized masses", then the equations of motion take the form „ W mat = - — . dqt

66

INTEGRATION OF VECTOR FIELDS

CH. 3

The substitution m,c/[ — ph the "generalized momentum", converts the equations of motion to p\ = -dV/dqi, and the substitution and equations of motion may be written together as Hamilton's equations , dH 4,==I7'

Pi =

dH -7-,

where the Hamiltonian H = T + V is the total energy as a function of p, and
(3.20) Example. Van der Pol's equation. This is, amongst other things, an important mathematical model for electronic oscillators. It has the form x"-ax'(l-x2)

+ x=0

where a is a positive constant. This is equivalent to the pair of first order equations x' = y,

y'= —x+ay(l—x

)

where a is a positive constant. The phase portrait of this vector field on R is topologically equivalent to the reverse flow of the flow in Example 1.27. It has a unique closed orbit, which is the co-set of all orbits except the fixed point at the origin (see Figure 3.20; a proof of this assertion may be found,

FIGURE 3.20

for example, in Hirsch and Smale [1] and in Simmons [1]). The system is said to be auto-oscillatory, since all solutions (except one) tend to become periodic as time goes by.

IV

LOCAL INTEGRALS

67

IV. LOCAL INTEGRALS Let p be a point of a Banach space E. Suppose that we have a vector field v with principal part / defined on some neighbourhood B' of p. We wish to show that for some neighbourhood B of p there are unique integral curves at each point of B. We must first make some comments. To stand any chance of proving uniqueness of an integral curve at the point x, we must fix the interval on which the curve is to be defined (since restricting to a smaller interval gives, strictly speaking, a different integral curve). If we wish to fit the integral curves together to form a local integral 4> at p, we may as well define them all on the same interval I. We cannot have I arbitrarily large, since the orbit of p may not be wholly contained in B'. Similarly, however small 7, we usually need B strictly inside B', since points x near the frontier of B' will leave B' at some time in I. Actually, we adopt a slightly different approach, for ease of presentation. We start with given closed sets B' and B, balls with centre/? in E. We insist that the radius J of B is strictly less than the radius d' of B'. Note, however, that the argument works equally well with B = B' = E, so we include this as an exceptional possibility d = d' = oo. We prove that there is some interval I on which the above situation holds (see Figure 3.21). For this we need to make

FIGURE 3.21

some assumption about the smoothness off. Continuity of/ensures existence of integral curves, but not their uniqueness. For uniqueness we need something stronger, and the relevant property is what is known as a Lipschitz condition (see Appendix C). This is about half way between continuity and differentiability; it implies continuity and it is certainly satisfied by any C 1 map on B'. Any Lipschitz map is bounded on a ball of finite radius. From now on we shall be making full use of the map space notation and theory of Appendix B

68

INTEGRATION OF VECTOR FIELDS

CH. 3

and the contraction mapping theorem of Appendix C. In particular, |/| 0 denotes sup {|/(x)|: xeB'}. Our version of Picard's theorem is: (3.22) Theorem. Let f: B' -» E be Lipschitz with constant K, and let I be the interval [-a, a], where a<(d'-d)/\f\0 ifd'X:I-*B' of fat x. Let (f>:IxB->B' send (t, x) to x(t). Then for all t e I, ': B -*• B' is uniformly (in t) Lipschitz, and is C'iff is Cr. Proof. We give a proof under the additional restriction a < \/K, which may be removed by an easy technical modification (see Exercise 3.24 below). We define a map x • B x C°(I, B') -* C°(I, B') by, for all t e I, (3.23)

x(x,y)(t)

= x + |"/V(" ) du, Jo

where C°(I, B') is as defined in Appendix B. We must check, of course, that x(x, y) is indeed in C (I, B'). It is continuous by elementary analysis, and, by the inequalities \x + \'fy{u ) du—p

-p\ + \t\

Jo

= d + a\

•:d\

its image is in B'. Now, for all x e B and y , f e C°(I, B'),

l^x(r)-^(f)lo = sup{|f'(/ r («)-/y(«))^ :tel U Jo

« s u p j l j \fy{u)-fy{u)\du

: re/j

-.Ka\y-y\0.

By the contraction mapping theorem (Theorem C.5) there exists a unique map g:B->C°(I,B') such that, writing g(x) = 4>x for all xeB, 4>x(t)= x+\

Jo

f(f>x{u) du

for all teI,or (equivalently) such that <j>x is an integral curve of / at x. The uniqueness of g gives the uniqueness of the integral curve at x, since another integral curve at x would give a possible alternative for the value g(x). Since x is uniformly Lipschitz on the first factor, with constant 1, it follows by Theorem C.7 that g is Lipschitz. Thus ', which is g composed with the continuous linear evaluation map ev':C°(I,B')^B', is Lipschitz, where

LOCAL INTEGRALS

IV

69

ev'(y) = y(t). Moreover, since \ev'\ = 1, the Lipschitz constant of g is also a Lipschitz constant for each map '. Suppose, finally, that / is C. We observe that x may be expressed as the sum of two maps. The first, taking (x, y) to the constant map with value x, is continuous linear (and hence C°°). The second is the composite of projection onto the second factor, the map

y^fy:C°(I,B')^C°(I,E) and the map fi^it^l

M(")^«):C0(/,E)^C°(/,E).

The outside maps of the composite are continuous linear, while the middle one is Cr, by Theorem B.10. Thus, by Theorem C.7, g, and hence <£', is C . D (3.24) Exercise. The assumption a < 1/K in the proof of Theorem 3.22 was made to ensure that the maps x* were contractions. They lose this property, for larger a, but regain it if they are conjugated with a suitable automorphism of the space C°(I, E). In fact, the map -y>->(f>-H>sech at. y{t)) has the desired effect for a > K. The exercise, then, is to eliminate the assumption a < 1/K, by considering instead of x, the map i(/:BxC->C defined, for all t e /, by ip(x, S)(t) = sech a n x +

/(cosh au . S(u)) du\,

where C is the closed subset {8 e C°(I, E): S(t) cosh at e B' for all 16 / } , the image of C°(I, B') under the above mentioned automorphism. (3.25) Exercise. Derive a local integrability theorem for non-autonomous ordinary differential equations (see Note 3.17). (3.26) Exercise. Show that the map / : R - » R sending x to x2/3 is not Lipschitz on any neighbourhood of 0, but that nevertheless / has a continuous local integral at 0. The way in which the local integral <j> depends separately on time t and initial position x emerges very clearly and easily from Proposition 3.15 and Theorem 3.22. One is naturally led to ask how <$> depends on t and x together. Here the answers are just as simple. As we shall now prove, the map is locally Lipschitz, and it is as smooth as the function /. (3.27) Theorem. If d' <<x> then the map of Theorem 3.22 is Lipschitz. In any case, it is locally Lipschitz.

70

Proof. For all (t, x) and (/', (3.28)

CH. 3

INTEGRATION OF VECTOR FIELDS

x')elxB,

\4>(t',x')-(t>(t,x)\^\\x'-x\

+ \At%

where A is, for all tel, a Lipschitz constant for ' (which is uniformly Lipschitz by Theorem 3.22). When d' is finite, / is bounded (being Lipschitz) and thus so is Dx(=f<j)x). Hence x is uniformly (in x) Lipschitz, and we deduce from (3.28) that is Lipschitz. If d' is not finite, then (3.28) nevertheless gives that <j> is continuous. Thus fcf> is locally bounded, cf>x is locally uniformly Lipschitz and hence is locally Lipschitz. • (3.29) Exercise. Find an example of a Lipschitz map / : E -» E for which there is no corresponding Lipschitz : I x E -» E, for any /. (3.30) Exercise. Suppose that y:[0, T ] - » R is continuous, and that, for all re[0,r], y(t)-K\ ) — K y(u)du^A y(u Jo

+ Bt + Ce°",

where K > 0, a # K and where A, B and C are all constants. Prove that y(t)^AeK'

+ BK-1(eK'-l)

+

C(a-Ky1(ae0"-Ke'<').

(This deduction is called "integrating the inequality".) Show that if :IxU -» V is a local integral of a map / : V -* E with Lipschitz constant K, then, for all tel, ' satisfies the inequality W{X)-4>'{X')\-K\

f \4>u(x)-4>u(x')\du

I Jo

Deduce that (/>' is Lipschitz with constant e

\x -x

.

We now turn to the situation when / is smooth. The only substantial point is to show that the partial derivatives of the local integral cf> with respect to x are continuous in t and x together. We deal with this first. (3.31) Lemma. Let f be C and let be the map of Theorem 3.22. Then the partial derivatives (D2Y<1>:IxB ->LS(E, E) exist and are continuous for 1 =s 5 =£ r. Proof. Since we may "slow down" / by multiplying it by a small positive constant a, we may assume that the Lipschitz constant K of / is less than 1. The local integral for / is recovered by "speeding up" the local integral of af. More precisely, we multiply the time parameter tbyl/a and then apply the local integral of af. This procedure does not affect the smoothness class of the integral. We leave the details as an exercise. We also continue to assume

IV

71

LOCAL INTEGRALS

that a < 1/K, having already indicated in Exercise 3.24 how to deal with this point. We observe that the relation (3.23) defines a map X:B*C\I,B')^C\I,B')

which is still a uniform contraction on the second factor. The extra calculation needed is

\D(xx(y)-xx(y))\o = l/V -fy\o^x\y - f|0, and, by assumption, K < 1. Moreover, \ y is still uniformly Lipschitz, and x is itself C. To prove this latter assertion, consider the inclusion t of Cl(I, B') in C°(I, E), followed by /*: im i -* C°(I, E). Since f* is composition with /, f^i is Cr, by Theorem B.10. Also integrating back into CX{I, E) is a continuous linear operation, s o ^ : B x CX{I, B')-* C1^, B') is C as claimed. Thus the fixed point map g:B->Cx(I,B') corresponding to our new ^ is C r . As in Theorem 3.22, we define cj> by {t,x) = g(x)(t). By the uniqueness of integral curves in Theorem 3.22, the present map <j> is the map of Theorem 3.22. Now for all t e I, ' is the composite of g: B -* Cl{I, B') and the continuous linear map ev': C1 {I, B')^>B'. Thus, for l=£s=sr, (D2)s exists, and is given by (D2)s{t,x) = ev'oDsg{x) (recall that Dsg(x) is an s-linear map from E to CX{I, E) so we may talk of its composite with ev': C\l, E)-»E). Thus {D2Y =comp (ev*Dsg), l where ev : I ^ L{C (I, E), E) is the continuous map of Corollary B.17 and comp is the continuous bilinear composition map from L{C\l, E), E ) x L , ( E , C\l, E)) to L,(E, E). Thus (D2)s is continuous. D Our main theorem on smoothness of local integrals now follows easily. (3.32) Theorem. If the map f of Theorem 3.22 is Cr(r^l), integral <j> is C. Moreover, D\(D2)r(j> exists and equals

then the local

(D2)r(#):/xE->L,(E,E). Proof. L e t / be C. We prove by induction on s the statement that is Cs, for 0=£.sssr. By Theorem 3.27, is locally Lipschitz, and hence C°. Now assume that is C", for some ;' with 0=£;'<s. We prove that 4> is C + 1 , by showing that its ;'th partial derivatives are C 1 . There are two cases. (i) With the exception of {D2)', all /th partial derivatives can be expressed, by the symmetry of D'cj>, as ( ; ' - l ) s t partial derivatives of D\. Since Di4>=f4>, which is C", these partial derivatives are C 1 .

72

INTEGRATION OF VECTOR FIELDS

CH. 3

(ii) By Lemma 3.31 we know that (D2)' is continuous, and also that (D2)'+14> exists and is continuous. Since (D2)'Di = ID2)'(/<£) is continuous, Di(D2)'4> exists and equals (D2)'Di. (To see this, observe that (D2)iD1cf>(t,x) = D1 f

{D2)iD14>{u,x)du

= D1(D2)1 f

D1<j>{u,x)du=Dl{D2)i{t,x).)

Hence, by induction, <j> is Cs, and hence C. Finally, since (D2)rDi(j> = (D2)r(f) is continuous, Di(D2Y<j) exists and equals (D2)r(f). • Since any C 1 vector field is locally Lipschitz (Proposition C.2), we obtain the following local result for manifolds: (3.33) Corollary. Any C vector field [r^ 1) on a manifold has a C local integral at each point of the manifold. • A very natural question arises at this stage. How does the local integral

and its derivatives small? We deal with this problem in the appendix to this chapter.

V. GLOBAL INTEGRALS Our main tool for extending local integrability results to global ones is the uniqueness of integral curves proved in Theorem 3.22. We first extend this into a global uniqueness theorem. (3.34) Theorem. Let vbea locally Lipschitz vector field on a manifold X, and let a: J^X and /3.J'-*Xbe integral curves of v. If, for some t0eJ, a (t0) = (3 (t0), then a =/3. Proof. Let a{t0) =/3(t0), where t0eJ. We prove that a(t) = /3(t) for all t eJ with tssto; the proof for t =£ t0 follows by reversing v (that is, replacing v by —«). Let T = sup {t e / : a = j3 on [t0, t]}, and suppose that T G int / . By the continuity of a and /?, a (T) = 0 (T) = p, say. Now identify a neighbourhood U of p in X with an open subset U' of the model space E of X, by a chart £:U^>U'. Choose balls B and B' in U with centre p and obtain a corresponding interval / = [-a, a ] as in Theorem 3.22. We may take a small enough for r + a to be in / . If a: I -» X and /3: / -» X take t to a (t + T) and /3(/ + T) respectively then a =/3 by Theorem 3.22, since both are integral

V

GLOBAL INTEGRALS

73

curves at p with domain /. Thus a = (3 on [t0, r + a], contrary to the definition of T. Thus T is the right end-point of /. If T £ / then, as above, a(T) = /3{T). • (3.35) Exercise. Let /:R->R take x to x2/3. Find two different integral curves of / at 0 with the same domain. (3.36) Exercise. Prove that all integral curves of a locally Lipschitz vector field v at a point p are constant functions if and only if p is a singular point of v (p is a singular point, or zero, of t> if i>(p) = 0 P ). Earlier in the chapter we commented that the velocity of a flow is independent of time, and so may be regarded as a vector field. Conversely, a local integral of a vector field is locally a flow in the following sense: (3.37) Theorem. Let 4>:IxU^*Xbea local integral of a locally Lipschitz vector field on a manifold X. Thenforallx e Uandalls, telsuchthats + tel and 4>(t, x)e U, (s, (f>(t, x)) = (s + t, x). Proof. Apply Theorem 3.34 to integral curves a: [0, s] -» X and /3: [0, s] -> X defined by a(u) = <£(«> (t, x)) and p(u) = (u +t,x). Since a and/3 agree at 0, they agree at s. • (3.38) Corollary. If : R x X -* X is a local integral of a locally Lipschitz vector field on a manifold X, then it is a flow on X. • (3.39) Theorem. Any Lipschitz vector field v on a Banach space E has a unique integral flow (f>.Ifv is C then 4> is C. Proof. For all a>0, v has a unique local integral :[—a, a ] x E - » E , by Theorem 3.22. Any two such agree on the intersection of their domains. Hence the collection of all such local integrals defines a local integral 0 : R x E ^ E , which is a flow by Corollary 3.38. If v is Cr then 4> is C by Theorem 3.32 and uniqueness. • (3.40) Exercise, (i) Prove that the vector field on R with principal part f(x) = x does not have an integral flow. {Hint: The vector field is so large that the orbits escape to infinity in finite time.) (ii) Prove that any C vector field v on a manifold X gives rise to a partial flow 4>:D^>X on X, such that, for all xeX, (j>x:Dx-*X is the maximal integral curve of v at x. We have been at some pains to prove that the local integrals that we have obtained in our existence theorems are as smooth as the vector fields we integrate. This behaviour is completely general, as we now show. (3.41) Theorem. Any local integral of a C vector field (r > 1) on a manifold X is C. Similarly any integral of a locally Lipschitz vector field is locally Lipschitz.

74

INTEGRATION OF VECTOR FIELDS

CH. 3

Proof. Let v be a C vector field on X, and let 4>:Ix U-*X be a local integral of v. Let a > 0 be the right end-point of I (possibly a = oo). Fix x 0 6 £/, and let T = {t>0: cf> is C at (K, X 0 ) for all u e [0, /]}. By Corollary 3.33, T is non-empty. Let T = sup T. It is enough to show that T = a, and that a e T if a e 7, for then a similar argument for negative t completes a proof that is C at all points of 7 x {JC0}. Suppose that r e i n t / . By Corollary 3.33 there exists a Cr local integral tl/:]-b, b[xV^X of v at (£(T, X 0 ), for some b>0. By continuity of ^)xoat T, there is some t0 e T w i t h f 0 > T - 6 and<£(?0, *o)e V. By continuity of '° at x 0 there is some neighbourhood W of x0 in £/ with '°(W) <= V. Let c = min {6, a -10], so that t0 + c>r. Then, for all i s W and all u with 0=£«(t0+u, x) = iff{u, (to, x)). To check this, we have only to fix x and apply Theorem 3.34 to integral curves a and /3 whose values at u e [0, c[ are respectively <£(/0 + u, x) and t/Kw, <£(f0, x)). Since i/f is C r and 4>'° is C r at xo, it follows that is Cr at (f 0 + «, xo) for all u e ]0, c[, which contradicts the definition of T. Thus T = a. II a el, we obtain i/s t0 and W as above, and observe that, for all x e W and all u with0=£« =£a -t0,(t0 + u,x) = ip(u, (t0, x)). This shows that is Cr at {a, xo), as required. An identical approach works for the locally Lipschitz case. • To complete our integration programme, we prove that every smooth vector field on a compact manifold has an integral flow. We make use of a lemma which is of some interest in its own right. (3.42) Lemma. Let v be a locally Lipschitz vector field on a manifold X. Suppose, that for some a> 0 and for all x eX, v has an integral curve at x with domain [—a, a]. Then v has a unique integral flow cf>, which is locally Lipschitz. Furthermore, if v is C ( r > 1) then is C. Proof. We have to show that for all x 6 X there is an integral curve 4>x: R -» X of v at x. Suppose that for some x this is not the case. Reversing v if necessary, we have that T = sup{? eR: v has an integral curve at x with domain [0, t]} is finite. By hypothesis r > 0. Choose t0 with T — a : R x X -> X, and its various properties come from Theorem 3.34, Corollary 3.38 and Theorem 3.41. • (3.43) Theorem. Any locally Lipschitz vector field v on a compact manifold

V

GLOBAL INTEGRALS

75

Xhas a unique integral flow 4>, which is locally Lipschitz. Ifv is Cr(r 3= 1) then <j> is C. Proof. For all xeX, there is, by Theorem 3.22, a local integral at x. Let this be 4>x'-IxxUx^X, where Ix = [-ax, ax], with ax>0, and Ux is an open neighbourhood of x in X. Choose a finite subcovering UXi,..., UXn of the open covering {Ux:xeX} of X, and let a = m i n { a x i , . . . , aXn}. Then any point x of X is in some UXt, and the corresponding <$>Xl provides an integral curve at x with domain [-aXi, aXj], and hence, by restriction, one with domain [-a, a]. The result now follows from Lemma 3.42. • (3.44) Exercise. One does not normally expect a vector field on a noncompact manifold to have an integral flow, since integral curves may not be defined on the whole of R (see Exercise 3.40). However, we can remedy this situation without affecting the orbit structure as follows. Let X be a (paracompact) manifold admitting C partitions of unity (r ^ 1). Let v be a C vector field on X. Prove that for some positive C function a: X -» R, the vector field x-*a(x)v(x) has an integral flow :RxX->X with the same orbits as v. (Here the orbit of v through x is defined to be thelimage of the integral curve of v at x with maximal domain.) A solution to this problem may be found in Renz [1]. By virtue of the correspondence between velocity vector fields and integral flows, the qualitative theories of smooth vector fields and smooth flows are one and the same subject. For example we define two smooth vector fields to be flow equivalent if their integral flows are flow equivalent and topologically equivalent if, after the modification in Exercise 3.44 if necessary, they have topologically equivalent integral flows. It is sometimes easier to describe the theory in terms of vector fields, sometimes in terms of flows. We are free to use whichever terminology we find the more convenient in any given situation.

Appendix 3

I. INTEGRALS OF PERTURBED VECTOR FIELDS We have already found spaces of maps to be a great technical convenience, and we now put them to a new use. We discuss the map that takes vector fields, regarded as points in a map space, to their local integrals, similarly interpreted. The properties of this map are clearly of considerable interest. For example, its continuity at a point (= vector field) corresponds to small perturbations in the vector field producing small perturbations in the integral. In this context, the precise nature of a "small perturbation" depends, of course, on the topology that we put on the map spaces. We do not attempt a comprehensive treatment of the subject, but give instead a couple of typical theorems that can be proved along these lines. Suppose that notations are as in Theorem 3.22. For the purposes of the following theorem ATr(rs=0) denotes the subset of Cr(B', E) consisting of Lipschitz maps h with \h\0< ((d' - d)/ a) -\f\0 when d' Cr(B, E) taking h to ip' -(f)' is continuous at 0. Proof. We may assume that the Lipschitz constant of / + h is strictly less than 1/a (otherwise we apply the technique of Exercise 3.24). We define a map r

X:N

xBx

C°(I, B') -* C°(I, E)

by x(h,x,y)(t)

= x+\

Jo

76

(f+h)y(u)du,

INTEGRALS OF PERTURBED VECTOR FIELDS

I

77

for t e /. For each h e Nr, we have, as in the proof of Theorem 3.22, a fixed point map gh :B-+ C°(I, B'). Now \h -\° is bounded by a\h\0, and D\h is C" 1 -bounded, by Corollary B . l l . Thus, by Theorem C I O , gh-g° is C-bounded. We may express xh ~X° a s a composite BxC°(I,

B')

>C°(I, B')-^-C°{I,

E)

>C°(I, E),

where the first map is the product projection, and the third is the continuous linear integration map sending y to (f >-> /„ y(«) du). By Corollary B.12, the map for Nr to C(C°(I, B'), C°(I, E)) taking h to h* is continuous linear. Hence by Lemmas B.4 and B.13, so is the map from Nr to C'{B x c°(I, B'), C°(I, E)) h

taking h to \ ~ X°- Suppose now that / is uniformly C'. Then \ is uniformly C\ by Theorem B.10. Thus we may apply Theorem C I O to deduce that a is continuous at 0. So also is /3 = (ev')^a, where ev':C°(I,B')

+ B'

is the continuous linear evaluation map. The situation with purely Lipschitz perturbations is not so clear. However, we shall only make use of the following results, which gives continuity at / of the operation of taking local integrals in the case when / is linear. (3.46) Theorem. Let f be a continuous linear endomorphism of E and let h:V^*E be Lipschitz, where V is open in E. Suppose that :IxU^V and ip: I x U -* V are local integrals off and f+ h respectively. Then, for all tel, iff' — 4>' is Lipschitz, with a constant that tends to zero with the Lipschitz constant of h. Proof. We may suppose that t > 0 (otherwise reverse the vector fields). Let 6' denote i/f'-'. For all x, x'eU, «f+h)u(x)-(f+h)i(,u(x')+f4>u{x'))du

\d'(x)-d'(x')\=\\ I Jo

*ZK\ \6"(x)-du(x')\du+\ Jo

f Jo

\ipu(x)-iP"{x')\du,

where K and A are positive Lipschitz constants for / and h respectively. Let fi = K +A. Then y. is a Lipschitz constant for f+h, and thus, by Exercise 3.30, i/r" is Lipschitz with constant eM". Thus \e\x)-6\x')\-K

f Jo

\du(x)-8u(x')\du^n-\e'"-l)A\x-x'\.

78

APPENDIX

CH. 3

Integrating this inequality (see Exercise 3.30 again), we obtain \6,{x)-d,(x')\^^1

-eK)

(^''"'"

A|x-x'| = (g"'-e"')\x-x'\,

and so 6' is Lipschitz with constant eM< — eKl. The proof is now complete, since /u. -* K as A -» 0. D

II. FIRST INTEGRALS Let u be a vector field on a differentiable manifold X and let g: X -* R be a C r function (rs^l). The derivative of g in f/ie direction of v is the C r _ 1 function L„g defined by Tg(v(x)) =

(g{x),Lvg(x)).

If y is an integral curve of v at x, then L„g(x) is, by the chain rule, the derivative with respect to t at t = 0 of fy(t). In fact, it is the rate of change of/ at x at any time t as we move along an integral curve through x with the prescribed velocity v. If X has a given Riemannian structure ( , ) then the gradient vector field Vg of g is defined as in Example 3.3, and in terms of this Lvg(x) = (v(x), Vg(x)). Thus, if X is Euclidean space R", Lvg(x) = vi(x)—+-

• • + v„(x)-—.

oXi

aXn

This explains the notation UiW—+• ••+ dXi

v„{x)-— dxn

sometimes used for a vector field on R". The notation is justifiable since we can recover v from Lv by applying Lv to the coordinate functions x <-» *,-. In fact, the notion of a vector field is quite commonly defined in terms of the directional derivative property (see, for example, Warner [1]). A first integral of v is a C 1 function g: X -> R such that L„g = 0, the zero function. Thus a first integral is constant along any integral curve of v. Of course all constant functions on X are trivially first integrals of any vector field X. For many vector fields these are the only first integrals. When a non-constant first integral g does exist, it is of considerable interest. The reason for this is that, by Sard's theorem the level sets of g are usually submanifolds of X of codimension 1 (see Theorem 7.3 and Example A.7,

II

FIRST INTEGRALS

79

below, and Hirsch [1]). Since the integral curve at any point lies in the level set through that point, v{x) is tangential to the submanifold at every point x. Thus for each non-critical value of g, we obtain a vector field on a manifold of dimension 1 less than X (assuming X to be finite dimensional). Since this ought to be easier to integrate than the original vector field v on X, we have made a reasonable first attempt at integrating v, hence the name first integral given to g. (3.47) Exercise. Prove that the vector field on R with principal part f(x) = x has no non-constant first integral. (3.48) Exercise. Find a non-constant first integral for the vector field on R 2 with principal part (i) f(x, y) = (x, -y), (ii) f(x, y) = (-y, x). (3.49) Exercise. Prove that g:R 2 -»R defined by g(x, y) = x2 + y2 is a first integral for the vector fields on R 2 with principal part (i)f1(x,y) = (-y(x22 + y2),x(x2 + y2)), (ii)/2(jc, y) = (~xy2, x2y), (m)f3(x,y) = (-x2y3,xY),

(iv) f4(x, y) = (*3y4> - * V ) •

Sketch the phase portraits of the vector fields. Are any of the vector fietds topologically equivalent? (3.50) Exercise. Let U = {(x, y) e R 2 : x # 0}. Prove that the function / : U -> R defined by f(x, y) = (x2 + y2)/x is a first integral for the vector field on R 2 with principal part v{x, y) = (2xy, y2-x2). Sketch the phase portrait of the vector field. Prove that the function g = R 3 -> R defined by g(x, y, z) = x2 + y 2 + z2 is a first integral of the vector field on R 3 with principal part w(x, y, z) = 2xyz, (y2-x2)z,

- ( x 2 + y 2 )y).

By comparing v and w, write down another non-constant first integral / i : f / x R - » R of w. Sketch the phase portrait of w on the hemisphere jt2 + y 2 + z 2 = l, x^O, as viewed from a point (a, 0, 0) for large positive a. (3.51) Example. The Hamiltonian H = T + V is a first integral of the Hamiltonian vector field of conservative mechanics (see Example 3.19). The principle of conservation of energy is the observation that H is constant along integral curves of the vector field. Although a vector field may have no non-constant global first integrals, it always has non-constant local first integrals in the neighbourhood of any non-singular point. This is an easy consequence of the rectification theorem (Theorem 5.8 below).

CHAPTER 4

Linear systems

In a naive approach to the theory of dynamical systems on Euclidean space R", we might hope to obtain a complete classification of all systems, starting with the very simplest types and gradually building up an understanding of more complicated ones. The simplest conceivable diffeomorphisms of R" are translations, maps of the form x >—>JC +p for some constant peRn. It is an easy exercise to show that two such maps are linearly conjugate (topologically conjugate by a linear automorphism of R") providing the constant vectors involved are non-zero. Similarly, any two non-zero constant vector fields on R" are linearly flow equivalent. The next simplest systems are linear ones, discrete dynamical systems generated by linear automorphisms of R" and vector fields on R" with linear principal part. Rather surprisingly, the classification problem for such systems is far from trivial. It has been completely solved for vector fields (Kuiper [2]) but not as yet for diffeomorphisms, although Kuiper and Robbin [1] have made a great deal of progress in this latter case. The difficulties that one encounters even at the linear stage underline the complexity and richness of the theory of dynamical systems, and this is part of its attraction. They also indicate the need for a less ambitious approach to the whole classification problem. Instead of aiming at a complete solution, we attempt to classify a "suitably large" class of dynamical systems. In the case of linear systems on R", it is easy to give a precise definition of what we mean by "suitably large", as follows. The set L(R") of linear endomorphisms of R" is a Banach space with the induced norm given by \T\ = sup{\T(x)\:xeRn,\x\ 80

= l}.

LINEAR SYSTEMS

81

A "suitably large" subset $f of linear vector fields on R" is one that is both open and dense in L(R") (from now on we shall not usually bother to distinguish between a vector field on R" and its principal part). The density of such a set Sf implies that we can approximate any element of L(R") arbitrarily closely by an element of Sf; the fact that y is open implies that any element of $f remains in y after any sufficiently small perturbation. In general, of these two desirable properties we tend to regard density as essential and openness as negotiable. The set GL(R") of linear automorphisms of R n inherits the subspace topology from L(R"). "Suitably large" in the context of discrete linear systems means open and dense in GL(R"). How, then, do we select a candidate for the set 9"> There are very good reasons for our choice, as follows. Suppose that X is a topological space and that ~ is an equivalence relation on X. We say that a point x e X is stable with respect to ~ if x is an interior point of its ~ class. The stable set S of ~ is the set of all stable points in X. Of course 2, is automatically an open subset of X. Moreover, if the topology of X has a countable basis, then 1 contains points of only countably many equivalence classes. Thus there is some chance that 1 will be classifiable. The snag is that £ may fail to be dense in X. In the present context, X = GL(Rn) (or L(R")), ~ is topological conjugacy (or topological equivalence), and 1 is called the set of hyperbolic linear automorphisms (or the hyperbolic linear vector fields). In each case the stable set 2 is dense and is easily classifiable, as we shall see. There is another vital reason for considering these particular subsets: in the next chapter we shall find that hyperbolic linear automorphisms (and hyperbolic linear vector fields) are stable not only under small linear perturbations but also, more generally, under small smooth perturbations. Now the basic idea of differential calculus is one of approximating a function locally by a linear function. Similarly, when discussing the local nature of a dynamical system we consider its "linear approximation", which is a certain linear system. The important question is whether this linear system is a good approximation in the sense that it is (locally) qualitatively the same as the original system. This question is obviously bound up with the stability (in the wider sense) of the linear system. In fact, we shall obtain a satisfactory linear approximation theory precisely when the linear system is stable. We begin with a review of linear systems on R". The bulk of the chapter, however, deals with stable linear systems on a general Banach space. For this we need a certain amount of spectral theory, which we include in an appendix to the chapter. The material of R" provides a background of concrete examples against which the more general results on Banach spaces may be tested and placed in perspective.

82

CH. 4

LINEAR SYSTEMS

I. LINEAR FLOWS ON R" Let T be any linear endomorphism of R". Then we may think of T as a vector field on R", in which case we call it a linear vector field. The corresponding ordinary differential equation is (4.1)

x'=T(x),

where x e R". One may immediately write down the integral flow. Since L(R") is a Banach space, it makes sense to talk of the infinite series exp(tT) = id + tT+h2T2 2

+ - ••+—tnTn n\

+ -- •

where id is the identity map on R" and f e R . This series converges for all t and T, and the integral flow 4> of T is given by (4.2) (term-by-term indicated later automorphism velocity vector

<j>(t,x) =

exp(tT)(x),

differentiation makes this plausible; another proof is in the chapter). We call a flow linear if ' is a linear varying smoothly with t. Thus is linear if and only if its field is linear.

(4.3) Example. If T = a(id) for some a e R then exp (tT) = id + at(id) + ••• + — (at)nidn + • • • = (l + at + - • •+ — (at)" + - • -)id n\ at-j

= e id.

(4.4) Example. If T has matrix [o

I)

* e n T 2 has matrix

[°

°],

so the series for exp (tT) terminates as id + tT. We may define a notion of linear equivalence for linear flows, as follows. Let and i/f be linear flows with velocities S and r e L ( R " ) respectively. Then ip is linearly equivalent to 4>, written if ~ L, if for some a e R, a > 0,

83

LINEAR FLOWS ON R'

I

and some linear automorphism h e GL(R") of R", the diagram (4.5)

RxR"

RxR commutes. That is, hc/>{t, x) = il/(at, h{x)) for all (t, x) e R x R". Equivalently (f/~ L4> if a ° d only if S = a(h~ Th). The problem of classifying linear flows up to linear equivalence is therefore the same as the problem of classifying L(R") up to similarity (linear conjugacy). Now the latter is solved by the theory of real Jordan canonical form (see Hirsch and Smale [1] or Gantmacher [1]) which we recall briefly. Every TeL(R") is similar to a direct sum Ti@- • -®Tq of linear endomorphisms 7} e L( V,), / = 1 , . . . , q, where Vi ® • • • © V, is some direct sum decomposition of R", and, for some basis of Vh the matrix of 7} takes the form l Ay

0 1

MrAy

1

0

Ay.

where Ay e C. For each M, there are two possibilities: (i) A, e R, in which case the entries of M, are real numbers, and 7} is associated with dim V} repeated eigenvalues A, of T. (ii) Aj• = a + ib, where a, b € R, b > 0, in which case the entries of M, are real 2 x 2 submatrices

and Tj is associated with \ dim Vj pairs of repeated complex conjugate eigenvalues Ay and A, of T. Notice that there may be more than one 7} having the same eigenvalue Ay. The set of numbers dim V, and the corresponding eigenvalues Ay (or Ay, Ay) determine the similarity class of T. It follows that the integral flow 4> of T can be decomposed as (t>i® • • • ©
where

4>,-(t, xt) = exp (f7y)(*y),

84

LINEAR SYSTEMS

CH. 4

and x = (xu ..., xq) e Vx ® • • • ® Vq = R". With respect to the above basis of V,; (4>j)': V,--> Vj has matrix t"1'1 eib' (m-1)!

te' te'

N, = ee

0

te' 0

where Ay = a + ib, a,beR, 2 x 2 block

and, if b > 0, (tr/r\) e"" must be interpreted as the tr [cos bt -sin6f"| f7« rlLsin rlLsi bt

cosbtj'

Thus, for example if dim V, = 4 and A; = /, then "0-1 1 0n 0 0

1 0n 0 1

0

and cos t sin t N,= 0 - 0

-sin t tcost cos t t sin t 0 cos t 0 sin t

-t sin f t cos t -sin t cos t

Let us examine the case a < 0, ft = 0, dim V, = 2. The phase portrait in the plane V, is illustrated in Figure 4.6. Every orbit has the point 0 as its unique w-limit point. This leads us to suspect that any two negative values of a give

FIGURE 4.6

LINEAR FLOWS ON R'

I

85

topologically equivalent flows. How can we prove this? Suppose it were true that for some value of a the distance \t. x\ decreased with t. Then we could define a map h:Vj-* V, as the identity on 0 and on the unit circle S 1 and, for all x € S 1 taking t. x to e~'x (see Figure 4.7). It is not hard to show that h is a

FIGURE 4.7

homeomorphism, and hence that it is a flow equivalence from the given vector field to the vector field -id on V}. Now \t.x\ decreases if and only if the scalar product of x and 7}(x) is negative definite. This quadratic form is a(xl+xl) + xix2 which is certainly not negative definite for small negative a. However we now note that, for any e > 0, a linear conjugacy with

[J I] Chang£S [o «] t0 [o «]' and, for given a, we can always take e small enough for the form a(xi +X2) + eX\X2

to be negative definite. Thus I

I is flow equivalent to

_

, and hence so is

since linear conjugacy trivially implies flow equivalence. Thus, as we suspected, negative values of a give topologically equivalent vector fields. We may go considerably further than this, using similar techniques. We may take the direct sum V- of all subspaces corresponding to eigenvalues £/ with negative real part, and show that the vector field on this subspace is flow equivalent to the vector field —id. Thus any two linear contractions on a subspace of R " are flow equivalent. Similarly for positive eigenvectors we get a vector field on a subspace V+ which is flow equivalent to the vector field id. We do not go into details at this stage, since we prove a more general version later in the chapter.

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There remains the linear subspace V0 obtained by combining those summands V, for which the real part of Ay is zero. If Vo = {0}, then R" = V+® V- and the vector field T is flow equivalent to the vector field on R" with matrix Ir@—Is =diag ( 1 , . . . , 1, — 1 , . . . , - 1 ) , where r = dimV+ and s = dim V— Such a linear vector field T is called a hyperbolic linear vector field and its integral flow (f> is called a hyperbolic linear flow. Hyperbolic linear vector fields are sometimes referred to as elementary linear vector fields, which explains our notation EL(R") for the set of all hyperbolic linear vector fields on R". We do not give a detailed analysis of the flow on V0, since it would involve a fair amount of work and the hyperbolic case is of more fundamental importance. Instead we state the main conclusion reached by Kuiper. The fact is that if (f> and tp are linear flows given by linear endomorphisms S and T of R" whose eigenvalues all have zero real part then is topological^ equivalent to if/ if and only if (j> is linearly equivalent to . More generally, two linear flows \U0 is linearly equivalent to tf/\V0, dim U+ = dim V+ and dim U- = dim V— For full details see Kuiper [2]. (4.8) Example. Linear flows on R. Any linear endomorphism of the real line R is of the form x >-» ax for some real number a, which is the eigenvalue of the map. Thus the map is a hyperbolic vector field if and only if a # 0. The integral flow is t.x=xeat, and there are exactly three topological equivalence classes, corresponding to the cases a < 0, a = 0 and a > 0 (see Figure 4.8). — > — • — - —

•••••••••••

——

• — > —

0

0

0

£7<0

£7-0

f>0

FIGURE 4.8

(4.9) Example. Linear flows on R 2 . The real Jordan form for a real 2 x 2 matrix is one of the following three types: r

(i)

A _0

0 UJ

where A, /J., a, b are real numbers and b > 0. The corresponding eigenvalues are A, /J, and a ± ib. It follows that a linear vector field on R 2 is hyperbolic if and only if A/* ^ 0 (case (i)), A ^ 0 (case (ii)) or a i* 0 (case (iii)). We have three topological equivalence classes of hyperbolic flows, and Figure 4.9

87

LINEAR FLOWS ON R'

I

FIGURE 4.9

illustrates representatives

H.3-

[I.?] - [J 3

of these classes. (N.b. Many classical texts on ordinary differential equations employ a finer classification than ours, giving rise to terminology such as proper node, improper node and focus.) There are also five topological equivalence classes of non-hyperbolic linear flows. Case (i) with fi = 0 gives three classes corresponding to A < 0, A = 0 and A > 0. If A = 0 we have the trivial flow; the other two cases are illustrated in Figure 4.10. These three are, of course, product flows. The remaining two, which are irreducible, are case (ii) with A = 0 and case (iii) with a = 0 (see Figure 4.11).

x»0

x<0 FIGURE 4.10

Case (ii), X=0 Case(iii), a = 0

FIGURE 4.11

(4.12) Exercise. Show that there are seventeen topological equivalence classes of linear flows on R 3 . Show also that, for all n > 3, there is a continuum of topological equivalence classes of linear flows on R".

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(4.13) Exercise. Classify up to topological equivalence affine vector fields on R" for n = 1, 2, 3.

II. LINEAR AUTOMORPHISMS OF R"

Discrete linear dynamical systems on R" are determined by linear automorphisms of R". There are two obvious equivalence relations to compare, topological conjugacy and linear conjugacy, which is similarity in the ordinary sense of linear algebra. Recall that if 5 and Te GL(Rn), S is similar to Tif f o r s o m e P e GL(R"), T = PSP~\ The relation between linear and topological conjugacy is more difficult to pin down than that between linear and topological equivalence for flows. Kuiper and Robbin [1] have made a detailed study of the problem. We content ourselves here with the observation that there is an open, dense subset HL(R") of GL(R") analogous to EL(R") in the flow case. Elements of HL(R") are called hyperbolic linear automorphisms of R". For any T e GL(Rn), T e HL(R") if and only if none of its eigenvalues lies on the unit circle S 1 in C. Note that T is a hyperbolic vector field if and only if exp T is a hyperbolic automorphism. It is unfortunate that the term "hyperbolic" must carry two meanings, but the context should always make clear which is to be taken. If T e HL(R") then we find that R" decomposes into two direct summands corresponding to eigenvalues of T inside S 1 and eigenvalues outside S . The classification of HL(R") up to topological conjugacy is bound up with the dimensions of these invariant subspaces, and so resembles the classification of EL(R") up to topological equivalence. In fact, elements S and T of EL(R") are topologically equivalent if and only if exp 5 and exp T are topologically conjugate. (4.14) Example. Automorphisms of R. As we have seen, any element T of GL(R) is of the form T(x) = ax, where a is a non-zero real number. There are exactly six topological conjugacy classes, given by (i) a < — 1, (ii) a = — 1, (iii) - 1 < a < 0, (iv) 0 < a < 1, (v) a = 1, (vi) a > 1. Of these, all but (ii) and (v) are hyperbolic. (4.15) Example. Automorphisms of R 2 . Let T e GL(R2). Referring to Example 4.9, we can suppose that the matrix of T is in Jordan form (i), (ii) or (iii). Now T is hyperbolic if and only i f A 2 ^ l # / u , 2 , in cases (i) and (ii), or a 2 + Z> 2 #l, in case (iii). There are eight topological conjugacy classes of hyperbolic maps. All are product of automorphisms of R, all occur in case (i) and six occur only there. Case (i) yields also eleven classes of non-hyperbolic maps. These are the products of the six classes of GL(R) with the identity

II

LINEAR AUTOMORPHISMS OF R"

89

map id on R, and of five of them (id being excluded) with the antipodal map —id on R. We get two further classes from case (ii) with A = ± 1 . The non-hyperbolic automorphisms of case (iii) give a continuum of different topological conjugacy classes, as we now show. When a2 + b2 = 1 in case (iii), T is a rotation through an angle 8, say, where 0 < 8 < IT. Rotation through 8 is topologically conjugate to rotation through —0, by conjugating with a reflection. The question is: Can rotations T\ and T2 through distinct angles 8i and 82 be topologically conjugate if 0 < 0, < 77, / = 1, 2? We first observe that if 8X = 2-rrp/q, with q > 0 and p coprime integers, then every point x ¥= 0 is a periodic point of T\ of period q. Thus T2 cannot be topologically conjugate to Tx unless 82 = 2irp'/q, where p' and q are coprime. Suppose, then, that 82 has this form, and let h: R -» R be a topological conjugacy from Tt to T2. Let Sr denote the circle with centre 0 and radius r. Choose r large enough for 5 r to contain the compact set h (Si) in its interior. By a highly non-trivial theorem of topology, there is a homeomorphism, / say, from the closed region D between h(S\) and Sr to the closed annulus A between Si and S 2 . We may assume that/takes Sr to S2 and preserves the standard orientation. Let g be the homeomorphism of A induced by / from T2 \D (i. e. g = /(T 2 \D )f~x). Then g is pointwise periodic of period q, since T2\D is. Let C = [1, 2] x R, and let w: C -> A be the covering defined by Tr(t, u) = te2mu. We may lift g to a homeomorphism G:C->C such that irG = gir, by a similar technique to that used in Exercise 2.4. By the periodicity of g, Gq{t, u) = (t,u+n(t, u)), where n(t,u) is an integer that varies continuously with (t, u). Now g\S2 has rotation number [p'/q], since it is topologically conjugate to r 2 |S r by f\Sr. Thus Gq(2, u) = (2, u + mq +p') for some fixed integer m and all u € R. One easily deduces that since n{t,u) is a continuously varying integer, n(t, u) = mq +p' for all (t, u). In particular the rotation number of g|Si is [p'/q]. But g|Si is topologically conjugate to Ti\Si by g/i|Si. Hence [p/q] = [p'/ql We have now proved that if 0 < 8t < TT {i = 1, 2), 8i/2n is rational, and 7\ and T2 are topologically conjugate then 8\ = 82. This leaves the case when di/2-jr and d2/2ir are irrational, and, fortunately, this is much easier than the rational case. If h is a conjugacy between T\ and T2 and if x e S 1 then ft maps the closure of the 7\-orbit of x onto the closure of the T^-orbit of h(x). But the first of these sets is Si and the second is S r for some r > 0. Thus h restricts to a conjugacy between Ti|Si and r 2 |S r . Therefore, 8\/2TT - 82/2TT, since these are the rotation numbers of the two circle homeomorphisms. As one might expect from the above details, it is rotations, and, in particular, periodic rotations that cause the most trouble when one attempts to deal with automorphisms of R" for large n. In fact, Kuiper and Robbin in their paper [1] reduced the classification of automorphisms of R" to the so called periodic rotation conjecture, that any two periodic orthogonal maps of

90

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CH. 4

R" are topologically conjugate if and only if they are linearly conjugate. It is this conjecture that is unsolved at the time of writing, although some results are known in the positive direction.

III. THE SPECTRUM OF A LINEAR ENDOMORPHISM We now embark on a study of linear dynamical systems on a Banach space E (real or complex). As we have seen above, when E = R" the eigenvalues of the system (vector field or automorphism) play a vital role in the theory. In infinite dimensions, the analogue of the set of eigenvalues of a linear endomorphism T is called the spectrum a{T) of T. In the complex case, it is the set of all complex numbers A for which T-\(id) is not an automorphism. It is a compact subset of C, and is only empty in the trivial case E = {0}. We shall need a certain amount of spectral theory for our investigation of linear systems. For those unacquainted with this theory, most of the necessary material is gathered in the appendix to this chapter. We give here a brief summary. Recall that the set L(E) of (continuous) linear endomorphisms of E is a Banach space, with norm denned by | r | = s u p { | r ( x ) | : x e E , |x|=£l}. The spectral radius r(T) of T is the radius of the smallest circle in the Argand diagram with centre 0 containing the spectrum
IV

HYPERBOLIC LINEAR AUTOMORPHISMS

91

uniquely as a direct sum E s ©E„ of T'-invariant closed subspaces, such that, if Ts: E s -> E s and Tu: EM -> Eu are the restrictions of T, then a(Ts) - crs(T) and cr(Tu) = aru(T). Moreover, the splitting of E depends analytically on l e L ( E ) . More precisely, the subspace E s , for example, is obtained as the image of a projection S e L(E) (projection means S2 = S), and 5 varies in L(E) as an analytic function of T in ( T e L ( E ) : cr(T)nD empty}.

IV. HYPERBOLIC LINEAR AUTOMORPHISMS Let T € GL{E), the open subset of L(E) consisting of all linear automorphisms of E. We say that T is a hyperbolic linear automorphism if cr(T) n S is empty, where S1 is the unit circle in the Argand diagram C. The set of all hyperbolic linear automorphisms of E is denoted by HL(E). (4.16) Example. The map T:R2^R2

given by

T(x,y) = (h,2y) 2

is in HL(R ). Its spectrum is the set {§, 2} of eigenvalues of T and neither of these points of C is on S . The effect of T is suggested in Figure 4.16, where

FIGURE 4.16

the hyperbolae and their asymptotes are invariant submanifolds of T. The x-axis is stable in the sense that positive iterates of T take its points into bounded sequences (in fact they converge to the origin 0). The y-axis is unstable (meaning, in this context, stable with respect to T _ 1 ). These axes together generate R 2 , of course. A direct sum decomposition into stable and unstable summands (one of which may, however, be {0}) is typical of hyperbolic linear automorphisms.

92

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(4.17) Exercise. Prove that if Te GL(Rn) and if a e R is sufficiently near but not equal to 1, then aT is hyperbolic. Deduce that HL(Rn) is dense in GL(R"). (4.18) Exercise. Let T e HL(E). Prove that 0 e E is the unique fixed point of T. Let TeHL(E) and let
HL(E) then there is an equivalent norm || || such that

(i) for all x = xs + xueE,

\\x\\ = max{||xs||, ||*„||},

(ii) m a x f l l r . U r ^ l l H a ^ i .

•

Thus, with respect to the new norm || ||, Ts is a metric contraction and Tu is a metric expansion. We call a the skewness of T with respect to || ||. The stable summand Es of E with respect to T is equivalently characterized as {xeE: Tn(x) is bounded as n-» oo} and {xeE: T"{x)->Qasn ^oo}. We also call E s the stable manifold of 0 with respect to T and Ts the stable summand of T. A similar characterization holds for the unstable summand Eu, with T~n replacing T". The main aim of this section is to prove that hyperbolic linear automorphisms of E are stable in L(E) with respect to topological conjugacy. A first step in this direction is: (4.20) Theorem. HL{E) is open in GL{E), and hence in L(E). Proof. Let TsHL(E). Choose a norm || || as in Theorem 4.19. Let T have skewness a. We assert that, for all T ' e G L ( E ) with | | 7 " - r | | < l - a , Te HL(E). Let A e S 1 . Then Ag cr(T), so T-\(id) is an automorphism. In fact we may explicitly write down the components of (T - A (id))~ in the E s and Eu directions; they are — Z r =o^ ""r-1(Ts)r and £ " 0 A ^ r j " ' - 1 . From these expressions it is easy to see that | | ( T - A ( / d ) ) _ 1 | | ^ ( l - a ) _ 1 . Thus, if T satisfies the given inequality, then the series

I (r-A(ia)) _1 ((r-r)(r-A(W)) _1 ) r r= 0

converges and is an inverse of T'-\(id). hyperbolic.

Thus \go-(T'),

and hence 7" is •

IV

HYPERBOLIC LINEAR AUTOMORPHISMS

93

Our next step is to study the direct sum decomposition of E associated with a hyperbolic linear automorphism. We say that T e HL(E) is isomorphic to 7" € HL(E') if there exists a (topological) linear isomorphism from E to E' taking E S (T) onto E'S(T') and EU(T) onto EU(T). Equivalently, T is isomorphic to 7" if and only if there are linear isomorphisms of ES(T) onto E'S(T) and of EU(T) onto E'U(T'). Thus, for example, there are exactly n +1 isomorphism classes in HL(R"). We prove: (4.21) Theorem. Any T e HL(E) is stable with respect to isomorphism. Proof. By the spectral theory of the appendix to the chapter, there is a continuous (in fact, analytic) map f:HL(E)-*L{E) such that for all Te HL(E),f(T) is a projection with image ES(T). Let TeHL(E). Then /(T)f( T) ^/(TfasT'^T in HL(E). Since the restriction of /(T) 2 ( = /(T)) t o E s ( T ) is the identity, we deduce that if T' is sufficiently near T then f(T)f(T') restricts to a (topological linear) automorphism of ES(T). Thus /(7") maps E S (T) injectively into E S (T). Similarly, since f(T')f(T) -f(T')2^ 0 as V-+T and since f{T'f is the identity on ES(T'), f(T')f(T) is an automorphism of E s (7") for V sufficiently close to T. Thus/(T") maps ES(T) surjectively to ES(T'). Thus ES(T) is isomorphic to ES(T'). A similar argument applies to the unstable summands. Thus T" is isomorphic to 71 D (4.22) Exercise. Let T<= GL(R")\HL(Rn). By comparing aT with bT, for 0 s, Tr(A) intersects TS(A) only when s = r +1, in which case the intersection is T r + 1 (Si). It is easy to show that any topological conjugacy h: E -» E' between T and some homeomorphism / of a Banach space E' is completely determined by its values on A. For, to find h (x) for any x e £'\{0}, we map x into A by some power T", then map by h and finally map by f~". Since h is a conjugacy, we have now reached h(x), since h =f~nhTn. (Because h is a conjugacy, fh = hT and so fh = hT" for all integers r.) Conversely, let T and T' be contracting linear homeomorphisms of E and E' respectively, and let A and A' be corresponding fundamental domains.

94

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CH. 4

FIGURE 4.23

Suppose that h: A-* A' is a homeomorphism such that (4.23)

for all x e Su

hT(x) = T'h(x).

Then we may extend h to a conjugacy from T to V by putting h(0) = 0 and, for all xeE\{0}, h(x) = (TY"hTn{x), where Tn(x)eA. By (4.23), if n n n n l n+ T {x) e Si then {T'y hT (x) = {T'Y - hT \x), and so h is well-defined. In like fashion we can construct an inverse /i _ 1 :E-»E from ft_1:A'-»A. x Continuity of h (and of h~ ) is in doubt only at 0. To deal with this, observe that h maps {Tn(B\)\ n e Z}, which is a basis of neighbourhoods of 0 in E, onto a similar basis {(T')"(Bi): n e Z} in E'. The result that we have now proved may be formulated, rather loosely, as follows: (4.24) Theorem. Contracting linear homeomorphisms are topologically conjugate if and only if they are topologically conjugate on fundamental domains. • In the following important case, we may explicitly construct a conjugacy between fundamental domains. (4.25) Theorem. LetT and T' belong to the same path component of GLi^E) and have spectral radii < 1. Then T and T' are topologically conjugate. Proof. Let T and T' be metric contractions with respect to norm | | and | |' respectively on E, and let A and A' be the corresponding fundamental domains (between 5i and T{S\) and between 51 and r ' ( 5 l ) ) . Pick e with 0 < e < m i n { l - | r | , l - | r | ' } , and let I = [l-e, 1]. Since I T 1 and the identity map id belong to the same path component of GL(E), there exists a continuous map g: I -» GL(E) with g(l - e) = T'T~l and g(l) = id. Identifying IxSi and 7 x 5 1 with annuli in E, by putting (t, x) = tx, we have a homeomorphism h:IxSi->IxS'i defined by ,(t

.

tg(t)(x)

IV

HYPERBOLIC LINEAR AUTOMORPHISMS

95

We extend to a homeomorphism h:A-*A', by mapping the line segment [(l-e)x, JC/|7 , " 1 (X)|] linearly onto the line segment ("

LU

, T'T-\x) TT-\x)-\ S)

\T'T-\x)\"

\T^(x)\'\

for each x e Si. Notice that for all y e Su T-\x) ~\T-\x)\

y

for some x e Si, and that

Thus (4.23) holds, and we can apply Theorem 4.24 to complete the argument. We leave the details of checking that h: A -» A' is a homeomorphism to the reader. D This result enables us to classify hyperbolic linear automorphisms up to topological conjugacy, in finite dimensional spaces. We concern ourselves particularly with R". In the first place, we have (4.26) Corollary. For each n > 0 there are exactly two topological conjugacy classes of contracting linear homeomorphisms of R", consisting respectively of orientation preserving and orientation reversing maps. Proof. GL(R") has exactly two path components, consisting of orientation preserving and orientation reversing maps (distinguished by the sign of the determinant). These give distinct topological conjugacy classes, since any topological conjugate of an orientation preserving homeomorphism is orientation preserving. This is most easily proved by extending such homeomorphisms to the one-point compactification S" of R", and using the fact that if / : R" -> R" is a homeomorphism, and / : S" -* S" is its extension to S", then the induced isomorphism of homology, maps a generator 1 of Hn(S") s Z to 1 or - 1 according as / is orientation preserving or reversing (see Greenberg [1] for details). • (4.27) Corollary. Two hyperbolic linear homeomorphisms of R" are topological^ conjugate if and only if (i) they are isomorphic, (ii) their stable components are either both orientation preserving or both orientation reversing, and (iii) their unstable components are either both orientation preserving or both orientation reversing.

96

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CH. 4

Proof. Suppose that T, T e HL(R") satisfy conditions (i) to (iii). For simplicity of notation, we put R" = E. Let L e GL(E) map ES(T) onto ES(T) and EU(T) onto EU(T'). By Corollary 4.26, the restrictions of T and L^T'L to ES(T) are topologically conjugate, and so are their restrictions to EU(T). This enables us to construct a topological conjugacy from T to L"1 T'L, and hence to T'. Conversely, let ft be a topological conjugacy from TeHL(E) to T'e HL(E). Then ft preserves 0, the unique fixed point of T and V. By the continuity of ft and of ft-1, r n (jc)-»0 as «-»oo if and only if hT"(x) = {T')nh{x)-> 0 as n -* oo. Thus ft maps E s ( r ) onto E s ( r ) , and similarly E „ ( r ) onto E u (7"). Since dimension is a topological invariant (§ 1 of Chapter 4 of Hu [2]) T and V are isomorphic. Conditions (ii) and (iii) now follow by Corollary 4.26. • It follows immediately from this theorem that there are exactly An topological conjugacy classes of hyperbolic linear automorphisms of R" (n 5* 1). (See Examples 4.14 and 4.15.) (4.28) Exercise. Prove that, for any n > 0 , GL(R") has, as asserted above, precisely two path components. (Hint: move T e GL(Rn) into the subspace GL(R)xGL(R"' 1 ).) (4.29) Exercise. Classify hyperbolic linear automorphisms of C" up to topological conjugacy. For infinite dimensional Banach spaces, the situation is less clear. Homeomorphic Banach spaces need not be linearly homeomorphic, so that topological conjugacy does not immediately imply isomorphism. Furthermore, the connectivity properties of GL(E) are not fully understood for an arbitrary Banach space E. In several important cases (for example if E is a Hilbert space—see Kuiper [1]) GL(E) is path connected, and so any two linear contractions are topologically conjugate. However our main stability theorem, which we now prove, is, happily, valid for all Banach spaces. (4.30) Theorem. For any Banach space E, any T e HL(E) is stable in L(E) with respect to topological conjugacy. Proof. Consider the ball B in L(E) with centre T and radius d > 0. Let T1 &B and, for e a c h / € [ 0 , 1 ] , define T' by T'= (l-t)T + tT1. As in the proof of Theorem 4.21, we choose d so small that for all T1 and f e [ 0 , 1 ] , the restrictions P':ES(T)^ES(T') and Q':ES(T')-*ES(T) of the projections f(T') and f(T) respectively are isomorphisms. We now compare Ts and ( P 1 ) - 1 ^ 1 ) ^ 1 , both of which are in GL(ES(T)) and have spectral radius < 1 . We show that these two maps are in the same path component of GL(ES(T)). In fact we assert that tH-» {P'yl(T')sP' is a path joining the one

V

HYPERBOLIC LINEAR VECTOR FIELDS

97

to the other. The only difficulty is whether this map of [0,1] into GL(ES{T)) is continuous. Since / is continuous the map from [0,1] to L(ES(T)) taking / to Q'(T')SP' (which is the restriction of f(T)(T')f(T')) is continuous. Similarly the map taking / to Q'P' is continuous. Hence the map taking t to (0'P')~ 1 0'(7 , ')s-P' is continuous, as stated. To complete the proof of the theorem, we apply Theorem 4.25, giving that Ts is topological^ conjugate to (P 1 r 1 (r 1 ) s i> 1 , and hence to (T1),. A similar argument holds for the unstable components. Thus T is topologically conjugate to T . • (4.31) Corollary. Let E be finite dimensional. Then HL(E) is the stable set of GL(E) with respect to topological conjugacy. Furthermore HL(E) is an open dense subset of GL(E). Proof. By Theorem 4.30 any hyperbolic map is stable with respect to topological conjugacy. If T e GL(E) is not hyperbolic, every neighbourhood of T meets two different isomorphism classes of HL(E) (by Exercise 4.22), and hence meets two different topological conjugacy classes (by Corollary 4.27). Thus T is unstable. This proves that HL(E) is the stable set of GL(E), and also that HL(E) is dense in GL{E). It is open in GL(E) by Theorem 4.20. • In the above theory we have not bothered to analyse the size of perturbation that a given hyperbolic map admits without change to its topological conjugacy class. We shall, in the next chapter, return to Theorem 4.30 and give it a new proof, which is better designed to deal with this point.

V. HYPERBOLIC LINEAR VECTOR FIELDS We now turn to a study of linear vector fields. We are interested in determining which ones are stable in L(E) with respect to topological equivalence, and once again we find we can give a complete description for finite dimensional E. Let T e £,(E). We say that T is a hyperbolic linear vector field if its spectrum a(T) does not intersect the imaginary axis of C. We denote the set of all hyperbolic linear vector fields on E by EL(E). As we have commented earlier, the notion of hyperbolicity depends on whether we are regarding T as a vector field or a diffeomorphism. It will, we hope, always be clear from the context in which sense we are using the term. The connection between the two notions is expressed in the following propositions, whose proofs are straightforward exercises in spectral theory (see Theorem 4.55 of the appendix). The special case E = R" has already been discussed above. The linear automorphism exp {tT) may be defined either by its power series or by a contour integral, as in (4.50) of the appendix.

98

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CH. 4

(4.32) Proposition. A linear vector field Tis hyperbolic if and only if, for some {and hence all) non-zero real t, exp (tT) is a hyperbolic automorphism. • (4.33) Proposition. The linear vector field T has integral flow cf>: R x E -» E given by (t, x) = exp (tT)(x). • (4.34) Example. The linear vector on R 2 defined by T(x, y) = (—x, y) is hyperbolic. The integral flow is illustrated by Figure 4.16, branches of the hyperbolae now being orbits of T, as are the positive and negative semi-axes and the point 0. Suppose that TeEL(E). Let as(T) = {\ ea(T): re A < 0 } and cru(T) = {A e a(T): re A > 0}. Let D be a contour in the Argand diagram consisting of the line segment [-ir, ir] and the semi-circle f

if, IT

3771

where r is large enough for D to enclose crs. Then the spectral decomposition theorem gives us, correspondingly, a ^-invariant direct sum splitting E = E S ©E U such that cr(Ts) = as(T) and a(Tu) = cru(T), where TSGL(ES) and Tu e L(EU) are the restrictions of T. Once again we call E s and Ts stable summands, and E u and Tu unstable summands. Similarly Es and E„ are again called stable and unstable manifolds at 0. (4.35) Exercise. Prove that, for all t > 0, the stable summand of the hyperbolic automorphism exp (tT) is exp (tTs). Similarly for unstable summands. The map exp: L(E)-> L(E) is continuous (see the appendix), and thus, since HL(E) is open in L(E), Proposition 4.32 implies that EL(E) is open in L(E). We define the term isomorphic for hyperbolic vector fields exactly as for hyperbolic automorphisms. Since the stable and unstable summands of E are the same with respect to T e EL(E) as they are with respect to exp T e HL(E) (by Exercise 4.35), the fact that TeEL(E) is stable with respect to isomorphism follows immediately from the corresponding result for HL(E). We sum up these observations in the following proposition. (4.36) Theorem. Let TeEL(E). Then T=TS@TU, where a{Ts) =
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HYPERBOLIC LINEAR VECTOR FIELDS

99

(4.37) Theorem. Let TeEL(E) have spectrum o-{T) = as(T). Then there exists a norm || || on E equivalent to the given one such that, for any non-zero xeE, the map from R to ]0, oo[ taking t to ||exp (fT)(x)|| is strictly decreasing and surjective. There exists b>0 such that, for all t>0, ||exp (tT)\\ =£ e~bt. Proof. Let d be the distance from a(T) to the imaginary axis. By the spectral mapping theorem the spectral radius r(exp (tT)) of exp (tT) is e~dt for all t e R. Choose b and c with 0 0 for t e [0,1]. Then, for all integers m and n with 0 =£ m < n,

f \eb'exp (tT)\ dt ^ \

fe(r+l)

e

-:AebY

-

e

e-r{c-b)

iAebe-mle-b)(l-eb-er\ which tends to zero as m -» oo. One easily checks that the relation W=

Jo

\eb'exp (tT)(x)\ dt

defines a norm || || on E. Moreover, for all x e E,

IWI 0, \\exp(tT)(x)\\^e-b,\\x\\, as required.

•

Theorem 4.37 gives a very precise hold on the phase portrait of T. Let Si be the unit sphere in E with respect to the norm || || of Theorem 4.37. Then,

100

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CH. 4

with the exception of the fixed point 0, every orbit of T crosses Si in precisely one point. Thus S x is a sort of cross section to the integral flow of T. We may sharpen this remark as follows: (4.38) Corollary. // maps R x Si homeomorphically onto E\{0}. Proof. We know that is continuous, and Theorem 4.37 shows that it maps R x Si bijectively onto E\{0}. Thus it remains to prove that the inverse of the restricted map is continuous. Consider, in E\{0}, a sequence (x„) converging to a point x. Then x„ = (tn, y„) for some tn e R and y„ e Si. The sequence (t„) is bounded, since ||x„||s=e~6r" for tn>0 and (||JE„||) converges to the non-zero number ||ac||. Let (t„k) be a convergent subsequence of (tn), converging to t, say. Then, as k -» oo, y„k =4>(-t„k, x„k) converges to y =(-t, x), which is thus some point of Si. If the sequence (t„) has some other cluster point t', we similarly deduce that tj>(t', x) is in Si, contrary to Theorem 4.37. Thus (tn) converges to t, and hence y„ = (-tn, x„) converges to y. This proves continuity of the inverse map at x. • Now suppose that T, T e EL(E) both have spectra in re z < 0. We can pick equivalent norms on E such that the unit spheres with respect to them are cross sections of the integral flows of T and T restricted to E\{0}. This situation is illustrated in Figure 4.39, in which Si and Si are the unit spheres.

FIGURE 4.39

Since Si and Si are homeomorphic, there is an obvious way of constructing a flow equivalence from T to T'. Once the proposed flow equivalence is given on the unit spheres (mapping Si to Si) its values elsewhere are determined up to the speeding up factor a in the definition of flow equivalence. (4.40) Theorem. Let T, T e HL(E) have spectra in re z < 0. Then Tand T are flow equivalent.

V

HYPERBOLIC LINEAR VECTOR FIELDS

101

Proof. We show that there is a homeomorphism h of E such that, for all (t, x ) e R x E , (4.41)

h(t,x) = il>(t,h(x))

where and ip are the integral flows of T and 7" respectively. Let || || be as in the statement of Theorem 4.37, and let || ||' have the same property relative to T". Let 5i and Si be the unit spheres_with respect to these norms. We define a homeomorphism h: Si -» Si by h(y) = y/||y||'. Let <£: R x Si -»• E\{0} and i/r: R x Si -> E\{0} be the restrictions of <£ and i/f. By Corollary 4.38, and i/f are homeomorphisms. We define h:E->E by putting /i(0) = 0 and letting /r be the homeomorphism if (id x h)4>~1 on E\{0}. Then /i extends £ Also (4.41) holds trivially for x = 0. For x ^ 0, x = <£(/, y) for some f e R and y e Si, and so, for all f e R, /t0(f, x) = h(t + t', y)

= ip(idxh)(t + t', y) = j(t + t',h(y)) = (t\ My)) = Ht,h(x)). Finally we show that h is continuous at 0 (the proof for h~1 being similar). Let W be any neighbourhood of 0 in E. By Theorem 4.37, there exists t0 > 0 such that, for all s^t0,4>s(S[)c: w. For all xeE with ||JC||< ||<£~'0ir , \\(~to, x)\\m 1 and so, by Theorem 4.37, the number 5 such that (-s, x)eSi satisfies s>t0. Thus, since h(x) = i//(s, h(-s, x)) and h<(>(-s,x)eS'uh(x)eW. D By applying this theorem to stable and unstable summands separately, we have: (4.42) Corollary. / / T and 7" 6 EL(E) are isomorphic, then they are flow equivalent (and thus topologically equivalent). • By Theorem 4.36, we now deduce our main stability result: (4.43) Corollary. Any TeEL(E) is stable in L(E) with respect to flow equivalence (and thus with respect to topological equivalence). • (4.44) Exercise. Let E be finite dimensional. Prove (i) EL(E) is dense in L(E),

102

LINEAR SYSTEMS

CH. 4

(ii) for all 7, 7" eI?7(E) the following statements are equivalent: (a) 7 is flow equivalent to 7", (b) 7 is topologically equivalent to 7', (c) 7 is isomorphic to 7', (iii) 737(E) is the stable set of 7(E) with respect to (a) flow equivalence, and (b) topological equivalence. Thus, for E = R" there are precisely n + \ flow equivalence classes in EL(E), classified by the dimension of the stable summand. (See Examples 4.8 and 4.9.)

Appendix 4

I. SPECTRAL THEORY In this brief survey of spectral theory we omit a number of proofs that can be found in our main reference for the subject, which is Dunford and Schwartz [1]. We shall also assume without comment easy generalizations of certain parts of the theory of functions of one complex variable from C-valued to E-valued functions, where E is any complex Banach space. For these we refer to § 14 of Chapter 3 of Dunford and Schwartz [1]. Let E be a complex Banach space, with E^{0}, and let T e L ( E ) . The resolventp{T) of T is the set {A e C: T - A (id) is a homeomorphism}. Clearly p(T) is an invariant of the linear conjugacy class of T. The spectrum a(T)ot T is the complement of p(T) in C. For finite dimensional E,
(T-A(irf))- 1 I

n

V"(T-\(id))-

.

n=0

Thus p(T) is open, and R is analytic. Notice the relation (4.45)

(r-A(W)r1-(T-/t(W)r1 = (A-Ai)(r-A(W)r1(r-/t(W)r1

resulting from the trivial manipulation L~1-M~1=M~1ML~1-M~1LL~1 103

=

M~1(M-L)L~\

104

CH. 4

APPENDIX

We may continue manipulating to obtain (L-1-M-1)(id

+ (M-L)L-1)

=

L-1(M-L)L-1

and then, for fixed L and small \M — L\, OO

L_1-M_1= I

(-l)rL_1((Af-L)L_1)r+1.

r= 0

We use this in the particular case (4.46)

(r-A(/J)r1-(r-A(/rf)r1 CO

= I (-mr-AOv/rtr-rxr-Aad))-1)"1 r=0

showing that i?(A) depends analytically on T, forfixedA. We define a non-negative real number r{T) by r ( r ) = l i m sup | r T " . n-*oo

Notice that r(T) =£ |T| (where | | denotes both the given norm on E and the induced norm on L(E)). The series —£" = 0 A " ~ l T " converges to i?(A) for | A | > r ( r ) , and diverges for | A | < / , ( T ' ) . Since this is a Laurent series for R centred on 0, we deduce that r{ T) is precisely the spectral radius sup{|A|: A eo-(r)} of T. Since cr(T) is bounded and closed it is compact. Moreover, since E^{0}, o-(T) is non-empty. For, otherwise, the domain of R is C, so the above Laurent series is a Taylor series, which is impossible, since the coefficient of A - 1 is id. It is possible to simplify the formula for r(T). We observe that, for all n > 0 , \ecr(T) implies \"ea(Tn), because T-X(id) is a factor of Tn-\n(id). In this case, | A " | « | T " | , and so \x\^\Tn\1/n. Thus r(T) =s lim„_<x, inf | T" \1/n, and it follows that lim^o, IT" | v " exists and equals r{T). We now give a very useful characterization of r(T), due to Holmes [1]. (4.47) Theorem. Let Nbe the set of all norms on L(E) induced by norms on E equivalent to the given norm \ \. Then r ( D = inf{||r||:||

\\eN}.

Proof. It follows from the definition of r(T) that r(r)=e||T|| for all || \\eN. Suppose that r(T) 0 and all n s= m,\T"\^Kn. We define a norm || || on E by II

II

f! I I

" I T

||jc|| = max 1|JC|, \K

1

/

\l

T{X)\,...,\K

I —m + 1 r^m — l/

\h

1 U)|j.

105

SPECTRAL THEORY

I

It is easy to check that || || is a norm. Moreover, \x\ =£ \\x\\, so, by the interior mapping theorem, | | is equivalent to || ||. Now ||T(x)|| = max{|r(x)|, k " 1 ^ ) ! , . . . , |/<- m + 1 r m (x)|} = K m a x d K ^ T U ) ! , . . . , \K-m+1Tm-\x)\

\K~mTm(x)\}

^K\\X\\, m m

since \K^ T \^l.

Thus

||T||»SK.

D

In particular: (4.48) Corollary. The spectral radius r(T) of T is strictly less than 1 if and only if there is an equivalent norm on E with respect to which T is a metric contraction. • (4.49) Exercise. Prove that if T is a linear automorphism then cr(r _ 1 ) = o-(T)~l( = {A -1 : A e o-(T)}). Deduce that o-(T) lies outside the unit circle S 1 of C if and only if T is a metric expansion with respect to some equivalent norm on E. Deduce that if E has a T-invariant direct sum decomposition E = E s ©E u , and if T is a metric contraction on E s and a metric expansion on E u , then T is hyperbolic (i.e. cr(T) n S 1 is empty). If p{z) is a complex polynomial, sayp(z) = X* = 0 a n .z n , where an e C , then we can define a function /?:L(E)-»L(E) (by an abuse of notation) by p(T) = Z«=o anT", and we can handle power series in the same way. The technique of contour integration allows us to carry this process one step further. Let / : C/-»C, where U is open in C, be an analytic function. Suppose that a(T)<^K <=• U, where the boundary dK of the compact set K is a union of contours {K need not be connected). We define f(T)eL(E) by (4.50)

/(T)=--M f(X)R(X)dX, 2 m JaK

where dK is positively oriented with respect to K in the usual way. Note that f(T) is well defined (by (4.45)), is independent of the choice of K, and that we obtain an analytic (by (4.46)) function / from the open subset { T e L ( E ) : c r ( r ) c U] into L(E). The proofs of the following propositions appear in § 3 of Chapter 7 of Dunford and Schwartz [1]). (4.51) Proposition. For all a, /3 6 C and all analytic functions f, g: U -* C, (af + Pg)(T) = af{T) + pg(T). • (4.52) Proposition. For all analytic functions f, g: U -» C, let g.f:U -» C be the map whose value at z is the product g(z)f(z). Then (g.f)(T) is the composite g(T)f(T) of the maps g(T) andf(T). D

106

APPENDIX

CH. 4

(4.53) Proposition. If f(z) = Y,r=0arZr converges on a neighbourhood of o-{T),thenf(T) = ZZo<*rTr. • (4.54) Proposition. Ifcr(T) is contained in the domain of the composite gfof analytic maps f and g then (gf)(T) = g(f(T)). • (4.55) Theorem. {Dunford's spectral mapping theorem). For all analytic mapsf:U^C,f(o-(T)) = o-(f(T)). • (4.56) Corollary. (Spectral decomposition theorem). Let Te GL(E), and let D be a contour in the Argand diagram C such that o-(T)r\D is empty. Let crs(T) be the part of cr(T) inside D and let o-u(T) be the part outside D. Then there is a direct sum decomposition E = E S ©E„ into T-invariant closed subspaces such that, ifTs e I ( E , ) and Tu eL(Eu) are the restrictions of T, then cr(Ts) = crs(T) and o-(Tu) = o-u(T). Proof. We define the analytic complex function / on the complement of D in C b y

*H; ;;

if A is inside D, A is outside D.

Let Ks (resp. Ku) be a compact subset inside (resp. outside) D and containing o-s(T) (resp. au(T)), and let K =KS u Ku. We suppose further that dK is a union of contours, so that f(T) may be defined by the formula (4.50). By Proposition 4.52 since f.f = f,f(T)2=f(T). That is to say, f(T) is a projection. Thus E = ES(T)@EU(T), whereE s (T) is the kernel of id -f(T) (or, equivalently, the image of f(T)), and EU(T) is the kernel of f(T). Since T commutes with/(T) (by Proposition 4.52 any two functions of T commute), ES(T) and E U (T) are invariant under T. Moreover Tf(T) = Ts®0 = h(T), where, by Proposition 4.52, MA) =

A if A is inside D, 0 if A is outside D.

By Theorem 4.55, o-(Tf(T)) = h(o-(T)) =
I

107

SPECTRAL THEORY

The space E has a natural complex vector space structure (in which scalar multiplication acts on the second factor C of E) and we may give it a norm |jt®l + y ® / | = max{|;c|, |y|}, which makes it into a Banach space. Then T induces a C-linear automorphism f e L(E) by its action on the first factor E. We define the spectrum a(T) of T to be cr(f). The crucial fact that emerges is that, if the spectrum of f decomposes as in Corollary 4.56 and if D is symmetric about the real axis, then the projection f(T) in the proof of the Corollary acts independently on the real and imaginary parts of E, and so decomposes the real space E. Once we have proved this, the real version of Corollary 4.56 follows immediately. (4.57) Theorem. Let the spectrum of T decompose as in Corollary 4^56. Suppose thatD is symmetric about the real axis. Then the projection P=f(T) is oftheformP{x®l + y®i) = P(x)®l+P(y)®i,forallx,yeE,wherePisa projection of E. Proof. Any R e L{E) may be written in matrix form as

Me l\ where the entries are in L(E). Let R denote the matrix

\ A ~Bl Y-c

D\

Note that, for all z e E,_the_complex conjugate Rz = Rz, and hence, for all R1,R2€Lm,R1R2 = R1R2. Now recall that P=f{f) is defined by (4.50), where K is as described in the proof of Corollary 4.56. Moreover we may choose K so that dK is symmetric about the real axis. Now, in the formula (4.50), R(A) denotes the resolvent (f-\(id))~\ By conjugating the relation (f-\(id))R(\) = id, we see that R(\) = R(\). Let BK+ and dK- be the portions of dK in the upper and lower half planes of the Argand diagram, with orientation induced by that of 3V. Then by the symmetry, dK- = -dK+. Thus P = ^—(\ R(k)dX+[ 2m\JsK+ ]aK = — (f 2m\iaK+

R{k)dk-\

= =-[ 2.TT J 3JC+ \l

JaK+

R{\)d\) I R{k)d\) /

(-R(x)d\+-R(\)d\). I

I

108

CH. 4

APPENDIX

Since the term in the bracket acts separately on the real and complex parts of E, so does P. Thus P, being a projection, has the form P(x®l

+ y®i) = P(x)®l

+ Q(y)®i.

But since P is C-linear P(iz) = iP(z), and this implies that P = Q.

•

It follows immediately that the stable summand T$: E s -» E s of T is the complexification of an R-linear automorphism TS:ES-*E,S restricting T. Similarly for the unstable summand. Summing up: (4.58) Corollary. (Real spectral decomposition theorem) Let E be a real Banach space, let T e GL(E) and let D be a contour in the Argand diagram, symmetric about the real axis, such that o-{T)nD is empty. Let o-s(T) be the partofa(T) inside D and letau(T) be the part outside D. Then there is a direct sum decomposition E = E S ©E U into T-invariant closed subspaces such that if Ts € l ( E j ) and Tu eL(E„) are the restrictions of T, then o-(Ts) = crs(T) and a(Tu) = o-u(T). •

CHAPTER 5

Linearization

We can hardly expect to make much headway in the global theory of dynamical systems on a smooth manifold X if we are completely ignorant of how systems behave locally (that is to say, near a point). We should like to attempt a local classification, but, bearing in mind our difficulties with linear systems, we do not expect complete success. However, regular (as opposed to fixed) points present no problems. As we shall see in the following section, there is essentially only one type, represented in the case of diffeomorphisms by a translation of E, the space on which X is modelled, and in the case of vector fields by any non-zero constant vector field on E. Thus, in both cases, interesting local behaviour occurs only at fixed points. As we have seen in the previous chapter, linear systems exhibit a wide variety of behaviour near the fixed point zero. Moreover, if we introduce higher order terms, we admit further possibilities. For example, the vector field v(x) = x2 on R has a "one-way zero" (see Figure 5.1), but no linear •

•

>

FIGURE 5.1

vector field has an orbit structure looking like this. In order to prevent our classification programme from getting out of hand, we pick on a class of fixed points that occur often enough to be thought of as "the sort that one usually comes across". In the case of linear systems we were able to give this rather vague notion a precise topological meaning in terms of the space of all linear systems on a given Banach space E. When we generalize to smooth systems on a given manifold X we are still able to topologize the set of all systems, and to discuss open and dense subsets of the resulting space (see Chapter 7 and Appendix B below). Of course the space of all systems is infinite dimensional even when X is finite dimensional. Although the connection 109

110

LINEARIZATION

CH. 5

may seem tenuous at the moment, that is, in fact, one reason why we try to make our theory work for infinite dimensional E whenever possible. Suppose that a given dynamical system has a fixed point p. If we are only interested in local structure, we may assume, after taking a chart at p, that X = E and p = 0. Suppose now that by altering slightly near 0 we always end up with another fixed point near 0, with a local phase portrait resembling that of 4> at 0. In particular, then, the linear approximation of <j> at 0 (if we can make sense of this term) must have a phase portrait near 0 resembling that of cf> near 0, as also must all nearby linear systems. This rough and ready argument, together with the stability theorems of Chapter 4, suggests that we consider fixed points whose linear approximations are hyperbolic systems (automorphisms or vector fields as the case may be). Such points are said to be hyperbolic fixed points. We now give some precise definitions. A fixed point p of a diffeomorphism / of X is hyperbolic if the tangent map Tpf: TPX^> TPX is a hyperbolic linear automorphism. Corollary 4.27 has supplied us with a classification of hyperbolic linear automorphisms, and it turns out that this amounts to a local classification of their fixed points. We extend this to a classification of hyperbolic fixed points using the result due to Hartman that any hyperbolic fixed point p of / is topologically conjugate to the fixed point 0 of Tpf (see Chapter 2 for the definition of local topological conjugacy). This theorem does not require X to be finite dimensional. (5.2) Exercise, (i) Prove that p is a hyperbolic fixed point of / if and only if it is a hyperbolic fixed point of f~ . (ii) Find an example of a diff eomorphism / having a fixed point p that is not topologically conjugate to the fixed point 0 of Tpf. As usual, an analogous situation exists with flows. A fixed point p of a local integral :IxU-^X of a vector field v on X (or equivalently, a zero p of v) is said to be hyperbolic (or elementary) if for some (and hence as we shall see any), t # 0 in /, p is a hyperbolic fixed point of '. Let us examine the implications of this definition. Any admissible chart $:U ^U' 2Xp induces a vector field w = {Tg)vg_1 on U'. We may assume £(p) = 0. Now p is hyperbolic if and only if 0 is a hyperbolic fixed point of i{/' = €'£~ , and by Theorem 3.11 ip is a local integral of w at 0. Let / be the principal part of w. By Theorem 3.32, T(t) = Dip'(0)eL(E) satisfies the differential equation DT = Df(0)T. Moreover Di(to(0) = id. By Proposition 4.33, Dip'(0) = exp (tDf(0)), and thus, by Proposition 4.32, p is hyperbolic if and only if Df(0) e EL(E). Summing up: (5.3) Proposition. The point p is a hyperbolic zero of a vector field v if and only if, for any chart £ at p, the differential atg(p) of {the principal part of) the induced vector field (7£)t>£_1 is a hyperbolic vector field. •

I

REGULAR POINTS

111

An equivalent, but rather more sophisticated, approach is to consider Tv: TX-» T(TX), which is a section of the vector bundle TTTX- T{TX)-+ TX, where irx '• TX -* X is the tangent bundle of X. Now Tirx may be identified with the tangent bundle on TX, TTTX '• T(TX) -» TX by the canonical involution (see Example A.38 of Appendix A) and Tv then becomes a vector field on TX. With this identification Tpv maps TpX into T(TpX). Thus Tpv is a linear vector field on TpX. We call this linear vector field the Hessian of v at p. Again, p is a hyperbolic zero of v if and only if the Hessian of v at p is a hyperbolic linear vector field. (5.4) Exercise. Justify in detail the statements made in the preceding paragraph. (5.5) Exercise. Let p be a critical point of a smooth function g:Rn->R. Relate the Hessian of g at p in the classical sense to the Hessian of the gradient vector field v = Vg (see Examples 3.3 and A.57) at p in the sense described above. We prove in Corollary 5.26 below that if p is a hyperbolic fixed point of a vector field v then it is flow equivalent to the zero of the Hessian of v at p. Thus, using Exercise 4.44, we obtain a complete classification of hyperbolic fixed points of flows on real ra-dimensional manifolds into n +1 classes, distinguished from one another by the dimension of the stable manifold (of the Hessian, let us say at this stage). Once again, the importance of the classification lies in the comparative ubiquity of the type of point under consideration (see Chapter 7). (5.6) Exercise. Give an example of a smooth vector field v having a zero p that is not topologically equivalent to the zero of the Hessian of v at p. We complete the chapter by discussing the structural stability of periodic orbits. In the case of a difTeomorphism / there is very little to be said beyond remarking that a periodic point of / is a fixed point of some positive power /". On the other hand, closed orbits of flows have a more interesting theory, and once again we are able to classify the types that "usually occur" in finite dimensions. I. REGULAR POINTS Recall that a point p of X is a regular point of a dynamical system on X if it is not a fixed point of the system. Equivalently, it is a regular point of a vector field v if v{p) ¥=• 0P. The theorems that follow show that regular points are uninteresting from a local point of view. See Chapter 2 for the definitions of the various types of local equivalence in use.

112

CH. 5

LINEARIZATION

(5.7) Theorem. If p is a regular point of a diffeomorphism f:X^>X and g is translation of the model space E of X by a non-zero vector x0 then f\p is topologically conjugate to g\0. Proof. Let f: [/-» U' be a chart at p with g(p) — 0. We may assume that f(U)r>U is empty, and that the diameter of U' is less than ||x 0 |. Then I together with (g| £/')£(/! £ 0 _ 1 is a (smooth) conjugacy from f at p to g atO. D Of course the above proof has little to do with the smoothness; it works in the context of homeomorphisms of Hausdorff topological spaces for any pair of regular points with homeomorphic neighbourhoods. Moreover, as we commented in Chapter 2, its simplicity could be taken as a criticism of our definition of local topological conjugacy. The analogous theorem for flows has a more secure status, and a correspondingly harder proof. (5.8) Theorem. (Rectification theorem) Letp be a regular point of a C 1flow4> on X. Letx0 be any non-zero vector of the model space E ofX, and let tp be the flow on E defined by ip{t, x) = x + txo- Then is the local integral of a vector field v on some open subset of E. Moreover we can assume after a linear conjugacy that E = R x F for some Banach space F a n d that c(0) = x o = ( l , 0 ) e R x F . Let h be the C 1 map defined on some neighbourhood of 0 in E = R x F by h(u,y) = 4>(u,(0,y)) (see Figure 5.9). {0},F

{0}„F

FIGURE 5.9

Then h preserves the action of R, since for all sufficiently small 5 and t e R and y e F, hUt,{s,y))

= h{s + t,y) = cf,(s + t,(0,y))

= (s,(fl,y))) = 4>(t,h(s, y)).

HARTMAN'S THEOREM

II

113

Since h is the identity on {0} x F, and since DM0)

= Di(0, 0) = v(Q) = (l, 0),

it is clear that Dh(0): E-> E is the identity. Thus, by the inverse mapping theorem (see Exercise C . l l of Appendix C), h is a C 1 diffeomorphism between suitably restricted neighbourhoods of 0 in E. Thus h is a (C ) flow equivalence from \\i at 0 to at 0. D (5.10) Remark. We emphasize that in both the above theorems the equivalence obtained is as smooth as the system under consideration.

II. HARTMAN'S THEOREM Let T be a hyperbolic linear automorphism of E with skewness a < 1 (see Theorem 4.19) with respect to some norm | | on E. Basically, we are interested in showing the stability of T with respect to topological conjugacy under small smooth perturbations, or, more generally, under small Lipschitz perturbations. The central pillar of this theory is the theorem of Hartman [1], which we present below in a version strongly influenced by work of Moser [1] and Pugh [3]. It is immediate from the definition of hyperbolicity that the map id - T is an automorphism. In fact, by the proof of Theorem 4.20 we have the further information that \(id - r ) _ 1 | =s (1 - a ) " 1 . The first statement implies that T has a unique fixed point (namely the origin), and we start by proving that this property is stable under Lipschitz perturbations of T. (5.11) Lemma. Let rf.E->Ebe Lipschitz with constant K < 1 - a. Then T+r\ has a unique fixed point. Proof. The fixed point set of T +17 is precisely that of (id - T ) - 1 T ? , which is a contraction. Thus the contraction mapping theorem (C.5 of Appendix C) gives the result. • (5.12) Exercise. Prove that the fixed point of T + r\ depends in a C fashion on the map 77 regarded as an element of the map space C r (E) (see Appendix B). (Hint: Consider the function from E x C ' ( E ) to E sending (x, rj) to

(M-rrM*).) (5.13) Exercise. Let B be the closed ball in E with centre 0 and finite radius b>0. Prove that if T J - . B ^ E is Lipschitz with constant K<\-a and if 117 |o ^ b(l — a) then T +17: B -> E has a unique fixed point. Recall (Appendix B) that C (E) is the Banach space of all bounded continuous maps from E to E, with the sup norm.

114

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CH. 5

(5.14) Theorem. (Hartman's linearization theorem) Let 17 e C°(E) be Lipschitz with constant K < min {1 — a, \TJ \~ }. Then T +17 w topologically conjugate to T. It is technically convenient to prove a more detailed statement, as follows: (5.15) Theorem. Let 77, £ € C°(E) 6e Lipschitz with constant K less than min {1 —a, ITJ11-1}. Then there exists a unique g e CU(E) such that (5.16)

(T + v)(id + g) = (id + g)(T + 0.

Moreover, id + g is a homeomorphism, and is thus a topological conjugacy from T + £to T + -q. Proof. By the Lipschitz inverse mapping theorem (Exercise C. 11 of Appendix C), the condition K < \ T~s 1 \ ~1 implies that T + £ is a homeomorphism. Thus we may write (5.16) as

(5.17)

(r+T?)(w+g)(r+^r1 = (r+^)(r+^r 1 + g.

We define maps f and ij of C°(E) into itself by

f&^TgiT+tr1 and

v(g)=v(id+g)(T+o~1-aT+a'1

so that (5.17) now reads {T + rj)(g) = g. Clearly 17 is Lipschitz with constant K. Equally clearly T is a linear automorphism with inverse g^*T~ g(T + £). We also observe that T is hyperbolic with skewness a, since it contracts and expands its invariant subspaces C°(E, ES(T)) and C (E, EU(T)) in the correct proportions, and C°(E) is the direct sum of these subspaces. We now apply Lemma 5.11 and deduce the existence of a unique fixed point / of T + rj. This gives the first part of the theorem. Finally, let g' e C°(E) be the unique map corresponding to g when 17 and £ are interchanged in (5.15). Thus (T + r,){id + g)(id + g') = {id + g){T + 0(id + g') = (id + g)(id + g')(T + r,). But also (T+r])id = id(T+-n). By the uniqueness in the first part of the theorem (applied with J\ = 0, (id + g)(id + g') = id. Similarly {id + g'){id + g) = id. Thus id + g is a homeomorphism, since g and g' are continuous.

•

(5.18) Exercise. Show that the hypothesis that g is bounded is necessary for uniqueness in Theorem 5.15. {Hint: Put E = R, and 17 = f = 0.)

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II

(5.19) Exercise. Obtain an alternative proof of Hartman's theorem by considering the map from C°(E) to C°(E) taking g = (gs, gu) to ((T,gs + r,,(id + g)-£s)(T

+ C)-\TZ1(Cu+gu(.T

+ £)-r,u(id

+ g)),

where C°(E) = C°(E, E S (D) x C°(E, E H ( D ) . It is natural to ask how smooth the conjugacy id + g is in Theorem 5.15, and tempting to suppose that it is just as smooth as the maps T + r\ and T + £ that it conjugates. However, a moment's thought shows that this cannot always be the case. If r), £ and g are C 1 , we may differentiate (5.16) at the unique fixed point p of T + £, and deduce that the differential of T + £ at p is similar to the differential of T + r) at its fixed point p + g(p). Thus we have a necessary condition for smoothness of g that is quite a heavy restriction on 17 and £. To see this, put £ = 0 and 17 (0) = 0, and observe that, however small we make 17 and its derivatives, the automorphism T+D-q(0) will not usually be similar to T. Moreover, this similarity condition is not sufficient for C conjugacy. We return to this and similar questions in the appendix to this chapter. Our main application of Hartman's theorem is to compare a diffeomorphism near a hyperbolic fixed point with its linear approximation at the point. The result is a local one, whereas, in Hartman's theorem as stated above, maps are defined on the whole of E. Thus the proof of the result consists of extending local maps to global ones. (5.20) Corollary. A hyperbolic fixed point p of a C 1 -diffeomorphism f: X -» X is topologically conjugate to the fixed point 0 of Tpf. Proof. We can assume, after conjugation with an admissible chart at p, that we are working in the model space E. We may suppose that p = 0 and that / is a C 1 diffeomorphism mapping some open neighbourhood U of 0 onto an open neighbourhood f(U) of 0. We are given that the differential Df(0) is hyperbolic. Let the skewness of Df(0) be a with respect to some norm | | on E, and let K be some positive number smaller than min {1 - a, [DfiO)^^1}. Since / is C 1 , there is some closed ball B centre 0 radius b on which \Df(x) -Df(fl)\ =£§K. This implies that / - D/(0) is Lipschitz with constant \K on B. Define 17: E -»• E by

)=

(f(x)-Df(0)(x)

^ \f(^Dmm I \\x\/

if\x\^b, \\x\J

ifH^.

We assert that 77 is Lipschitz with constant K. We must prove that, for x and x' in E, \iq(x)-ri{x')\^K\x -x'\, where (as we may suppose) |*'|=s|x|. The case \x\^b is trivial, and the case 6 < | x ' | reduces to the case Ixlssftssbtl,

116

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CH. 5

since multiplying the vectors by b/\x'\ leaves the left-hand side unaltered and decreases the right-hand side. Suppose now that \x'\ =£ b =s |*|. Then |TJ(*)-I7(*')| =

•€)-«

~ ^ 2 ^

I

I ~~ • *

|JC

|

^kK{\x-x'\

+ \x\-b)

=^2/C(|JC — JC'| -»-|JC| — |JC'|) *^K\X—X'\,

as required. Now TJ is also bounded. By Hartman's theorem there is a topological conjugacy h, say, from .0/(0) to Df(0) +17. Let V be a neighbourhood of 0 such that h(V)<=B. Then, on V, fh = (Df(0) + 71)h = hDf(0). Thus / is conjugate at 0 to Df(0) at 0.

•

(5.21) Corollary. There are An topological conjugacy classes of hyperbolic fixed points that occur on n-dimensional manifolds. Proof. This follows immediately from Corollary 5.20 and the classification of hyperbolic linear homeomorphisms of R" in Corollary 4.27. • We call a fixed point p of a diffeomorphism / : X -* X structurally stable if for each sufficiently small neighbourhood U of p in X there is a neighbourhood V of / in Diff X such that, for all geV,g has a unique fixed point q in U and q is topologically conjugate to p. Even although we have not discussed the topology of Diff X in this chapter, we have assembled all the facts that are needed to prove that fixed points are structurally stable if (and, in finite dimensions, only if) they are hyperbolic. Since we are discussing local phenomena, we can work in the model space E. The basic essentials are contained in the following exercise. (5.22) Exercise. Let / : U -*f(U) be a diffeomorphism of open subsets of E with a fixed point at 0. Suppose that 0 is hyperbolic, and that .0/(0) has skewness a with respect to the norm | | on E. Choose K, with 0 < K < 1 - a, so small that, for all T e L(E) with | T - D f ( 0 ) \ « K, T is hyperbolic and topologically conjugate to Df(0). Let B be a closed ball in E with centre 0 and radius b{< 1) such that \Df(x)—£>/(0)| =£ K/2 for all xeB. Use Exercise 5.13 to prove that, for all C 1 maps g: U^>E with |g-/|i=£/<&/2, g has a unique fixed point p in B. Show that g\p is topologically conjugate to /|0, and hence that 0 is structurally stable under C 1 -small perturbations of/

Ill

HARTMAN'S THEOREM FOR FLOWS

117

Conversely, prove that if E is finite dimensional and if 0 is structurally stable under C 1 -small perturbations of / then 0 is hyperbolic. We have already discussed at some length in Chapter 4 the stability of linear automorphisms with respect to topological conjugacy under linear perturbations. As we commented there, it is possible to prove that hyperbolic automorphisms are stable in this sense (the result of Theorem 4.30) as a corollary of Hartman's theorem. This is just a matter of getting a local conjugacy at 0 between the original and the perturbed maps (one first doctors the perturbation to get it into C (E)) and then extending the local conjugacy to a global one, using the conjugacy relation much as in the proof of Theorem 4.24. We leave the details to the interested reader in the following pair of exercises. (5.23) Exercise. Let / : E -» E be Lipschitz with constant K > 0. Given e > 0, find a map g e C°(E) such that g is Lipschitz with constant K+S and g = / on the unit ball in E. (Hint: Assume /(0) = 0, and choose N>0 with \/N< S/2K. Prove that the map p: E -» R defined by p{x) = \(N + l-\x\)/N

(o

ifl«|jc|ssAT + l

ifiV + l s s H

is Lipschitz with constant 1/N. Deduce that g = p. f is locally Lipschitz with constant K + S, and hence Lipschitz with constant K + e.) (5.24) Exercise. Let T e HL(E) have skewness a with respect to the norm | | on E. Prove that any 7" € L(E) with 1 -1

|r-r|<min{i-a, Ir; ! }

is topologically conjugate to T.

HI. HARTMAN'S THEOREM FOR FLOWS Hartman's linearization theorem was independently discovered, in the flow context, by Grobman [1, 2]. We can readily deduce the global Banach space version for flows from its analogue for diffeomorphisms, as follows: (5.25) Theorem. (Hartman and Grobman) Let T e EL{E). For all Lipschitz maps T) e C°(E) with sufficiently small Lipschitz constant, there is a flow equivalence from T to T+r\. Proof. Let ^ : R x E ^ E be the integral flow of T. Thus \= exp T) is a hyperbolic automorphism. Suppose that it has skewness a < 1 with respect

118

CH. 5

LINEARIZATION

to some equivalent norm | | on E. We recall a few necessary facts from Chapter 3. Firstly, the vector field T + tj has an integral flow, ip say (Theorem 3.39). Next, the map ip1-^ is in C°(E) (Theorem 3.45). Finally, the Lipschitz constant of tp1-^1 can be made arbitrarily small by decreasing the Lipschitz constant of 17 (Theorem 3.46). Thus, for the latter constant sufficiently small, we may apply Hartman's theorem to the hyperbolic automorphism (p1 and the perturbation \px -<pr, and obtain a topological conjugacy h from
= (~'-id = (')h4>-' + '(.h-id)<(>~'

is C°-bounded. Thus, by the above mentioned uniqueness of h, h - ip'hcp~', and h is a flow equivalence from (p to \p. • Just as with diffeomorphisms, we may now deduce the theorem for hyperbolic fixed points of flows on manifolds. The proof, which we leave to the reader, is that of Corollary 5.20 with the obvious modifications. (5.26) Corollary. A hyperbolic zero p of a C equivalent to the zero of its Hessian at p.

vector field on X is flow •

(5.27) Corollary. Flow equivalence and orbit equivalence coincide for hyperbolic zeros ofC1 vector fields on an n dimensional manifold X. There are precisely n + 1 equivalence classes of such points with respect to either relation. Proof. Immediate from Corollary 5.26 and Exercise 4.44.

•

(5.28) Exercise. Show that the flow equivalence h of Theorem 5.25 is unique subject to the condition h-ideC (E). (5.29) Exercise. Let v be a C 1 vector field on E with a zero at p. Prove that if p is hyperbolic then it is stable with respect to flow equivalence under C 1 -small perturbations of v. Prove, conversely, that if E is finite dimensional and if p is stable with respect to topological equivalence under C -small perturbations of v then p is hyperbolic. It is worth emphasizing at this stage that hyperbolicity of fixed points is not an invariant of topological equivalence or of flow equivalence. (We might well have made a similar remark earlier in the context of topological

119

HYPERBOLIC CLOSED ORBITS

IV

conjugacy.) That is to say, it is perfectly possible for a non-hyperbolic fixed point to be topologically equivalent, or even flow equivalent, to a hyperbolic fixed point. For example, the non-hyperbolic zero 0 of the vector field v(x) = x3 on R is clearly flow equivalent to the hyperbolic zero 0 of the vector field w (x) = x. Suppose that a fixed point p of a flow is topologically equivalent to a hyperbolic fixed point q of a flow i/f. There are three possibilities: (a) ( r ^ 1 ) , = Tqijj\ (b) {J^)u = T^/1, and (c) neither (a) nor (b). We call p (and also, of course, q) in case (a) a sink, in case (b) a source and in case (c) a saddle point (see Figure 5.30). We use the same terminology for fixed points of a diffeomorphism that are topologically conjugate to hyperbolic fixed points.

(i?) sink

(b) source

(c) saddle

FIGURE 5.30

Any sink p has the property that any positive half orbit R + . x = {t. x: t s= 0} starting at a point x near p stays near p and eventually ends at p, in the sense that co (x) = p. The point p is said to be asymptotically stable in the sense of Liapunov. In the language of differential equations, this type of stability is concerned with the way an individual solution of a system varies as the initial conditions are altered. This should not be confused with structural stability, which deals with the way the set of all solutions varies as we alter the system itself. We shall say some more about Liapunov stability in the appendix to this chapter, which also contains a section on the index of a fixed point. This last is an important integer valued invariant of local topological equivalence.

IV. HYPERBOLIC CLOSED ORBITS Let p be a point on a closed orbit of a C flow ( r ^ l ) o n a manifold X. Suppose that the orbit T = R . p of <j> throughp has period r. Then <j}r:X^X is a C diffeomorphism with a fixed point at p. The tangent map Tpr is a

120

LINEARIZATION

CH. 5

linear homeomorphism of TpX. It keeps the linear subspace (v(x)) of TpX pointwise fixed, where v(x) is the velocity of at x, because 4>T keeps the orbit T pointwise fixed. We say that T is hyperbolic if TpX has a TpTinvariant splitting as a direct sum rpX = 0F p T

where Tp(t,p) then Tq<j>T = (Tp') ( 7 > T ) {Tp'Y\ so Tq4>T and TP4>r

are linearly conjugate. Thus hyperbolicity for a closed orbit means that the linear approximation to (j>T is as hyperbolic as possible, bearing in mind that it cannot be hyperbolic in the direction of the orbit. We can get a clear geometrical picture of what this means in terms of Poincare maps. Let Y be some open disc embedded as a submanifold of X through p, such that (v(x))@ TPY = TpX. We say that Y is transverse to the orbit at p, and call it a cross section to the flow there. Now {p{y), y)e Y. The function p is a first return function for Y. Any two such functions agree on the intersection of their domains; we could have phrased this in the language of germs (see Hirsch [1]). Intuitively p(y) is the time that it takes a point starting at y to move along the orbit of the flow in the positive direction until it hits the section Y again. We call the point at which it does so f(y) (see Figure 5.31). Thus / : £/-> Y is a map defined by f(y) = cf>(p(y),y). We call / a Poincare map for Y. As with p, it is well defined up to domain.

FIGURE 5.31

IV

HYPERBOLIC CLOSED ORBITS

121

Note that these definitions do not require T to be hyperbolic. In fact, we now show that T is hyperbolic precisely when the Poincare map for some section Y at p has a hyperbolic fixed point at p. (5.32) Theorem. For sufficiently small U, the first return function p is well defined and C, and the Poincare map f is a well defined Cr diffeomorphism of U onto an open subset of Y. IfTpXhas a TP4>T-invariant splitting (v{p))®Fp then Tpf is linearly conjugate to Tp<j> T\FP. The orbit T is hyperbolic if and only if p is a hyperbolic fixed point off. Proof. Notice that by Theorem 5.8 and Remark 5.10 we may identify p and some neighbourhood of it with the origin in the Banachspace E = R x F and some neighbourhood of it, with the flow given locally by 4>(t,(s,y)) = (s + t,y). This is because a C flow equivalence affects tangent maps at p only by a linear conjugacy. Under this identification the map T satisfies, near p, 4>T(s,y) = ct>s(r(0,y))

= T(0,y) + (s,0).

Thus if g and h are maps defined on some neighbourhood V of 0 in F by f ( 0 , y ) = (g(y),ft(y))eRxF then cj)T is locally of the form T(s,y) = (s +

g(y),h(y)).

Notice that g and h are Cr and vanish at y = 0. Moreover if V is sufficiently small, h is a diffeomorphism of V onto an open subset of F, by the inverse mapping theorem (Exercise C . l l of Appendix C). We first prove the theorem for the particular cross section {0}xF (or, rather, some neighbourhood of p = (0, 0) in it). In this case, the first return function p is the map (0, y)*-+r-g(y) and the Poincare map / is (0, y)>-> (0, h(y)). If TpX splits as in the statement of the theorem, we can suppose after a linear automorphism of E that Fp = {0} x F. Then the differential of 4>T at p = (0, 0) is given by (5.33)

DT(p)(s,y) =

(s,Dh(0)(y)),

and it is clear that Tp(f>T\Fp equals Tpf. It follows immediately that if V is hyperbolic then p is a hyperbolic fixed point of /. Suppose conversely that p is a hyperbolic fixed point of/. It is sufficient to prove that TpX splits as in the statement of the theorem. Now the differential of T at p is given by (5.34)

DT{p)(s, y) = (s+Dg(0)(y),

Dh(0)(y)).

122

LINEARIZATION

CH. 5

Since Dh(0) is hyperbolic there is some a > 1 for which iterates \Dh(0)n(y)\ grow as a multiple of a'"'|y|, either for n positive or, if y is on the stable manifold of Dh(0), for n negative. Thus (s, y) is in the eigenspace of Dcf>T(p) corresponding to 1 if and only if y = 0. One easily checks directly that D-+(TT2\Y)~ h-rr2(y). Since the tangent map to the latter at p is linearly conjugate to Dh(0), the proof is complete. • (5.35) Remarks. The requirement in the statement of Theorem 5.32 that TpX should split, is not so very inconvenient. One may, in general, speed up or slow down the flow <j> near p along individual orbits so that the formula (5.34) becomes (5.33). The new flow has the same orbits as the old one and in addition has the splitting property (see the proof of Theorem 5.40 below). When X is finite dimensional it is clear from (5.34) that det (DT(p)-\(id)) = ( 1 - A ) det(£>fc(0)-\(id)). Thus, whether or not the splitting occurs, the eigenvalues of Tpcf>T are those of Tpf, with the correct multiplicity, together with an extra eigenvalue 1. The eigenvalues of Tpf are called the characteristic multipliers of I\ Thus (5.36) Corollary. The orbit T is hyperbolic if and only if none of its characteristic multipliers lies on the unit circle in C. • In order to analyse further the structure of a flow in the neighbourhood of a closed orbit, we recall the notion of the suspension of a diffeomorphism (Construction 1.23). The situation is now rather more complicated, as we wish to suspend diffeomorphisms which are only locally defined (in particular, Poincare maps for cross sections of flows). Let Y be a manifold and let / b e a C ' diffeomorphism (r3= 1) of a connected open subset U of Y onto an open subset f(U) of Y, with a fixed point p. Let / be the open interval ]—e, 1 + e[ for some small positive e (certainly e < 2). Consider the quotient space of R x Y under the equivalence relation ~ given by (s, y) ~ (s1, y') if and only if y e U, y' = /(y) and s = s'+l. Let V be the quotient subset Ix U/~ (see Figure 5.37). Then V is a differentiable manifold, and the standard unit vector field in the direction of the first component of R x Y induces a vector field on V. The suspension 2(/) of / is the local integral of

IV

HYPERBOLIC CLOSED ORBITS

123

tolxr

4

nu)

FIGURE 5.37

this vector field. Thus, if [s, y] denotes the ~ equivalence class of (s, y) and t is sufficiently small, Z(f)(t,[s,y])

= [s + t,yl

One identifies y e Y with [0, y]. Notice that £(/) is C, and that the orbit through p is periodic of period 1. The set Y is a cross section of £(/) with first return map the constant function with value 1, and / is a Poincare map for Y. Now let / ' : £ / ' - > / ' ( U') be a diffeomorphism of open subsets of a manifold Y'. A topologicalconjugacy from f to f is a homeomorphism h:Uuf(U)^> t / ' u / ' ( £ / ' ) such that, for all yeU, hf(y)=f'h(y). The following result considerably simplifies the problem of classifying suspensions. (5.38) Proposition. / / two diffeomorphisms f: U-*f{U) and f: U'^-f'(U') are topologically conjugate, then their suspensions are flow equivalent. Proof. Let h be a topological conjugacy from / to /'. Let S(/) and £(/') be the suspensions, local integrals on V = (I x U)/~ and V = (I x U')/~' respectively, where J is the interval ]-e, l + e[. Define a map H: V^> V by ^([*» y]) = [s, h(y)]' where [ ]' denotes the equivalence class with respect to ~'. Provided that H is well defined (i.e. respects the identifications under ~ and —') it is clear that it is a flow equivalence from S(/) to £(/'). But, for all s e ]-e, e[ and for all y &U, (s + l,h(y))~'(s,f'h(y)

= (s,hf(y)),

and so the representatives (s +1, y) and (s, f(y) of [s +1, y] both lead to the same value for H. Thus H is well defined. •

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LINEARIZATION

CH. 5

(5.59) Remark. Topologically conjugate could be replaced by C" conjugate above, in which case the flow equivalence H would be C" (1 «£/ =£ r). The main connection between the flow near a closed orbit r of a C flow on a manifold X and the Poincare map at a cross section Y of the flow at the point p of T is made by the following theorem: (5.40) Theorem. Letf: U-*f(U) be a Poincare map at p for the cross section Y. Then there is a Cr orbit preserving diffeomorphism hfrom some neighbourhood of the orbit R . p of the suspension 1(f) to some neighbourhood of the orbit r of (f> such that h(p) = p. Proof. Let e be as in the definition of 1(f) above, and let A be a C°° real increasing function on the closed interval [e, 1 - e ] such that A(e) = 0, A (1 - e) = 1 and all derivatives of A of order 5= 1 vanish at e and 1 - e. Let A be the maximum value of the first derivative A' on [e, 1 - e ]. We may assume, by changing the time scale if necessary that T has period 1. We may also assume that, provided U is sufficiently small, the first return function p satisfies p(y) > max { 1 - 1 / A , 2e}for ally e Uuf(U). We define a C orbit preserving diffeomorphism h from the domain V = (I x U)/~ of £ ( / ) to a neighbourhood of T in X by

4>{s, y) h([s, y]) = l(s + (p(y)-l)\(s), (f>(s+p(y)-l,y)

for s e ] - e , e[, y € U y)

for s e[e, 1 - e ] , y e U f o r s e ] l - e , l + e[, yeU.

Notice that, since for 5 e ]1 - e, 1 + e[ < M * + p ( y ) - i , y ) =
•

Now if T is hyperbolic, then any Poincare map / has a hyperbolic fixed point at p. Since by Corollary 5.20 / is topologically conjugate at p to Tpf at 0, we deduce from Proposition 5.38 that 1(f) is topologically equivalent to the suspension 1(Tpf) on some neighbourhood of its unique closed orbit. Putting all this together, we have: (5.41) Corollary. / / T is hyperbolic then the flow is topologically equivalent at T to 1(Tpf) at its unique closed orbit. • In the finite dimensional case, the classification of hyperbolic linear maps up to topological conjugacy in Corollary 5.21 yields immediately a

IV

HYPERBOLIC CLOSED ORBITS

125

classification of hyperbolic closed orbits: (5.42) Corollary. There are, up to topological equivalence, precisely 4n different hyperbolic closed orbits that can occur in a flow on an (n + 1)dimensional manifold (n 3= 1). Proof. The different types are distinguished by the dimensions and orientability (or lack of it) of the pair of submanifolds whose a-set or w-set is the closed orbit. These are the so-called unstable and stable manifolds of the closed orbit, and we shall discuss them further in the next chapter. • (5.43) Exercise. Visualize the above types of hyperbolic closed orbit for n = 1 and for n=2. Hyperbolic closed orbits are structurally stable, in the sense that, if T is such an orbit of a C 1 vector field v on X, and if w is a vector field on X that is C -close to v (see Appendix B), then for some neighbourhood U of T in X, w has a unique closed orbit in U, and this closed orbit is topologically equivalent to I\ To construct a proof of this result, use Theorem 3.45 to show that, for a given cross section Y, the Poincare map of w is C 1 -close to the Poincare map of v and then use the structural stability of hyperbolic fixed points of diffeomorphisms (see Exercise 5.22).

Appendix 5

I. SMOOTH LINEARIZATION Recall that in Hartman's theorem we altered a hyperbolic linear homeomorphism T by a perturbation rj and found a topological conjugacy h = id + g from T to T + 77. We pointed out at the time that h is not necessarily C even when the perturbation 77 is C°°, since differentiating the conjugacy relation would place algebraic restrictions on the first derivatives of T +77. The question now arises as to whether further differentiation places further restrictions on higher derivatives, and whether, even if these algebraic restrictions are satisfied, the smoothness of 77 has any effect on that of h. It turns out that, in finite dimensions at any rate, further restrictions are the exception rather than the rule, and that positive results on smoothness can be obtained. We state here, without proof, the major theorem to this effect, due to Sternberg [1,2], and add, also without proof, two relevant theorems of Hartman [1]. We also set an exercise to show that Holder continuity of the perturbation implies Holder continuity of the conjugacy. This seems to be the only completely general result in which a property is transferred from the perturbation to the conjugacy. (5.44) Theorem. {Sternberg's Theorem) Let TeL(Rn) A i , . . . , A„ (possibly complex or repeated) satisfying

have eigenvalues

A,-^Ar*...A^

for all 1 =£ i =£ n and for all non-negative integers mi,..., mn with £" = 1 m, 3= 2. Let 77: £/ -»R" be a Cs map (s 3= 1) defined on some neighbourhood U of 0 withrt(0) = Dr)(0) = 0. Then (T+ TJ)|0 is C conjugate to T\0, where, for given T, r depends only on s and tends to 00 with s. • In particular, if 17 is C°°, the maps T and T + 77 are C°° conjugate at 0. Notice that the eigenvalue condition implies that T€HL(R"). There is a vector 126

I

SMOOTH LINEARIZATION

127

field version of Sternberg's theorem, where the above multiplicative eigenvalue condition is replaced by the additive one Aj # miAi + - • - + m„A„,

the conclusion being that, near 0, the vector field T +17 is induced (in the sense of Theorem 3.11) from T by some Cr diffeomorphism (provided s is sufficiently large) and that r tends to 00 with 5. See Nelson [1] for a good proof of this theorem and a more precise statement of how r depends on s. (5.45) Theorem. (Hartman) Let T e L(R") be a contraction and 17: U-> R" be a C map defined on some neighbourhood U of 0, with TJ(0) = £>i7(0) = 0. Then {T + r\)\Q is C conjugate to T\0. • (5.46) Theorem. (Hartman) Let T e HL(R"), where n = lor2, as in Theorem 5.45. Then (T + h)\0 is C1 conjugate to T\0.

and let r\ be •

(5.47) Exercise. A map g: E -> E is Holder continuous with constant A (>0) and exponent a > 0 if, for all x, jc'eE, \g(x) - g(x')\m \\x - x'\a. Prove that, for fixed A and a, the subset $f(A, a) of C°(E) consisting of all Holder continuous maps with constant A and exponent a is closed in C (E). Let TsHL(E), and let 17 e C°(E) be Lipschitz and Holder continuous with exponent a < 1. Prove that for any A > 0 the map of C°(E) to itself defined in Exercise 5.19 (with £ = 0, for simplicity) takes $f(A, a) into itself, provided that a and the Lipschitz and Holder constants of -q are sufficiently small. Deduce that in this case the map g of Theorem 5.15 is in 5if(A, a). (Note that, if we are only interested in the local behaviour of T + T J , the Holder condition on 17 is a consequence of the Lipschitz condition, since \x~x'\<\x-x'\a

for

|*-*'|<1.)

In the negative direction, we give some counter examples, which emphasize the value of the foregoing theorems. (5.48) Exercise. Prove that two linear automorphisms of R are Lipschitz conjugate at 0 (i.e. with local conjugacy a Lipeomorphism) only if they are equal. (5.49) Exercise. L e t / : R 2 -» R2 be defined by f(x, y) = (a2x+y2,ay) where a is fixed and positive. Prove that / is not C2 conjugate to Df(0) at 0. (Hint: Differentiate the relation hf = Df(0)h twice at 0.) (5.50) Exercise. Let T : R 3 ^ - R 3 be defined by T(x, y, z) = (ax, acy,cz), where fl>l>c>a_1>0 and let ?7:R 3 ->R 3 be defined, for some fixed positive e, by r\(x,y,z) = (0, eacxz, 0). Prove that there is no Lipschitz local conjugacy from T to r + 17 at 0. (Hint: Show that, near 0, any local conjugacy h preserves the z-axis. Now suppose that h is Lipschitz. Prove

128

APPENDIX

CH. 5

that, near 0, h preserves the x-axis, and satisfies c~"h2(x, 0, cnz)-anh2(a~"x,

0, z) = nehi(x, 0, c"z)h3(a~"x, 0, z),

where h = (hi, h2, h3) and n is any positive integer. Obtain a contradiction by letting n -»oo. As we have seen the smoothness of the perturbation r? is not always fully echoed by the smoothness of the conjugacy h from T to T + TJ. However, it is in other ways. For instance, the image under h of the stable manifold of T is a submanifold which is always as smooth as TJ: this is one of the main theorems of the next chapter. Another question that arises naturally is the dependence of h on 17 when both are regarded as points in map spaces. Here again the smoothness of 77 shows itself. For example, let S5r denote the set of Lipschitz maps in UCr(E) (see Appendix B) with constant less than min {I-a, I r ^ T ^ . W e may drop the U of UC if r = 0 or if E is finite dimensional. With the notations of Theorem 5.15, we then have: (5.51) Theorem. For fixed £, the map 6 sending r\ e 5ft° to the corresponding g e C°(E) is Lipschitz, and its restriction to 2ft' is C. Proof. We define a map y • ®° x C°(E) ^ C°(E) by x(v,g) = (id-f)-\(r1(id + g)-()(T + £r1), where T is as in the proof of Theorem 5.15. Then x is a uniform contraction on the second factor, and uniformly Lipschitz on the first factor with constant (1 — a)~ . Thus, by Theorem C.7 of Appendix C, the fixed point map is Lipschitz, and this map is precisely 6. Now restrict x to $/&' x C°(E). The restricted map is C (see Theorem B.19 of Appendix B), and thus, by Theorem C.7 again, the restriction of 6 to 88r is C. • (5.52) Exercise. Investigate the dependence of the map g of Theorem 5.15 on the pair (17, £) together. (5.53) Exercise. Investigate the dependence of h — id on 17 in (the proof of) Theorem 5.25. II. LIAPUNOV STABILITY Let p be a fixed point of a flow <j> on a topological space X. We say that p is stable in the sense of Liapunovt (or Liapunov stable) if, given any neight The results of this section date back to Liapunov's doctoral thesis, written at the end of the last century. See Liapunov [1] for a French translation.

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bourhood U of p, there is some neighbourhood V of p such that for all x e V, R + . i c [/, where R + . x = {t. x: t3= 0}. If this is not the case then p is unstable. We say that p is a asymptotically stable if it is Liapunov stable and, in addition, for some neighbourhood W of p, x e W implies t.x-*past^>°o. We may similarly talk of Liapunov stable and asymptotically stable zeros of a vector field on X (see Exercise 3.44). (5.54) Example. The flow t. x =xe~' on R has an asymptotically stable fixed point at 0. The flow t.z~ze"on R 2 (= C) has a fixed point 0 that is stable but not asymptotically stable. The flow t. (x, y) = (xe', y e~') on R 2 has an unstable fixed point at (0, 0). Figure 5.54 illustrates a flow on S2 with a

FIGURE

5.54

fixed point p that satisfies the additional condition for asymptotic stability without being Liapunov stable. It is the one-point compactification of the integral flow of a constant vector field on R (with p the point of compactification oo). Of course one could start with R" for any n s* 1 and obtain a similar example. Now let X be a finite dimensional smooth manifold. Let / : X - » R b e a smooth function with an isolated critical point at p. Suppose that / has a (necessarily strict) maximum at p. Our experience of elementary several variable calculus leads us to expect that the level surfaces {xeX:f(x) = constant} of / near p should be, as in Figure 5.55, a sequence of smoothly embedded spheres of codimension 1 enclosing p. If V/ is the gradient vector field of/ (with respect to some Riemannian metric on X, see Examples 3.3 and A.57) then p is a zero of V/. Moreover the orbits of V/ intersect the level surfaces of / orthogonally going inwards, the direction of increasing f(x).

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FIGURE 5.55

Thus p is asymptotically stable, as a zero of V/. (N.b. we shall shortly obtain a completely rigorous proof of this last statement.) Suppose now that 4> is the integral flow of a vector field v on X, and that, for all x ¥=• p nearp, (Vf(x), v(x))>0. This condition says that v(x) makes an acute angle with Vf(x), and thus the orbit of v through x crosses the level surface at x in the same direction as the orbit of V/, namely inwards. Thus p is also asymptotically stable as a fixed point of (see Figure 5.56). If we

FIGURE 5.56

weaken the above inequality to 3=, we allow orbits of v to be tangential to the level surfaces. It is, of course, possible for an orbit to be tangential to a level surface at a point where it crosses over the surface. However it is difficult to see how an orbit can make much progress in an outwards direction if neither it nor nearby orbits may cross any level surface transversally outwards. We are led to suspect that, in this case, p is Liapunov stable as a fixed point of . This is, in fact, correct, but it is not easy to make the above approach rigorous. We leave it as a useful picture to have in mind, and start again. Let g:X^R be a smooth function"!". Recall from the section on first integrals in Appendix 3 that the directional derivative Lvg:X-*R is the t Actually, here, and in what follows, g need only be defined on some neighbourhood of p. However we do not wish to overcomplicate the notation.

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continuous function denned by Tg(v(x)) = (g{x), Lvg(x)) e R 2 = T(R). For all t0e R, Lvg{t0 • x) is the derivative with respect to t of g(t.x) at t0. With respect to any Riemannian metric ( , ) o n X , Lvg(x) = (v(x), Vg(x)). We say that g is positive definite at p if it has a strict local minimum at p and g(p) = 0. We define positive semi-definite by dropping the word "strict", and negative definite and semi-definite in the obvious way. We say that g is a Liapunov function for v {or for ) at p if g is positive definite at p and Lvg is negative semi-definite at p. It is strong if L„g is negative definite. Notice that a strong Liapunov function for v at p has an isolated minimum at p, since L„g vanishes at any critical point of g. Thus, modulo a constant, g takes the place of - / in the above geometrical description; this change of sign is traditional, and has no significance. (5.57) Theorem. (Liapunov's Theorem) If there exists a Liapunov function g for at p then p is Liapunov stable as a fixed point of cf>. If further g is strong then p is asymptotically stable. Proof. Let g be a Liapunov function for 4> at p, and let U be any neighbourhood of p. We wish to show that for some neighbourhood V of p, x e V implies R + . x <= JJ. By taking a smaller neighbourhood if necessary we may assume that U is compact and that, for all xtU, g(x)>0 and Lvg(x)^0. The frontier d U of U is compact and non-empty (provided U # X; if U = X then, of course, we put V = X). Thus the infimum, m say, of g\dU is strictly positive. By continuity of g, there exists a neighbourhood V of p on which g(x) is strictly less than m, and we assert that this V has the above property. This is because if, for any x e V, R + . x leaves U, it does so at some point t0 . x of dU(to>0), but since g(t.x) is non-increasing while t. x remains in U it can never attain a suitable value g{t0.x)^m. To be more precise, Jx(int U) is open in R, and hence is a union of open intervals. If ? 0 at p, and that U and V are neighbourhoods as above, with Lvg(x) < 0 for allx^p in U. Then g(t.x) decreases as t increases, and therefore tends to some limit / 5=0 as ?->oo. If / > 0 , then, by continuity of g, t.x never enters some open neighbourhood W of p. Thus, for all 13= 0, d(g(t. x))/dt =s M < 0, where M is the supremum of Lvg on the compact set U\ W. However this inequality implies that g(f.x)-»-oo as ?-»oo, a contradiction. We deduce that / = 0. Now let k = inf {g(y): y e C/\ W}. Since fc is strictly posi-tive, g{t. x) < k for all sufficiently large t, and hence t.x is eventually in W. Since this last argument holds equally well for any open neighbourhood W of p, t. x -» p as ? -» oo, and hence /? is asymptotically stable. •

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The attractive feature of Liapunov's theorem is that one does not need to integrate the vector field (i.e. solve the differential equations) in order to apply it. For this reason, it is sometimes called Liapunov's direct method. For example, if v is the vector field v(x, y) = (-2y 3 , x 3 - y4) on R 2 and g is the positive definite function g(x, y) = x + 2y 4 , then it is a trivial observation that Lvg(x, y) = 4x 3 (-2y 3 ) + 8y V - y4) = - 8 y 4 is negative semi-definite, and so (0, 0) is a Liapunov stable zero of v. The converse to Liapunov's theorem also holds (see Antosiewicz's survey [1] for details), and so we know that a Liapunov function (resp. strong Liapunov function) exists at any Liapunov stable (resp. asymptotically stable) fixed point. The snag about applying Liapunov's theorem is that in any specific case it may be very hard to recognize whether such a function does exist, and to find one if it does. It would be rather a tall order to prove that a given fixed point is unstable by showing that no Liapunov functions exist there. The following instability theorem also due to Liapunov is sometimes useful: (5.58) Theorem. //there exists a C 1 function h:X->R with h(p) = 0 such thatL„h is positive definite atp and h is strictly positive on a sequence of points converging to p, then p is unstable as a fixed point of 4>. Proof. Let U be a compact neighbourhood of p such that Lvh(x) > 0 for all x # p in U, and let V be any neighbourhood of p. Then V contains a point x^p such that h(x)>0. Suppose that R+ ,X<=U. Then h(t.x) is strictly increasing with t, so by continuity of h, there is some open neighbourhood W of p that contains no points of R + . x. Since U\ W is compact, the infimum of Lvh on U\W is strictly positive, and thus /j(r.x)-»oo as f-»oo. But h is bounded on U\W, so we have a contradiction. • (5.59) Exercise, (i) Use the function g(x, y) = x6 + 3y 2 to prove that (0, 0) is an asymptotically stable zero of the vector field v (x, y) = (—3x - y, x - 2y ) on R 2 . Prove that the vector field v(x, y) = (3x 3 + y, x 5 - 2 y 3 ) has an unstable zero at (0, 0). (ii) Prove that the function /(x, y) = x 2 + 4y 2 + 2xy 2 + y 4 is a Liapunov function for the vector field v(x, y) = (-2xy 2 , xy - 2 y ) at (0, 0)e R2. (5.60) Example. If a C 1 function / : X •+ R has an isolated critical point at p, and this is a (necessarily strict) local maximum of /, then the gradient vector field V/ (with respect to any Riemannian metric on X), has an asymptotically stable fixed point at p. For, let g(x)=f(p) -f(x). Then g is positive definite at p, and L V /g(x) = (-V/(x), V/(x)) is negative unless V/(x) = 0, which

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implies that x is a critical point of /. Thus L^fg is negative definite at p, and so g is a strong Liapunov function for V/ at p. Similarly if / has an isolated critical point at q and this is not a local maximum of /, then V/ has an unstable fixed point at q by the instability theorem. (5.61) Example. Recall (Example 3.49) that in conservative mechanics the Hamiltonian energy function H = T + V is a first integral for the Hamiltonian vector field. Thus, rather trivially, H satisfies the condition that its directional derivative with respect to the vector field is negative semidefinite at any zero of the vector field. These zeros correspond to equilibrium positions of the system, and since the kinetic energy T is a positive definite function of the momenta, H is positive definite at an equilibrium position if and only if the potential energy V has a strict minimum there. For dissipative systems, H decreases along the orbits at points where the momenta are non-zero, and so the directional derivative is still negative semi-definite, and the above conclusion continues to hold. Liapunov stability can be defined for orbits other than fixed points, and analogues of the above theorems can be obtained in this broader context. Equivalently one may discuss stability of a fixed point of a time dependent vector field, since stability of the solution y of x' = v(x) is equivalent to stability of the zero solution of y' = v(y + y{t))-vy(t). See, for example, Hale [1].

III. THE INDEX OF A FIXED POINT Let v be a continuous vector field on an open subset V of R 2 , and let C be a simple closed curve in V. Suppose that v does not pass through any zero of v. Then we can associate with C an integer, called its index with respect to v which we may describe intuitively as follows. Consider a variable point x starting at some point x0 e C and moving around C in the positive (anticlockwise) direction. The (anticlockwise) angle 6{x) that D(JC) makes with the positive x axis is only defined up to a multiple of 2TT. However if we start, say, with 0^6(x0)<27r, we can choose a representative 6{x) that varies continuously with x. When we return to x0 after one trip round C, 6{x) may not return to the original value 6(x0): it takes up a value that differs from 6(x0) by 2mr, for some integer n. Thus 2mr is the total angular variation of the vector field around the curve C. The number n, which obviously does not depend upon the starting point x0 and the speed with which x moves round C, is the index of C. In Figure 5.62, C has index 1 in (i)-(iii), but index 0 in (iv), index 2 in (v) and index - 1 in (vi).

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CH. 5

(i)

(ii)

(Mi)

dW

(v)

(vi)

FIGURE

5.62

(5.63) Exercise. For any integer n, positive or negative, visualize a vector field and curve for which the index is n. (5.64) Exercise. Let v be C and let C be parametrized by the C map g: [0, T] -» R 2 . Up to a multiple of IT, dg(t) =

tan-1(v2g(t)/v1g(t)).

Prove that the index of C is f (vg(t), -iDv(g(t))(g'(t)))

Jo

dt/27r\vg(t)\2

(identifying R 2 with C in the usual way). Use the formula to prove that the index of the unit circle with respect to v(x, y) = (x, —y) is —1. Suppose that C is deformed continuously into a curve C", through a family of curves none of which contains a zero of v. Then it is intuitively obvious that the total angular variation and hence the index, changes continuously, and, since the index is an integer, this can only mean that it remains constant. Similarly if the vector field v is deformed continuously into a new vector field v', then the index of C with respect to both these vector fields is the same

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provided that at no intermediate stage does the perturbed vector field have a zero on C. These two facts enable us to make some interesting observations. For example, C bounds a topological 2-dimensional ball B in R 2 . If B is in V and if B contains no zero of v, then C may be deformed until it lies in a small neighbourhood N of a given point x0 of B. Since v(x) is near v(x0) for xeN, the total angular variation is now small, and hence zero, and hence the original curve C had index zero with respect to v (see Figure 5.65). Again, if

FIGURE 5.65

p is an isolated zero of v, and if C\ and C2 are circles with centre p, small enough to contain no other zeros of v and no points of R 2 \ V, then Ci has the same index as C2, since we may deform the one radially to the other. Thus we may unambiguously define the index ofp with respect to v to be the index of any sufficiently small circle with centre p. Moreover, by an argument familiar in contour integration and illustrated by Figure 5.66, the index of C is the sum of the indices of the zeros p, in the interior domain B, provided that they are finite in number and that B is in V.

FIGURE 5.66

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CH. 5

As an example of deforming the vector field, we may prove the fundamental theorem of algebra, which states that any complex polynomial p(z) has a zero. For suppose that p(z) = z" + aiz"~1 + - • - + a„, and let r>\ be large enough for the inequality rn >\ai\rn~1+ • • - + \a„\ to hold. Then p,(z) = zn + t(aiz"~1 + • • - + a„), for 0=sf *sl, defines a deformation from the vector field z ^*p(z) on C = R 2 to the vector field z ^>p0(z) = z", and p, has no zeros on the circle z = r. Using the formula of Exercise 5.64, the index of the circle with respect to p0 is r 2Tr h

(rn ein', -inrn~l

ei(n~X)tir ek) dt/27rr2n,

which simplifies to n. Thus the index of the circle is n with respect to p. Since, by our above remarks the index would be zero if p had no zeros inside the circle, we deduce that p has zeros. It is possible to generalize the concept of index to higher dimensions. Let us think of the Jordan curve C as being the image of a continuous map g: S1 -> V which preserves orientation (i.e. as x moves anticlockwise round S 1 , g(x) moves anticlockwise round C). Then the direction vg(x)/\vg(x)\ of the vector field at g(x) e C is a point 6(x) of S (note that 6 now has domain. S 1 rather than C), so we have a continuous map 0: S 1 -»S 1 . The index of C is the number of times the new map 6 wraps S round itself (in the anticlockwise direction). At first sight this new approach is even vaguer than the previous one, but it may be made completely precise using homology theory. The index is just the degree, deg 6, of the map 6. For the definition of degree, and for other bits of algebraic topology used in this section, we recommend the reader to Greenberg [1], Hu [2] and Maunder [1]. Now suppose that v is a continuous vector field on an open subset V of R" and that g:M-> V is a continuous (not necessarily injective) map of a compact (connected) oriented (n -1)-dimensional manifold, and that no zeros of v lie on g(M). Then we may define a continuous map d :M'-* 5 " " 1 by 6{x) = vg(x)/\vg(x)\ and the index of g with respect to v, ind„ g, to be deg 6. Equivalently, we proceed as follows. The map vg:M^>R"\{0} induces a homology group homomorphism (ug)*://„_i(M)-»//„_i(R"\{0}). Both groups are isomorphic to Z, and the given orientation of M and the standard orientation of R"\{0} give generators which we identify with 1 e Z. In this case, ind„ g is just (ug)*(l). If g is homotopic to h by a homotopy that avoids the zero set Fix v of v, or if v is homotopic to w through vector fields v, with no zeros on g(M), then we obtain a homotopy from vg:M^> R"\{0} to vh or wg, as the case may be. Hence, (ug)* = (vh)* = (wg)*, and: (5.67) Proposition. Ind„ g = ind„ h = ind w g.

•

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137

In particular, since a constant map clearly has index 0 with respect to any vector field: (5.68) Corollary. / / g is homotopic to a constant map in V\Fix v then ind„ g = 0. • (5.69) Exercise. Prove that if w: V -* R" is a vector field that never takes Jhe opposite direction to v at any point of g(M) (i.e. (wg(x), vg(x))> -\wg(x)\. \vg(x)\ for all x e M) then ind„ g = indw g. Let /• R" -*• R" be a continuous map taking the closed unit ball into itself. By putting V = R", M = S" _ 1 , v = id —f, w = id and g = the inclusion, prove Brouwer's theorem that any continuous map of a closed ball into itself has a fixed point. If g is a topological embedding of M in V, and N = g(M), then R"\N has two connected components, one of which, D say, is bounded. If p is any point of D, and g*: H„-i(M) -*• H„-i(Rn\{p}) = Z is the induced map, then g*(l) is independent of p and takes the value ± 1 . If it is +1 we say that g is orientation preserving, otherwise orientation reversing. If g\: Mi -* V is another embedding of an oriented manifold Mi with image N, then gh = gx defines a homeomorphism h:Mi-*M, and, since the degree of the composite of two maps is the product of the degrees of the maps, indv gi = (ind„g). (deg/i). Note that deg/i = ± l . It is +1 (i.e. h is orientation preserving) if and only if g and gi either both preserve or both reverse orientation. Thus we may define the index ofN with respect to v, ind„ N, to be indu g for any orientation preserving embedding g:M^*V with image N. As in the n = 2 case, we may now define the index of an isolated zero p of v, mdvp, to be ind„iV for any sufficiently small (n — l)-sphere with centre p. Equivalently indvp = u*(l), where v*:Hn(B, B\{p})->//„(R", R"\{0}) is the induced homomorphism of relative homology groups, restricting v to some neighbourhood B of p that contains no other zeros of v. We make the same definition if p is a regular point of v. Since N is homologous to zerot in N u D , where D is the bounded component of R"\N, we deduce: (5.70) Proposition. If DczV and v has no zeros in D then ind„ N = 0. If D <=• V and v has only finitely many zeros p\,..., p, in D then md0N =

X-=1ind„p,.

•

(5.71) Corollary. If p is a regular point of v then ind„ p = 0.

•

(5.72) Exercise. Let n = 2, and let C be an integral curve of v. Prove that if C is a closed orbit then ind„ C = 1. (Hint: Take x e C with minimal second coordinate x2. Let T be the period of C, and identify S 1 with R / T Z . Consider t At least, this is clear if N\JD is polyhedral. More generally the proposition can be proved using Exercise H.2 of p. 361 of Spanier [1], together with Alexander duality (Theorem 16 of p. 296 of Spanier [1]).

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the homotopy G: S1 x [0, r] -+ R2\{0} defined for all t e [0, T], by 'v(v{x)) f((2t).x-x) v(u.x-x) G(Lt],u) = -v(v(x)) v(u.x-{2t-u).x) .v(v(x))

for? = 0, for 0 < f < M/2, foT t = u/2,0
where v : R2\{0}-> Sl is defined by v(x) = x/\x\. Prove that [t]-> G([t], r) has degree 1.) (5.73) Exercise. Let V = R" and let i; be a linear automorphism. Prove that ind„ 0 is 1 if det v > 0 and - 1 if det v < 0. Thus ind„ 0 = ( - l ) m , where m is the number of eigenvalues with negative real part, counting multiplicities. We wish to prove that the index of a fixed point with respect to a vector field is a topological invariant. For this statement to make sense, we must be able to talk about orbits of the vector field, so we now assume that v integrates to give a partial flow on V (see Exercise 3.40). Actually there is no loss in assuming that the partial flow is a flow (see Exercise 3.44), so we make this assumption. We reformulate the definition of index in the context of flows. Let 4> be any flow on V, let M be as above and let g: M-* V be any continuous map such that g(M) contains no fixed points of . Then g(M) does not contain periodic points of arbitrarily small period, for if there were a sequence of periodic points in g(M) whose periods converged to 0, then g(M) would contain the limit of some subsequence, and this would necessarily be a fixed point. Let a be any continuous positive valued function on M such that, for any xeM with g(x)ePer , R"\{0} be given by a (x) = a{x). g(x) - g(x). We define the index of g with respect to 4>, ind^ g to be the integer a*(l), where a*://„-i(M)-».f/„_ 1 (R' , \{0}) is the induced homomorphism of homology. It is easy to check that the definition is independent of choice of the map R"\{0} defined by /3(X) = T{X) . g{x)-g(x), and hence P*(l) = a*(l). Notice that if is an integral flow of v, then v(y) = lim (t.y-y)/t. Therefore, putting a(x) = t, small and constant, gives us a map a such that ind^, g = a ^ l ) = ((l/f)a)#(l), and, for all u e [0,1], putting .

(((ut).g(x)-g(x))/ut \.v(g(x))

for0<««l forw=0

gives us a homotopy from vg to (!//)«• Thus ind„ g = ind^, g in this case.

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Let i/f be a flow on an "open subset W of R". In proving our topological invariance result, we need to assume that the domains V and W in question are homeomorphic to the open n-ball. (5.74) Theorem. Let V be a topological n-ball and let h:V-*W be a topological equivalence from

to the induced flow t. f(x) = f(t. x) on R". Since hf~x and / _ 1 are topological equivalences with domain R", and h factorizes as the composite (hf~ )('/), we may assume from the start that V = R". Let a: M -> R be as in the definition of ind^ g. By definition of topological equivalence, for each y e V there exists an increasing homeomorphism a y : R - » R such that' h(t. y) = ay(t). h(y) for all teR. For all xeM, put T(X) = agM(cr(x)). Note that T is positive valued. Moreover, since agM is bijective and a(x), T(X)< per hg(x) if hg(x)e Per ip. We assert that T is continuous. To see this, note that h maps the compact set F = {t.g(x):xeM, t e[0, a(x)]} homeomorphically onto the set G ={t. hg(x):x e M , ; G [ 0 , T(X)]}. Let x0 e M Then, since a{x). g(x) tends to a(x0). g(x0) as JC -» x0 and h is continuous, T(X) . hg(x) -» T(X 0 ) . /ig(*o) as x -* x0. If T is not continuous at x0 then there exists some sequence (x,)r^\ such that xr -» x 0 as r -> oo and r(x r ) is bounded away from r(x0). We may assume that either. (i) r(xr) -> some limit / < T(X0), or (ii) r(xr) > T(X0) for all r 3= 1. If (i) holds, then r ( x r ) . hg(x)->l. hg(x0) as r-»oo, and thus l.hg(x0) = T(X0) • hg(x0), contrary to our above remark that r ( x ) < p e r hg(x). If (ii) holds, we choose t with T(x0)
-u)g(x)).

This gives a homotopy from y0 = y to yu where y1(x) = h(o-(x).g(x)-g(x))-h(0)

=

ha(x)-h(0),

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CH. 5

APPENDIX

and so ind^, hg = (yi)#(l). If h is orientation preserving, the homomorphism A*: /f„-i(R"\{0})-» W„-i(W\{/i(0)}) is just id :Z->Z. So is the map back to H„_i(R"\{0}) induced from translation through —h{0). Thus indj,hg = a^(\) = \nd^,g. Similarly, • if h is orientation reversing h*=—id, and ind^,/ig = -a ; i c (l) = -ind < jg. • The corresponding result for the index of a map g:M->V vector field is:

with respect to a

(5.75) Corollary. Let V be a topological n-ball and let h:V^W be a topological equivalence from a C vector field v on V to a C vector field w on W. Then ind w hg = indi, h if h orientation preserving, and indw hg = — indt, h if h is orientation reversing. D (5.76) Exercise. If V and v are as shown in Figure 5.76 (i), find a map g:S -» V and a diffeomorphism h:V^W for which the conclusion of Corollary 5.75 does not hold when w = (Th)vh~l. If V and v are as shown in Figure 5.76 (ii), explain how to find, for given integers r, s with r — s even, a map g:S -» V and a diffeomorphism h:V->W such that ind„g = r and ind w % = 5, where w = ( 7 7 I ) D A - 1 .

(i)

(id

FIGURE 5.76

If p is an isolated fixed point of a flow on V then we may, as for vector fields, define ind^p to be ind^g where g:Sn~ -*• V is an orientation preserving embedding with image a small sphere with centre p. In this case the index is preserved by any topological equivalence h. We do not get a change of sign when h is orientation reversing, since, although ind^, hg = - i n d ^ g , it is also true that ind^A(p) = -ind^/ig, since hg:S"~ ->W is orientation reversing. That is to say: (5.77) Corollary. If an isolated fixed point p of a flow <j> on Vis topological^ equivalent to a fixed point qof a flow iponW then ind^, p = ind^, q. •

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THE INDEX OF A FIXED POINT

141

Notice that if is the integral flow of the vector field v then ind^ p = ind„ p. Thus Corollary 5.77 also holds for vector fields. We are now able to define the index ind^, p of an isolated fixed point p of any flow cf> (or C vector field v) on any finite dimensional smooth manifold X by taking a chart £: U -*• U' <= R n at p and defining ind^ p to be the index of £(p) with respect to the induced (partial) flow on U'. Since another chart 17 at p gives rise to a fixed point 17(p) that is topologically equivalent to g(p), the integer ind^ p does not depend on the choice of chart at p. Corollary 5.77 continues to hold when V and W are general finite dimensional manifolds. By Corollary 5.77, Exercise 5.73 and Corollary 5.20 we have: (5.78) Proposition. The index of a hyperbolic fixed point p of a flow is (—l)m where m is the dimension of the stable manifold at p. • If a flow 4> on a finite dimensional manifold X has only finitely many fixed points, its index sum is Xiind^p (summing over peF\xcf>). Similarly for vector fields. Thus, for example, the index sum of the north-south flow on S" is 2 if n is even, and 0 if n is odd. If X is not compact then the index sum of may take any integer value, but if X is compact its topological structure puts stronger restrictions on the geometrical properties of the flows it can carry. In this case we have the celebrated theorem, due to Poincare in dimension 2 and to Hopf in general, that the index sum is independent of the flow 4> and depends only on- the manifold X. (5.79) Theorem. (Poincare-Hopf) The index sum of a flow 4> (resp. C vector field v) on a compact manifold Xis independent of 4>(resp. v), and equals the Euler characteristic of X. The proof of this theorem needs some transversality theory and it is essentially contained in Hirsch [1]. Milnor [2] and Guillemin and Pollack [1] give nice versions of the proof for smooth v. The whole of the above theory goes over perfectly well to diffeomorphisms / : X -> X, and even to continuous maps f:X-*X.If V is open in R", / : V-»R" is continuous, and g:Mn~l-» V is a map of a compact oriented (n - l)-manifold whose image contains no points of Fix/, then the index of g with respect to f, ind/g, is a*(l)> where a*:H„-i(M)-*Hn-i(R"\{0}) is induced by the map a (x)=fg(x)-g(x). So, in fact, ind^ g = indt, g where v is f-id regarded as a vector field. (5.80) Exercise. Prove that if h: R" -»R" is a homeomorphism then mdhfh-' hg = ind/g if h is orientation preserving, and - i n d / g if not. The theory now develops as for flows. The index of an isolated fixed point p of / is more commonly known as the Lefschetz number, Lefp (/), of f at p. Thus if p is an isolated fixed point of the flow then ind^, p = Lefp (') f o r

142

APPENDIX

CH. 5

any t > 0 with t < per q for all q e Per n 5, where 5 is a small sphere with centre p. The index sum of a continuous map / : X -» X is called the Lef schetz number, Lef (/), of /. For compact X, the analogue of the Poincare-Hopf theorem is not that this number is independent of /, but rather that it is an invariant of homotopy. This actually generalizes the Poincare-Hopf theorem, since, for any two flows and ip on X, <)>' and tp' are homotopic to the identity and hence to each other.

CHAPTER 6

Stable Manifolds

If we look at a picture of a saddle point (for example Figure 6.1), and try to analyse what qualitative features give it its characteristic appearance, we are bound to pick out the four special orbits that begin or end at the fixed point. These, together with the fixed point itself, form the stable and unstable manifolds of the dynamical system at the fixed point. We have noted in the last chapter the importance of hyperbolicity in the theory of dynamical systems, and a hyperbolic structure always implies the presence of such manifolds. If we know the "singular elements" of a system (in some sense which must include periodic points for diffeomorphisms, and fixed points and closed orbits for flows), and if we also know the way in which their stable and unstable manifolds fit together, then we have a pretty good hold on the orbit structure of the system. In this chapter we develop a theory for such manifolds, first for hyperbolic fixed points and then for more general invariant sets.

FIGURE 6.1

143

144

STABLE MANIFOLDS

CH. 6

I. THE STABLE MANIFOLD AT A HYPERBOLIC FIXED POINT OF A DIFFEOMORPHISM Let f:X^>X be a diffeomorphism, and let p be a fixed point of /. The stable set (or in-set) off at p is the set {xeX:f"(x)->p as n-»oo}. Notice that this set is always non-empty, since it contains p. The unstable set {or out-set) of fat p is the stable set of/ - at p. (Thus results about stable sets can generally be restated in terms of unstable sets; we leave this to the reader.) Given any open neighbourhood U of p, we define the local stable set off\Uat p to be the set of a l l x e U such that (f(x): n 3= 0) is a sequence in U converging to p. Notice that any point of the global stable set can be taken to a point of the local stable set by a suitable power of the diffeomorphism /. Our main theorem asserts that if p is hyperbolic then the global stable set is an immersed submanifold of X which is at least as smooth as the diffeomorphism /. We call it the stable manifold of / at p, Ws(p). It need not be an embedded submanifold, however. For example, there is nothing to prevent the situation illustrated in Figure 6.2, where the stable and unstable manifolds of p coincide. Moreover, things look far worse when the stable and

FIGURE 6.2

unstable manifolds at p have a point of transverse intersection elsewhere than at p. In such a situation, the point of intersection is said to be transversally homoclinic. This phenomenon is worth examining in more detail. First of all, we consider the situation (shown in Figure 6.3) where the

I

THE STABLE MANIFOLD

145

FIGURE 6.3

stable manifold at a hyperbolic fixed point p intersects the unstable manifold at another hyperbolic fixed point q transversally at some point x, say. The point x is then said to be transversally heteroclinic. The sequence (/" (x): n 3= 0) tends to p along the stable manifold at p, and is contained in a sequence (/"(/): n s= 0) of images of a small segment / of the unstable manifold at q. The members of the latter sequence are transverse to the stable manifold at p and eventually "press themselves up against" any local unstable manifold at p. Thus any point of the local (and hence of the global) unstable manifold of p is a limit of points on the unstable manifold of q. Similarly every point of the stable manifold at q is a limit of points on the stable manifold at p. This is an interesting situation, but does not make any extreme demands on the topology of the submanifolds in question. However, if we now put p = q, then we have that each point of the stable manifold is a limit of points on other branches (that is, local connected components) of the stable manifold, and similarly for the unstable manifold. Thus these immersed submanifolds are not, globally, copies of the real line with the usual topology. The points of intersection of the stable and unstable manifold of p, other than p itself, are transversally homoclinic points. If the reader is sceptical as to whether such behaviour can actually occur, he should take another look at the toral automorphisms of Example 1.30. We now return to the general theory. We shall deduce the global version of the stable manifold theorem from the local version. Roughly speaking, once we have planted a local stable manifold, we can easily grow it to obtain a global stable manifold; the difficulty lies in establishing it locally. As usual, we take a chart at the point in question, and thus transfer the problem to the model space E. Thus we obtain a local diffeomorphism with a hyperbolic fixed point at the origin, say, and the differential there is a linear approximation for the map itself nearby. We have been through this routine already in Corollary 5.20. The difference now is that we are interested in

146

CH. 6

STABLE MANIFOLDS

smoothness of maps, and whereas before we had no difficulty in extending a local Lipschitz map to a global one, a similar extension for smooth maps presents problems when E is an arbitrary Banach space. It seems a good idea, then, to give our basic results on perturbations of hyperbolic linear maps a local character. We start out with a hyperbolic linear automorphism T of a Banach space E. Thus E splits into stable and unstable summands E = E S ( D © E „ ( T ) = E , ( D x E U (T). We use the max {| |„ | |„} norm on E. Suppose that T has skewness a (see Theorem 4.19), and let K be any number with 0 < K < 1 - a. Let B = Bs x Bu be the closed ball with centre 0 and radius b (possibly b = oo) in E. We have, for T, the picture on the left in Figure 6.4. We shall show that, after a

~A

f

N

•*—B„

FIGURE 6.4

Lipschitz perturbation 17 with constant K satisfying | T J | 0 ^ M 1 — a), the local stable set of (T + -q)\B is, as in the picture on the right, the graph of a map, h say, from Bs to Bu. Moreover the map h is as smooth as the perturbation 17. By Exercise 5.13, T +17 has a unique fixed point in B, and by transferring the origin to this point we may assume, for simplicity, that 17 (0) = 0. We may now state: (6.5) Theorem. {Local stable manifold theorem) Let T\:B -»E be Lipschitz with constant K, where K<\—a. Suppose 77(0) = 0. Then there is a unique map h:Bs-*Bu such that graph h is the stable set of {T + TJ)\B at 0. Moreover the map h is Lipschitz, and C when r\ is C Proof. The idea is to identify in Sf0(B), which we recall (Appendix B) is a closed subset of the Banach space 5^0(E) of sequences in E converging to 0, those members of the form ((T4-TJ)"(*): n s=0) for some x = (xs, xu)eB.

147

THE STABLE MANIFOLD

I

This is a good idea because the first members of such sequences are precisely the points of the local stable manifold. We show first that y e Sf0(B) is of this form if and only if (6.6) where ,*,, ,

y(n) =

x(xs,y)(n)

M •, \^J + 7])sy(n-\\T-u\yu{n^\)-y\uy{n)))

(6.7) x(.xs,yw) = \

i

n.

forn>0

,nw

,

(.(*« Tu (y„(l)-i7„y(0)))

„

for n = 0,

and T; = (T/S, t j „ ) : 5 - * E s x E „ . This is clear, since the relation Vu(n) = T~ur (yu(n + l)-Tj M y(w)) holding on the unstable component for n 3= 0 may be rewritten as yB(« + l) = (T + Tj)By(«), and this combines with a similar relation on the stable component to give y(n + l) = (T+v)y(n) for n s= 0. The useful feature of (6.6) is that it exhibits y as the fixed point of a contraction of Sf0(B). For, given y and y' e Sf0(B), \x(xs,y)-x(xs,y')\ = sup {|(r+ r,) s y(n) - (T + T,).y'(n)\, \TZ1(yu(n + l)-yL(n

+ l) +

rluy'(n)-71uy(n))\:n^0}

1

«sup{(|rs|+Lipr,)|y-y'|,|r: |(l+Lipr,)|y-y'|} =s(a + /c)|y — y'|. Since a + K is < 1 , ^ contracts the second factor uniformly. Putting y' = 0 proves that x(x„ y) takes values in B. A similar estimate on (6.7) shows that x(xs, y) converges to 0. We have shown that (6.7) defines a map x'Bs x9,0(B)^y0(B). Notice that x is Lipschitz on the first factor, since \x(x's, y)-x(xs,y)\

=

\xs-x's\.

Thus by Theorems C.5 and C.7 of Appendix C, there is a unique map g: Bs -* 5^0(5) such that, for all xs e Bs, (6.6) holds with y replaced by g(xs), Correspondingly there is for each xssBsa unique point g(xs)(0) of the local stable set with stable component xs. Thus the map h of the theorem is given byh(x,) = (g(x,)(0))u.

148

STABLE MANIFOLDS

CH. 6

Notice that by Theorem C.7 the map g is Lipschitz. Now h is g composed with the map from y0(B) to B that evaluates sequences at 0, followed by the product projection. Since the latter maps are both continuous linear, we deduce that h is Lipschitz. Finally suppose that 17 is Cr. We can split x into the sum of two functions, one of which factors through Bs (this one takes (xs, y) to the sequence whose only non-vanishing term is (xs, 0) in the 0th place) and the other through y0(B). The first of these functions is trivially C°°, and the second is C\ by Lemma B.4 and Theorem B.20 of Appendix B. This is because Ts, T" 1 , product projections, and maps of Sf0(B) that move sequences to the left or right are all continuous linear. Thus x is C. We deduce from Theorem C.7 that g, and hence h, is Cr. • (6.8) Remark. The formula (6.7) defines equally well a contraction on 38(B), the space of all bounded sequences in B, and one obtains a unique fixed point map : Bs -» 53(5) which, by uniqueness, is the above map g. Thus the local stable manifold may be characterized as the set of all points x whose iterates (T + TJ)"(JC) for n s= 0 form a bounded sequence in B. Notice that the stable manifold theorem works under substantially weaker hypotheses than Hartman's theorem. We do not necessarily assume 17 to be bounded when b = 00, nor do we impose a condition to ensure that T +17 is a homeomorphism. In the appendix to this chapter we further investigate the dependence of h on 77. There are two other features of the local stable manifold that we would like to establish. Firstly the tangent space at 0 to the local stable manifold is the stable summand of the tangent map T0f, where we are now writing / = T +17. Secondly, iterates under / of points of the local stable manifold do not drift in gently towards 0; they approach it, and one another, exponentially. (6.9) Theorem. (/) Iff] is C1 in Theorem 6.5, then the tangent to graph hatO is parallel to the stable manifold of the hyperbolic linear map T + Dri(0). (ii) The maps h and f\graph h are Lipschitz with constant A, where f=T+r\ and A = a + K < 1. For any norm || || on E equivalent to \ \, there exists A > 0 such that, for all n^O and for all x and y e graph h, \\fn(x)-f(y)\\^AXn\\x-y\\. Proof, (i) Differentiating the relation g(xs) = x(xs, g(xs)) at 0 gives Dg(0) = Dx(0,0)(id,Dg(0)). We compute that, for all (xs, y)&Bs + 9>0(B), Dx(0, 0)(xs, y) is the sequence ( ( ( r + £ M 0 ) ) s y ( / i - l ) , r - 1 ( y B ( n + l)-£>7j ll (0)y(/i))) " ~ I(xs, T~ul (y„(l)-Dr,„(0)y(0)))

for » > 0 for n = 0.

THE STABLE MANIFOLD

I

149

Thus Dg(0): ES -» 5"0(E) is the fixed point map for x when 17 is replaced by Z?TJ(0). Thus the stable manifold map for T+DriiO) is xs>-*(Dg(0)(xs)(Q))u. But this is the map Dh{0). (ii) Recall that we are working with the norm |x| = max{|x s |, |xH|}. Let x and y e graph h, with x # y. Suppose that |x - y | = \xu - yu\. Consider fix)-fiy)

= iTsixs-ys)

+ Vsix)-Vsiy),

Tuixu -yu) +

rjuix)-T/U(y)).

The norm of the first component is no greater than (a + K) \x - y |, while that of the second component is at least ((1/a) —K)|X —y|. Since a + x
K>

1 K 1 -= >1, a aya + K) a + K

the norm of the whole expression is the norm of its second component. Thus, by induction, for all n s= 0, \fnix)-fniy)\^ia

+

K)-n\x-y\.

Since (a + K)~" -» °o as n -» 00, the sequences if"ix)) and (/"(y)) are not both bounded, which contradicts the fact that x and y are on the local stable manifold. We deduce that | x u - y M | < | x s - y s | . But now observe that, since fix) and fiy) e graph h, \fix) - / ( y ) | is also the norm of its first component, and hence, as above, not greater than (a + /c)|jc —y|. Thus f{graph h is Lipschitz with constant A. The given inequality follows immediately by changing the norm in

\fix)-fniy)\^\n\x-y\. Finally, if | x „ - y „ | > A | x s - y s | , then | / u ( x ) - / u ( y ) | > ( ( A / a ) - / c ) ) | x - y | , which is a contradiction, since (A/a) - K = 1 + /c(l - a)/a >l>a + K. Thus h is Lipschitz with constant A. • (6.10) Exercise. Prove an unstable manifold theorem for T +17 by using the formula ( \( ^l^T + risyin + ViT^iyuin-D-Vuyin))) for n > 0 , X(xmy)(n) \i{T + r))M1)>Xu) f o r „ = 0 (6.11) Exercise. We call Ta-hyperbolic (a > 0) if the circle of radius a in C does not intersect the spectrum of T. Thus a~lT is hyperbolic, and we deduce from Theorem 4.19 that there is a decomposition E = E S ©E U into T-invariant summands and an equivalent norm I | = max{| \s,\ |„}onEwith respect to which the restrictions Ts and Tu satisfy \TS\ < a and \T~ul \ < I/a. Let B be a ball with centre 0 in E (if a > 1 we require B = E), and let X _1 T / : B - » E be a Lipschitz map with constant K < m a x { a — |T"S|, |T~ | - a } and with 17(0) = 0. Replace SfQ{B) by di£f0(B)) (see Exercise B.22 of

150

STABLE MANIFOLDS

CH. 6

Appendix B) in the proof of Theorem 6.5, and thus prove that there is a unique map h:Bs^Bu such that, for all x e B, x e graph h if and only if {a~n{T+r))"(x): n ^ 0 } is well defined and converges to 0. Prove that h is Lipschitz. Prove also that if a =s I t and 17 is C then h is C. We call graph h the local a-stable manifold ofT + -qatO. Convince yourself that this result is not an immediate application of Theorem 6.5 to the hyperbolic automorphism a'1 T. We now prove the stable manifold theorem for fixed points of a diffeomorphism of a smooth manifold X. We start with the local version. The proof is completely straightforward, but we give it in full to establish some notation. (6.12) Theorem. Let p be a hyperbolic fixed point of a C diffeomorphism (/•s= l)fofX. Then, for some open neighbourhood Uofp, the local stable set of f\U at p is a Cr embedded submanifold of X, tangent at p to the stable summand of Tpf Proof. Let £: V-> V be an admissible chart at p, with £(p) = 0. Let g be defined, on some small neighbourhood W of 0 by g(y) = £/£ _1 (y). Then g is C and has a hyperbolic fixed point at 0. Pick a norm | | on E with respect to which Dg(0) has skewness a < 1. Then on some open ball B with centre 0, \Dg(y)-Dg(0)\^KBS is projection onto the stable summand. Then the family {Yr.i^O} covers Y. For all /,/ with i^j^0,£i(YinYi) = Bs and tji(Yi n Yj) = 77-sg'~J(graph h). Moreover the coordinate transformation 6/ = tti1 :6(Y, n Y,)•* £(Y, n Y,) t The reason for the failure of this approach in the case a > 1 is that Exercise B.22 does not cope successfully with smoothness. However there are some positive results in this case; see, for example, Irwin [2] and Hirsch, Pugh and Shub [1].

II

STABLE MANIFOLD THEORY FOR FLOWS

151

and its inverse £,-,- are the Cr diffeomorphisms given by £//(JC) = Trsg'~'(x, h(x)) and £a(x) = -n-sg'~'(x, h(x)). Notice that Bs is open, and hence, by the inverse function theorem vsg'~'(graph h) is open. Hence {£:/s=0}isa Cr atlas on Y. Let y e Y,. To show that the inclusion is a C r immersion at y, it suffices to show that its composite with / ' is an immersion at y (since / ' is a C dhTeomorphism). But the representative of this composite, with respect to the charts f, at y and £ at /'(y) is the Cr embedding {id, h). • The nature of the charts £• in the above theorem strongly suggests that the global stable manifold of / at p is an immersed copy of the stable manifold of the linear approximation. This is borne out by the following exercise. (6.14) Exercise. By extending the map 17 (in the proof of Theorem 6.12) to the whole of E, construct a locally Lipschitz bijection of the Banach space E s (Dg(0)) onto the stable manifold of /. Show that if / is Cr and if E admits C bump functions then this can be done so that the bijection gives a Cr immersion of E s (£)g(0)) in X. One may extend the above theory from fixed points to periodic points of a diffeomorphism with no extra effort. If p is a periodic point of a diffeomorphism / : X -» X, then p is a fixed point of the diffeomorphism fk, where k is the period of p. Notice that d(fm(x),fm(p))-*0 as m -»oo (where/is some admissible distance function on X) if and only if f"k(x) -» p as n -* 00. Thus, if we define the stable set of / at p to be (6.15)

{xeX:d(fm(x),fm(p))-+Oasm^oo},

and say that p is hyperbolic if it is a hyperbolic fixed point of fk, then Theorem 6.13 becomes: (6.16) Theorem. Let p be a hyperbolic periodic point of period k of a C diffeomorphism f of X, where rs=l. Then the stable set of f at p is a Cr immersed submanifold of X tangent at p to the stable summand of Tpf . D Once X is given a distance function d, our new definition of stable set makes sense for any point peX. We shall shortly be proving a stable manifold theorem of a more general nature that involves non-periodic as well as periodic points.

II. STABLE MANIFOLD THEORY FOR FLOWS The stable manifold theorem for a hyperbolic fixed point of a flow is a simple corollary of the corresponding theorem for diffeomorphisms. We first

152

STABLE MANIFOLDS

CH. 6

give a definition of stable set which will serve for any point of the manifold. Let 4> be a flow on a manifold X, and let d be an admissible distance function on X. The stable set of <j> at p e X is the set {xeX:d((t>(t, x), cf>(t, p))->0 as f-»oo} and the unstable set of(f>atp is the set {x&X:d{<j){t, x), (t,p))->0 as f-»-oo} The connection between stable manifolds of dirreomorphisms and flows is made by recalling the result of Exercise 1.39, that if p is a fixed point of 4> then 4>{t, x)^>p as t-»oo if and only if 4>(n, x)^>p as n ->oo, where f e R and n 6 Z. Thus the stable set of the flow <j> at /? is precisely the stable set of the diffeomorphism (f)1 at p, or, equally, of <£' for any other t>0). We deduce: (6.17) Theorem. (Global stable manifold theorem) Let p be a hyperbolic fixed point ofa Cflow (rs^l) 4> onX. Then the stable set of

in U is not necessarily the whole of the local stable manifold of l in U because orbits may leave U at non-integer values of / and then come back again. Nevertheless the former is an open subset of the latter, and thus a submanifold of it. One can get a Lipschitz version of the theorem, corresponding to part of Theorem 6.5, and also a local version when a C vector field does not have an integral flow defined for all t. Both are completely straightforward to prove. The theory for a hyperbolic closed orbit r = R . p is slightly more complicated. Let T have period T. It is an easy consequence of continuity of and compactness of T that d{(t, x), (t, p))-*0 as r-»oo in R if and only if 4>(nT,x)-*((>(nT,p) = p as n->oo in Z (see Exercise 1.39). Thus we are interested in the stable set of the diffeomorphism T at its fixed point p. Now p is not a hyperbolic fixed point of T; we have the splitting TpX = E s © (v) ® E„, where E s and EM are the stable and unstable summands at p and v is tangent to T at p. If /3 is the spectral radius of TpT\Es, we may choose any a with /3 < a < 1 and deduce from Exercise 6.11 the existence of a Cr local a-stable manifold W" (p) of 4>T at p. This submanifold is independent of the choice of a, by uniqueness, and is, as above, contained in the stable set of (j> at p. We may extend it backwards into a global Cr immersed submanifold Ws(p) modelled on E„ just as in the proof of Theorem 6.13. The images under ' of Ws(p), for all t e R, give a family of submanifolds, one through each point q = '(p) of I\ We write Ws(q) for '(W,(p)) if q = 4>\p), and WS(F) for {.re Ws(q): qeT}. Note that, since 4>{nr, x)-*p as n ->oo for all

II

STABLE MANIFOLD THEORY FOR FLOWS

153

x € Ws(p), 4>{nr, x)->q as n ->oo for all x e Ws(q) (see Figure 6.18, where the double arrows indicate orbits of the diffeomorphism (id x h), where h is the local a-stable manifold function at p. Since (v) does not lie in E„ which is tangent to the a-stable manifold at p, this map is a Cr immersion, and it follows that globally WU(T) is a C r immersed submanifold modelled on R x E s and C foliated (see Appendix A) by the family { Ws(q): q e T}.

We call Ws(r) the stable manifold of at T. We emphasize that we have not yet ruled out the possibility that (i) Ws(q) depends on choice of p, or that (ii) for some point x, cj> nT(x) -* p as n -» oo, but so slowly that x is not on the a-stable manifold for any a < 1, or that (iii) for some point x, the distance from {t,x) to T tends to 0 as" t-*oo without 4>"T(x) tending to any particular point q of T. If any of these phenomena were actually to occur, we would have points outside W s (r) that would nevertheless have a good claim to being called members of the stable set of I\ However it is easy to see, in finite dimensions at any rate, that (i)-(iii) cannot occur. For, let dim E s = d, and take a cross section Y to the flow at p. Let f:U->Ybea Poincare map at p, and let W be the (^-dimensional) local stable manifold of f\ U at p. If p : U -* R is the first return function, then the positive orbit R +. W = {t. y: ?s=0, y e W} is a (d + l)-dimensional submanifold consisting of all points x in the neighbourhood V = {t. y: y € U, 0 =£ ? =£ p (y)} of T whose positive orbits R + . x are wholly in V. All points w of R + . W have the property that the distance from t.w to T tends to 0 as t-»oo, and any orbit whose points have this

154

STABLE MANIFOLDS

CH. 6

property contains points of R + . W. But now observe that A = {t. y: y € W"{p), 0«.f SST} is a ( at p. In the infinite dimensional case, we are still able to prove that A is a neighbourhood of T in R + . W. It is not hard to show that the two submanifolds have the same tangent space E s +(i>) at p, and the result follows from this. Our theorem, then, may be stated as follows: (6.19) Theorem. {Stable manifold theorem for closed orbits) J^et r = R . p be a hyperbolic closed orbit of a C"flow{r ^\)ona manifold X with distance function d. Then the stable set Ws(p) of atp is a Cr immersed submanifold tangent at p to the stable summand of TPXwith respect to TP4>T, where T is the period of I\ If q = \p), then Ws(q) = '(Ws (p)). The stable set WS(T) of at T is a C immersed submanifold which is Cfoliatedby {Ws{q): qe T). • The submanifold Ws(p) is called the stable manifold of (f> atp. Since by the above remarks Ws(p) is also the stable manifold of the diffeomorphism T at p, it is independent of the distance function d, as also is WS(T) = U
III. THE GENERALIZED STABLE MANIFOLD THEOREM Let AT be a finite dimensional Riemannian manifold (see Appendix A) and let f:X-*X be a diffeomorphism. We have already commented that our definition (6.15) of the stable set of / at a point p is valid for any p e X, and we have seen that, when p is periodic and has a hyperbolic structure with respect to /, its stable set is a manifold. The question now arises as to whether we can extend our notion of hyperbolicity to non-periodic points p and get a stable manifold theorem for such points. The difficulty, of course, is that Tpf" does not map TpX to itself for n > 0, and so we cannot define hyperbolicity in terms of eigenvalues of this map. If we concentrate for the moment on the expanding and contracting properties of hyperbolic maps rather than their eigenvalues, we can see how a definition might go, but it will involve the whole orbit of p (as indeed does the definition of stable set). In fact, we may as well define the term hyperbolic structure for an arbitrary invariant subset of X, since many applications of stable manifold theory need this degree of generality.

HI

THE GENERALIZED STABLE MANIFOLD THEOREM

155

Let A be any invariant subset of X, and let TAM be the tangent bundle of X over A (that is, {TXX: x e A}). We say that A has a hyperbolic structure (with respect to /) if there is a continuous splitting of TAX into the direct sum of r/-invariant subbundles Es and Eu such that, for some constants A and A and for all v e Es, w e Eu and n s* 0, (6.20)

|2T(u)|«AA"|o|,

|7r n (w)|=sAA"|w|,

where 0 < A < 1. Thus one may say that Tf is eventually contracting on Es and eventually expanding on Eu. A hyperbolic subset of X (with respect to /) is a closed invariant subset of X that has a hyperbolic structure. There is a temptation to make "eventually" into "immediately" by putting A = n = l in the above definition, but the result would not be invariant under differentiable conjugacy. If A is compact, one may always introduce a new Riemannian metric on X with respect to which (6.20) does hold with A = n = l. Such a metric is said to be adapted to f. The proof, which resembles that of Theorem 4.47, is due to Mather [1], and also appears in Shub [4], Hirsch and Pugh [1] and Nitecki [1]. However we shall not need this result. The idea behind the generalized stable manifold theorem is as follows. If A is a compact hyperbolic subset, the space C°(A, X) of all continuous maps from A to X may be given the structure of a smooth manifold modelled on the Banach space YA{X) of C° sections of TA(X), given the sup norm derived from the Riemannian Finsler | \x on X (see Appendix A). One considers the map f: C°(A, AT)-* C°(A, X) defined by f*(,h)=fh{f\k)~\ Clearly this has a fixed point at i (the inclusion of A in X). It turns out that the hyperbolic structure of A makes i a hyperbolic fixed point. Thus / has a stable manifold at t, consisting of heC°(X,X) such that (f#)(h)->i as n n -» oo. For such h, f hf~"(x) -> x as n -» oo for all x e A, or, equivalently, by compactness of A, d(fh(y), f"(y)) -» 0 as n -* °o for all y e A. Here d is the distance function on X derived from the Riemannian metric (see Appendix A). So if we evaluate h at y e A, we get a point on the stable set of y. If we do so for all h near t, we get the local stable manifold at y, and properties (such as smoothness) of the stable manifold at t yield, after evaluation at y, properties of the stable manifold at y. Moreover, if we let y vary, we can get information about how the stable manifold at y varies with y. There is no need for us to give a full proof of all the statements in the previous paragraph. Since we are only applying the stable manifold theorem at a single point of C°(A, X) we can, provided we have a chart at i, remove the problem to the model space r°A(X), and, indeed, doing so makes some of our arguments easier to apply. For the manifold structure of C°(A, X) and related theory we refer the reader to Franks [2], Eliasson [1], Foster [1]. Notice, however, that for a e FA(X) near the origin, composing with the

156

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CH. 6

exponential map exp:TX-»X gives an element h of C°(A, X) near the inclusion, and that all such h come about in this way. Reversing the process gives us a chart at t. Moreover the tangent map Tf* is, in terms of this chart, the map o-^>{Tf)o-(f\hY . The decomposition r°A(X) =

C°(Es)®C°(Eu)

is TJ* -invariant, and, if we write Ts and Tu for the restrictions of TJ* to the two summands, we have that |r"|=sAV" and \T~"\^A\". Thus, by our remarks about the spectral radius in the appendix to Chapter 4, the spectra of Ts and Tu are separated by the unit circle, and so c is hyperbolic, as asserted. We now make some definitions needed for the statement of our main theorem. Let B(x, a) be the open ball in X with centre x and radius a, with respect to the Riemannian distance function d, and let S(JC, b) denote the set {yeX: d(fn(x),f(y)):P-> TAX defined by 4>{v) = (exp / ( x ) ) _ 1 /exp (v), where v is in the fibre Px. We may define a map : Cb(P)-* Cb{TAX) by

4>(p) = jP(f\Ar\ where Cb(TAX) is the Banach space of bounded sections of TAX (using the sup norm) and Cb{P) is the subset of sections with values in P. We observe that the map taking p to (77)p(/|A) _1 is continuous linear, and that the map on sections induced by the Fr map ${Tf~l) is Cr (see Remarks B.26 of Appendix B). Thus (f> is Cr.

Ill

THE GENERALIZED STABLE MANIFOLD THEOREM

157

The differential of <j> at OeCb(P) is the linear automorphism p>-» (r/)p(/|A) _ 1 of Cb(T^X). As indicated earlier, this automorphism is hyperbolic, with stable summand Cb(Es) and unstable summand Cb (Eu). We may, by Theorem 4.19, give those subspaces norms (equivalent to the previous ones) with respect to which the stable component of D (0) and the inverse of its unstable component have operator norm < 1 . We may then apply Theorem 6.5, with 17 = -D(j>(0), and obtain a C stable manifold map tp:Bs^> Bm for some small balls Bs and Bu of equal radii (see Theorem 6.9 (ii)) and with centre 0 in Cb(Es) and Cb(Eu) respectively. This ip is the unique map with the property that r = ip{a) if and only if "(cr, T) 6 Bs x Bu for all n 3= 0. To get back to the original norm on Cb(TAX), choose balls B(0,a) and B(0, b) with respect to that norm, and balls Ds and Du with respect to the new norm, all with centre 0, such that B(0,a)cDsxD„c5(0,A)c5sxBu. We have a stable manifold function :DS-*DU which is the restriction of i(i:Bs-*Bu. Let G denote its graph {(a, iA(er)): creDs}. If peB(0, a)n G, then <j>n{p) stays in Ds x£>u, and hence in B(0, b) for all n 2*0, whereas if p € B(0, a)\G, then n(p) leaves Bs x 2?u, and hence leaves B(0, b), for some n > 0 . Let V denote the image of B(0,a)r\G in Cb(Es) by the product projection. Clearly V is open in Cb(Es). Now consider the restriction tp-.V-* Cb(Eu). It is the unique map with the property that r = tjj(cr) if and only if (
for y 5* x

is in V, and ip{cr){x) = tf/(a-v)(x). Thus, if W denotes the open subset {cr(x): x e A, a- 6 V} of £ „ we may define a map g: W^>EU by g ( « ) = l/'(or I ,)(x),

where v e W*, such that ip is induced by g. That is to say, V = Cb (W) and, for all o- 6 V, ip(a) = go: In the notation of Appendix B, ip = g*.

158

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The map h of the theorem is defined by h(v) = exp (v, g(v)). Thus if g is F\ so is h. We would like to deduce the result for g from the fact that g* is C. This is not immediate. However if we now restrict our attention to the subspace C (TAM) of all continuous sections of TAM, we obtain as before a C stable manifold map from C°(W)= Vr^C°(Es) to C°(£„) which agrees with ip: W-* Cb(Eu) by uniqueness of the latter. Thus the C map 4>: C°( W) -* C°(EU) is induced from g, and this fact, by Theorem B.27 of Appendix B, ensures that g is Fr. Now that we know that g is Fr, we may work with bounded sections again. By Theorem 6.9, the differential Di)/(0) of ip: V-* Cb(Eu) is the zero map from Cb(Es) to Cb(Eu). The fibre differential of g at v e Wx is the linear map from {Es)x to (Eu)x whose value at w is the value at x of Dij/(av)(o-w)e Cb(Eu), where <x„ is the section defined above (see Remarks B.26 of Appendix B). Thus the fibre differential atOeWx is the zero map from Es to Eu, and this gives the required tangency property. D (6.22) Corollary. There exists constants A and A, with 0 < A < 1, such that, for all x e A, for all y and z e Wl°c (x) and for all n 3= 0, d(f"(y),fn(z))^A\nd(y,z). For all x and x' e A, W]°c (x) n W°c (x') is open in W°c (x). Proof. The inequality follows easily from Theorem 6.9 (ii) applied to the stable manifold of the map cf> in the above proof. Now let y 6 Wl°c (x) n W'soc (*'), and let z e W' oc (x). By the first part d(fn(z),fn(x'))^A\nd(z,x') ^A\n(d{z,x)

+ d(x,y) + d(y,x'))

«3aAA", and so, for all n 5* some n0, d(fn(z), fn(x')) < b for all z e Wl°Q (z). But, for z sufficiently near y,d(y,z)
wlr(x').

a

We shall not give a detailed account of the generalized stable manifold theorem for flows. We could obtain a version by following through the proof of Theorem 6.21, and replacing stable manifolds by a-stable manifolds (for a < 1) as in the proof of Theorem 6.19. There would remain the difficulty of proving that the a-stable manifolds so obtained are precisely the stable

Ill

THE GENERALIZED STABLE MANIFOLD THEOREM

159

sets of the points of the invariant set A. The generalized stable manifold theorem for flows is actually a corollary of an even more general theorem, Theorem 6.1 of Hirsch, Pugh and Shub [1], which is the reference for further reading on the subject.

Appendix 6

I. PERTURBED STABLE MANIFOLDS We extend the proof of the local stable manifold theorem information about the dependence of the stable manifold function the perturbation 17. We continue with the notations of Theorem addition, let Nr (r^O) denote the subset of Cr(B, E) consisting of that are Lipschitz with constant K and satisfy 17 (0) = 0.

to give h upon 6.5. In maps 77

(6.23) Theorem. For all 17 e Nr, the stable manifold function hnofT+-qis r r v C"-bounded. The map6:N -» C (Bs,Bu) sending T\ to h is continuous at T/0 if T/O is uniformly C. Proof. We regard r\ as another independent variable on the right-hand side of (6.7), so that x becomes a map from NrxBsx Sf0(B) to £f0(B). We obtain a fixed point map f:Nr xBs -> Sf0(B). Using Corollary B . l l of Appendix B, Xv -x° is C°-bounded and Dxv is C r l - b o u n d e d . Thus by Theorem C I O of Appendix C, gn —g° is C-bounded, and hence hv -h° is Cr-bounded. Since h°, the stable manifold function of T, is zero, hv is C r -bounded. Now let 770 e Nr be uniformly C, and let 17 e Nr. Then the map from Nr to C(BsxSf0(B), #o(E)), taking T? to A 4 1 1 - ^ " 0 is continuous at TJ0 (this is essentially Corollary B.12). Using Theorem B.20, we see that the map A-"0 is uniformly C. Hence, by Theorem C I O , the map from Nr to CT (Bs, &0{B)) taking rj to gv -gv° is continuous at 770. Thus so is 6, which is this map composed with (nu ev0)*, where tru: E -> E„ is the product projection. •

160

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Stable Systems

We have, with the generalized stable manifold theorem, reached the border between elementary theory and recent research. In this final chapter we describe one direction in which the subject has developed. We have already aired the general philosophy behind this development in earlier chapters. The guiding idea is to find a family of dynamical systems which, on the one hand, contains "almost all" of them and, on the other hand, can be classified in some qualitatively significant fashion. For some years it was conjectured that structurally stable systems would fulfil both requirements. Although this turned out not to be the case except in very low dimensions, structural stability is such a natural property, both mathematically and physically, that it still holds a central place in the theory. The two ingredients of structural stability are the topology given to the set of all dynamical systems and the equivalence relation placed on the resulting topological space. The latter is topological equivalence for flows, and topological conjugacy for difreomorphisms. The former is the C topology (where 1 =£ r =s oo). This topology is explained in detail in Appendix B, but the idea is clear enough. For example, two diffeomorphisms are C r -close when their values, and also the values of corresponding derivatives up to order r, are close at every point. One may think of the derivatives as being defined via charts on the manifolds. Since the derivatives depend on the charts there is some ambiguity, but, for compact manifolds at any rate, this is unimportant provided we work with suitable finite atlases. Once we have defined the Cr topology we may be more specific about what we mean by "almost all" systems. Certainly an open dense subject of this topological space would be admirable for our purpose. Even the intersection of countably many such sets (called a Baire or residual subsett; think of the irrationals as a subset of the reals) would satisfy us. t Such a subset would itself be dense, since the C r -topology makes the space of dynamical systems a Baire space. 161

162

STABLE SYSTEMS

CH. 7

Let Diffr X (1 =s 7-ssoo) denote the set of all C diffeomorphisms of the smooth manifold X, provided with the C topology. We say t h a t / e DifTX is structurally stable if it is in the interior of its topological conjugacy classt. That is to say, / is structurally stable if and only if any C r -small perturbation takes it into a diffeomorphism that is topologically conjugate to /. Similarly, let TrX denote the set of all C vector fields on X topologized with the C topology. The vector field v e TrX (or the corresponding integral flow ) is structurally stable if it is in the interior of its topological equivalence class. The definition of structural stability is due to Andronov and Pontrjagin [1].

I. LOW DIMENSIONAL SYSTEMS As the simplest possible example, consider a vector field v on 5 1 , with no zeros. Then there is precisely one orbit (exercise: why?) and this is periodic. We may identify TS1 with S ' x R . Let v : S1 -» R be the principal part of v. Since v is continuous and S1 is compact \v(x)\ is bounded below by some constant a > 0. Any perturbation of v with C -size less than a does not introduce any zeros. Thus the perturbed vector field still has only one orbit, and is topologically equivalent to v. So v is structurally stable (C° structurally stable, in fact). Now suppose that v has finitely many zeros, all of which are hyperbolic. We call such a vector field on S Morse-Smale (a definition for higher dimensions follows later). Then there is an even number 2n of zeros, with sources and sinks alternating around S 1 , as in Figure 7.1. The hyperbolic zeros are individually structurally stable (see Exercise 5.22), so a sufficiently

F I G U R E 7.1

t Notice that this means that we have (apparently) more than one notion of structural stability for /, since we may also regard it as in DifF X for any s < r. One ought, really, to talk of C structural stability, and we occasionally will. At the time of writing it is not known which, if any, of these notions are equivalent.

I

LOW DIMENSIONAL SYSTEMS

163

C -small perturbation of v leaves the orbit configuration unaltered on some neighbourhood of the zeros. But a sufficiently C°-small perturbation introduces no further zeros outside this neighbourhood. Thus the orbits flow between the components of the neighbourhood as before, and the perturbed vector field is topologically equivalent to the original one. Thus t) is C 1 structurally stable. It is important to note that there are vector fields v which are not Morse-Smale but which nevertheless have the above orbit structures. In this case the principal part v has zero derivative at one or more of the zeros, and the inward or outward motion near the zero is given by "higher order terms". One may also have vector fields on S1 with "one-way" zeros, as in Figure 7.2. However, provided that v has only finitely many zeros and is not

I

) V

>

FIGURE 7.2

Morse-Smale, one can easily alter its orbit configuration. Choose a nonhyperbolic zero p. By a C^-small perturbation of the vector field near p, alter p to a source if it was not originally one, or to sink if it was originally a source. Since we leave all the other original zeros of v unaltered, we have changed the number of sources in either case. We have shown, then, that structurally stable vector fields on S1 with finitely many zeros are MorseSmale. However, a stronger result is true: (7.3) Theorem. A vector field onS1 isC1 structurally stable if and only if it is Morse-Smale. Morse-Smale vector fields form an open, dense subset of Y'S1 (1 =£/•=£ oo). Proof. We have proved sufficiency above. Let c b e a vector field on S1. Suppose that v is not the zero vector field. The tangent bundle TS is isomorphic to the trivial bundle S 1 x R, so the image of v may be pictured as a curve on the cylinder (see Figure 7.3). The zeros of v are the points where the curve crosses the zero section, identifying the latter, as usual, with S 1 . The zero is hyperbolic provided the curve is not tangent to the horizontal there.

164

STABLE SYSTEMS

CH. 7

FIGURE 7.3

Now the image of the principal part v of v is a closed subinterval [a, 6] with a < b. The critical points of v (over which the curve is tangent to the horizontal) may form an infinite subset of S1 but, by Sard's theoremt, their image under v (the set of critical values) has measure zero in R. Thus arbitrarily near 0 e R is a number c which is not a critical value of v. We perturb v(x) to v (x) — c and get a new vector field with only hyperbolic zeros. These, being isolated, are finite in number. Thus the perturbed vector field is Morse-Smale. We have proved the density part of the theorem, as the perturbation is arbitrarily C°°-small. Note that if v were the zero vector field, we would first make a C°° small perturbation to get something else. Now suppose that v is structurally stable. Then v is certainly not the zero vector field. Since v can be made Morse-Smale by an arbitrarily small perturbation, it is topologically equivalent to a Morse-Smale diffeomorphism. But we showed in the run-up to the theorem that this implies that v is Morse-Smale. • The above theorem is a simple illustration of a fruitful approach to the classification problem. One attempts to prove that certain properties are open and dense (i.e. hold for systems in an open, dense subset of DifTX or FrX) or generic (the same thing with "Baire" replacing "open, dense"), using transversality theory. This is a theory that investigates the way submanifolds of a manifold cross each other, and how a map of one manifold to another throws the first across a submanifold of the second. Sard's theorem is the opening shot in this theory; for further details see Hirsch's t See Hirsch [1]. The proof in the present case is by dividing S 1 into n equal subintervals and letting n -> oo. By Taylor's Theorem, if a subinterval contains a critical point of v then its image under v has a length which is bounded by a fixed multiple of the square of the length of the subinterval. Thus the total length of all such images tends to zero as n -> oo.

165

LOW DIMENSIONAL SYSTEMS

I

book [1]. If a property is dense, then automatically a stable system is equivalent to a system with the property. One then anticipates, and attempts to prove, that the stable system itself has the property (since the property will often be one that is not shared by all equivalent systems). For example, the property of having only hyperbolic zeros is open and dense for flows on a compact manifold of any dimension, and any structurally stable flow possesses this property. The proof is essentially that for S given above. Similarly for dift'eomorphisms of a compact manifold X the following properties are generic, and satisfied by structurally stable diffeomorphisms: (i) all periodic points are hyperbolic, (ii) for any two periodic points x and y, the stable manifold of x and the unstable manifold of y intersect transversally. (This means that the tangent spaces to the two submanifolds at any point p of intersection generate the whole tangent space of the manifold X at p : TpWs(x)+TpWu(y)=TpX. See Figure 7.4. We denote this by the standard notation Ws(x)t\\ Wu(y).) The above result is known as the KupkaSmale theorem (Kupka [1], Smale [3], Peixoto [1]). Kiy)

• K(y)

* 1

11

>

Ws{x)

1 ll

1

•

r

1 i (

Non-transverse intesection at p

Transverse intersection at p FIGURE 7.4

Theorem 7.3 also holds for vector fields on compact orientable 2manifolds and for diffeomorphisms of S 1 , a result due to Peixoto [2]. Of course we must extend our definition of the Morse-Smale property to cover these cases. In fact, we give a definition for dynamical systems on manifolds of any dimension. We say that a dynamical system is Morse-Smale if (i) its non-wandering set is the union of a finite set of fixed points and periodic orbits, all of which are hyperbolic, and (ii) if G.x and G.y are any two orbits of the non-wandering set then

W,(G.x)hWu{G.y). Thus in the case of a 2-dimensional flow the non-wandering set consists of at

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CH. 7

STABLE SYSTEMS

most five types of orbit, viz. hyperbolic sources, sinks and saddle points, and hyperbolic expanding and contracting closed orbits. The second condition is vacuous unless both x and y are saddle points, and, when they are, it reduces to the condition that Ws(x) and Wu(y) do not have a common orbit (apart from {x} in the case x = y). One says informally "no orbit joins x to y". So the configurations in Figure 7.5 are not allowed in Morse-Smale flows. It is

x=y

F I G U R E 7.5

easy to get some idea of why condition (ii) should be necessary for structural stability. If it does not hold, we can take some point p on the offending orbit, and by making a slight (downwards in Figure 7.6) deflection of the vector

W,(x)

•y

*Jy)

FIGURE 7.6

I

LOW DIMENSIONAL SYSTEMS

167

field in a thin flow tube containing p, we can divert Ws(x) and Wu(y) into different paths. Thus we have altered the orbit configuration. Or have we? It certainly looks as though we have in our picture, but it is dangerous to lean too heavily on pictures. For example we might, in our alteration of the vector field, join up two perfectly harmless orbits which happended to pass through the tube, and thus create a new orbit joining some other pair of saddle points, x' and y' say. In this case the new system might possibly be equivalent to the old one by a homeomorphism taking x' to x and y' to y. Or again, suppose that condition (i) were not in force and that there were infinitely many orbits joining pairs of saddle points. It is far from obvious that obliterating one of them would alter the topology of the picture as a whole. In fact we have to be a bit careful, and we have to assume that condition (i) holds. We take a neighbourhood U of the non-wandering set sufficiently small not to contain p. Since there are only finitely many separatrices (as we call orbits that begin or end at a saddle point) and since each is in U for all but a bounded period of time, each will pass through a flow tube about p in the complement of U only finitely many times. By making the tube thinner, we can ensure that R.p passes through the tube only once and the other separatrices miss it altogether. If we perform the perturbation in this tube we do indeed decrease the number of saddle point connections by one, and hence alter the orbit configuration topologically. To show that a 2-dimensional Morse-Smale vector field v is structurally stable, one builds up a topological equivalence between it and any small perturbation of it first on a neighbourhood U of the non-wandering set of v (on which v is structurally stable by hyperbolicity), then on the components outside U of the separatrices (these have not been moved far by the perturbation and so still join corresponding components of U) and finally on the homogeneous looking blocks of orbits that remain. We shall not embark on the details of this programme, nor deal with the density problem. This is rather technical and special to two dimensions, and we refer the reader to Peixoto's paper [21. We just quote the result: (7.7) Theorem. A vector field on a compact orientable 2-manifold X (resp. a diffeomorphism ofS ) is C structurally stable if and only if it is Morse-Smale. Morse-Smale systems are open and dense in T'{X) {resp. DiffS ) for 1 ^/-ssoo. The sufficiency of the conditions in the case when X is a 2-dimensional disc, was first suggested by Andronov and Pontrajagin [1] and proved for an analytic vector field by De Baggis [1]. In a recent paper [3] Peixoto has developed his theorem into a classification of structurally stable systems. The theorem for non-orientable 2-manifolds is still an open problem, except for some cases which have been proved by Gutierrez [1].

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STABLE SYSTEMS

CH. 7

II. ANOSOV SYSTEMS For some time it was hoped that the above results might generalize to higher dimensions. Unfortunately neither the characterization nor the density of structurally stable systems goes over. It is, then, a relief to find that, for all dimensions: (7.8) Theorem. A Morse-Smale system on a compact manifold is C1 structurally stable. In fact, it is doubly welcome, because, since every compact manifold admits a Morse-Smale systemt, every compact manifold admits a structurally stable system. This was an early conjecture which turned out to be hard to prove, and survived until Palis and Smale's paper [1]. By the time this paper appeared it was known that there were other structurally stable systems besides Morse-Smale systems. The first of these to emerge were the toral automorphisms (see Example 1.30). These are structurally stable, but their non-wandering sets are the whole of the tori on which they are defined, so they certainly fail (i) of Morse-Smale (in fact, they are the opposite extreme to Morse-Smale systems in this sense!). Toral automorphisms were first put forward by Thom as a counterexample to the density of Morse-Smale diffeomorphisms. The justification of this is by proving them structurally stable, and this was first done by Anosov. His proof was for a wider class of systems which we now define. A diffeomorphism f:X^X of a manifold X is Anosov if X has a hyperbolic structure with respect to /. Recall that this means that the tangent bundle TX splits continuously into a 77-invariant direct-sum decomposition TX = ES®EU such that Tf contracts E and expands Eu (with respect to some Riemannian metric on X). Trivially hyperbolic linear maps / possess this property, since one has the identification TR" = R " x R " and Tf(x, v) = (f(x),f(v)), so that the splitting of R" into the stable and unstable manifolds of/, Ws and Wu say, gives a splitting of TR" as (R" x W S )©(R" x Wu). In the case of toral automorphisms this splitting is carried over to the torus when we make the identification, and so toral automorphisms are Anosov. Similarly a vector field on X is Anosov if X has a hyperbolic structure with respect to it, and as examples of such we have all suspensions of Anosov diffeomorphisms. The main result due originally to Anosov [1, 2] is: (7.9) Theorem. Anosov systems on compact manifolds are C stable.

structurally

t Take the unit time map of the gradient flow of almost any smooth real function on the manifold (see Smale [1]). Note also that Morse-Smale is an open property (see Palis [3]).

II

169

ANOSOV SYSTEMS

The diffeomorphism case of the theorem has a nice functional analytic proof due to Moser [1] (see also Franks [1] or Nitecki [1]). Recall, from the preamble to Theorem 6.21, that the set C°{X,X) of all C° maps of the compact manifold X to itself has the structure of a Banach manifold modelled on r°(X). Let f:X-*X be an Anosov diffeomorphism and let g:X-*X be C'-near /. As in Chapter 6, one considers the map f: C°(X, X) -> C°(X, X) sending h to fhf~\ Let G: C°(X, X) -* C°(X, X) be the C'-nearby map sending h to ghf1. Since, as in Chapter 6, f* has a hyperbolic fixed point at the identity idx, G has a fixed point, h say, near idxSo h = ghf~l, or, equivalently, hf = gh. Now the map / is expansive, meaning that the orbits of different points of X are never close together. More precisely there is a number a > 0 such that d(f"(x), /"(y)) < a for all n e Z implies x = y. This is clear from the fact that the map : Cb(P) ^ Cb(TX) denned as in the proof of Theorem 6.21 has a hyperbolic fixed point at 0, so that only iterates of 0 under <j> stay near 0. Suppose that h(x) = h(y). Then hfn(x) = gnh(x) = gnh(y) = hfn(y) for all neZ. Since h is near idx, fn(x) is near/"(j) for all n eZ, and thus since/is expansive, X = y. This shows that h is injective. But it is a standard result of algebraic topology that a continuous injection h.X-^X of a compact manifold is a homeomorphism. Thus / i s structurally stable. It is worth while getting as much insight as possible into the effects of hyperbolic structure, and so we give a more down-to-earth description of the existence of h and the expansiveness of / (due to Robinson [4]). The existence of a hyperbolic structure on X implies that, for any xeX, Tf-'Mf throws a product e-disc neighbourhood Ds x£)„ of 0 in Tr\x)X across a similar e-disc neighbourhood of 0 in TXX as shown in Figure 7.10. This £"»('"'(*))

£,(x) DS*DU

B

s*Bu

/V-i,./ •EuV'W)

0

£"„(*)

t T

ri{x)nD,*

FIGURE

D„)

7.10

behaviour is echoed by / in the manifold X, provided that e is small; it throws Df-\x) = exp/-i (x) (Ds xDu) across Dx = expx (Bs xBu), as shown in Figure 7.11. We have similar product neighbourhoods Dy for all y eX. We observe that / throws Dr*M across Dr\x) as shown in Figure 7.11, and hence that C\fn{DrnM): 0=£ n =£ 2} is as shown (shaded). It is, of course, possible that

170

STABLE SYSTEMS

CH. 7

FIGURE 7.11

/ (Df-2(X)) has further intersections with Dx, and we have attempted to indicate this in the diagram. If g is C 1 -close to / then we have a similar picture replaced by gn(Dr*M). It is fairly clear from the way that the (s-) height of the sets D™{g) = f\gn(Dr-M):0^n^ m) decreases, that £>"(g) is a disc stretching across Dx in the w-direction, as shown in Figure 7.12.

FIGURE 7.12

Since D™(g) in the set of points of Dx whose negative g-iterates stay near their corresponding /-iterates, g _ 1 maps DfM{g) into D " ( g ) . Moreover it contracts it, since the discs are in the u-direction. Thus f\g~n{Df"(x){g))\ n 3=0} is a single point, which we denote h(x). We observe that h(x) is the unique point of Dx such that every g-iterate of it stays near the corresponding /-iterate of x. Expansivity of / follows by putting g = /. In general, since gn(gh(x)) £ Df»+\X) = DffM for all n e Z, we deduce that hf(x) = gh(x). There are, at the time of writing, several unsolved problems about Anosov diffeomorphisms. For example, is their non-wandering set always the whole

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manifold? Do they always have a fixed point? Not all manifolds admit Anosov diffeomorphism (in contrast to Morse-Smale diffeomorphisms). Do all n -dimensional manifolds which do admit them have R" as universal covering space? One positive thing that we can say is that Anosov diffeomorphisms of tori are well understood; they are all topologically conjugate to the toral automorphisms (see Manning [1]).

HI. CHARACTERIZATION OF STRUCTURAL STABILITY The main problem now facing us is how to characterize structural stability, bearing in mind that we have to reconcile such apparently dissimilar systems as Morse-Smale and Anosov systems. The link comes when one recognizes, as Smale did, that if we generalize the Morse-Smale definition by replacing the term "closed orbits" by "basic sets" (definition shortly) then Anosov systems, and others as well, are allowed in. For example, toral automorphisms have only one basic set, the whole torus, so they trivially satisfy the new conditions. We say that a dynamical system has an Ci-decomposition if its non-wandering set Ci is the disjoint union of closed invariant sets VL\, il2, • • •, n k . If the system is topologically transitive on fl ; (that is, n , is the closure of the orbit of one of its points) for all i, we say that fl,- u . . . u Qfc is a spectral decomposition, and that the Q,- are basic sets. One could define the concept of a basic set in isolation by saying that a closed invariant set, A c O is basic if the system is topologically transitive on A but A does not meet the closure of the orbit of fl\A. Note that a basic set is indecomposable, in that it is not the disjoint union of two non-empty closed invariant sets. In general there is no reason why the non-wandering set of a system should have a spectral decomposition. For example in the flow illustrated in Figure 7.13, the figure eight consisting of two orbits joining a saddle point is an

F I G U R E 7.13

isolated part of the non-wandering set, but there is no way of splitting it into basic sets. However, Smale [5, 6] made the following definition:

172

CH. 7

STABLE SYSTEMS

A dynamical system is (or satisfies) Axiom A if its non-wandering set (a) has a hyperbolic structure, and (b) is the closure of the set of closed orbits of the systemt, and proved a fundamental theorem: (7.14) Theorem. (The Spectral Decomposition Theorem) The non-wandering set of an Axiom A dynamical system on a compact manifold is the union of finitely many basic sets. Proof. To get some idea of how Axiom A brings about this decomposition, let us examine the local structure of the non-wandering setfiof an Axiom A diffeomorphism / of a compact manifold X. We know that O has a generalized stable manifold system, and the first point to realize is that H is given locally as the set of vertices of a sort of grid formed by the family of stable manifolds crossing the family of unstable manifolds (this is the so-called local product structure of fl). To see this, note that if x and y are nearby points of H then, locally, by continuity of the unstable and stable manifold systems, Ws(x) intersects Wu(y) transversally in a single point z, and similarly Wu(x) intersects Ws(y) transversally in a single point t. We assert that z and t are in fl (a slightly more general result is known as the cloud lemma). By Axiom A(b), continuity of the stable manifolds system and the fact that ft is closed, we may assume that x and y are periodic, or even fixed (since periodic points are fixed by some power of/). Under positive iterates of /, any neighbourhood U of z presses up against and is spread out along Wu(x), until it intersects Ws(y) (see Figure 7.14). It is then pressed against Wu(y) and spreads out along it until it eventually intersects U. Also note that if N is a small open neighbourhood of x in fl and if U is a non-empty open subset of N then the orbit of U is dense in N. For, given

t,

\\\v

u \vv\S\ \\ ^ f(V) fr(u)

JC

\/C f(U)

'///J///////////AV7> ~-.r

F I G U R E 7.14

t It was for several years conjectured that A(a) implied A(b). This conjecture is now known to be true for diffeomorphisms of two-dimensional manifolds (Newhouse and Palis [1]) but false for diffeomorphisms of all higher dimensional manifolds (Dankner [1]).

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peN,v/e may, as above, assume p fixed, and also assume that there is a fixed point q in U. Then Wu{q) intersects Ws(p) in a point z which is, by local product structure, in fl. Some negative power of / takes z into U, and some positive power takes it arbitrarily near p. Thus p is in the closure of the orbit oft/. We now define a relation ~ on fl by x ~ y if and only ify is in the closure of the orbit of every neighbourhood of x in fl. We assert that ~ is an equivalence relation. Reflexivity is trivial. Suppose x~y. Let P be any open neighbourhood of y in fl, and let N be a small open neighbourhood of x in fl (i.e. small enough for the density property mentioned in the previous paragraph to hold). Since x~y, P intersects some f(N), and hence Nnf~r(P) is a non-empty set, U say. Since the orbit of U is dense in N, so is the orbit of P, and thus y ~x. See Figure 7.15. Similarly, suppose that x ~ y and y ~ z. Let

N

X

• orbit approaches

V

\

Su

X

I

/f-'(p) FIGURE 7.15

P be any neighbourhood of x in fl, and let M and N2 be small neighbourhoods in fl of y and z respectively. Then since y ~ z, some f'{Ni) intersects N2, and hence Ni nf~r(N2) is a non-empty open subset, V say. Also, since x ~ y, some f(P) intersects Nx in a non-empty open set C/i, say. Since the orbit of U\ is dense in Nu some/'(C/i) intersects V. Thus/ r + '([/i) intersects iV2 in a non-empty open subset U2. Since the orbit of U2 is dense in N2, so is the orbit of P. Thus * ~ z. See Figure 7.16. Let fl, be the equivalence classes of Cl under ~ . By the proof of symmetry, we get a typical fl ; by taking any open set Nt on fl small enough to have the density property and taking the closure of its orbit. Thus there are only finitely many fi„ because we can take a finite covering of the compact set fl by small enough open sets Nh and, of course, the associated fl„ some of which may coincide, cover fl. The fl, are, by the above description, closed

174

STABLE SYSTEMS

CH. 7

FIGURE 7.16

invariant subsets. We need finally to show that each fl, is topologically transitive, and this is a standard deduction from the fact that it contains an open set £/, with dense orbit. For, let V} be a countable basis for the topology of fl„ and let O, be the orbit of V,. Then 0 , is open and dense in H, (because fi, is an —equivalence class), and hence (~]jOj=D (say) is residual, and hence dense in H,. Let x e D. Then the orbit of x is dense in fl„ because any open subset V of O, contains some Vh and x ssome /"(V,), so/""0c)e VtcV. • To visualize a general Axiom A system, then, one thinks of a system with finitely many fixed points and periodic orbits, all hyperbolic (such as the gradient of the height function on the torus, or a Morse-Smale system, for example Figure 7.17 for S2), but in higher dimensions and with the fixed

FIGURE 7.17

points and periodic orbits replaced by more general basic sets. The basic set could, for example, be a torus on which the system (a difTeomorphism) restricts to a toral automorphism. For example, if one takes the product of a

Ill

CHARACTERIZATION OF STRUCTURAL STABILITY

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toral automorphism on S1xS1 with the (North pole)-(South pole) diffeomorphism of S1, one gets an Axiom A diffeomorphism of S1 x 5 1 x S1 with two basic sets, S1 x S 1 x {North pole} and S ' x S ' x {South pole}. There is, however, no need for a basic set to be a submanifold. We shall describe two examples for which it is not. Such basic sets are usually termed strange or exotic, and a good deal of work has been done on their structure (see, for example, Williams [1,3], and Sullivan and Williams [1]). The first of these is the expanding attractor described by Smale [5]. An attractor of a diffeomorphism is a subset A that has a compact neighbourhood N such that /(AT)cint N and f l ^ / W = A. For a flow 4> the corresponding conditions are 4>'{N) <= N for all t => 0 and (~),^0 '(N) = A. We visualize a map / of the solid torus SixB2 into itself, stretching the Sl factor to approximately twice its length, contracting the B2 factor into something under half its width, and wrapping it twice around the target copy, as shown in Figure 7.18. We insist that each cross-sectional 2-disc {[6]}xB2

FIGURE 7.18

is mapped into {[26]}xB2. Imagine the domain S1xB2 as embedded in a 3-ball B3 in some 3-manifold X. Certainly / extends to a diffeomorphism of X. One could even produce a deformation of X, fixed outside B3, which would slide S1 xB2 into / ( S ' x B 2 ) so that x finishes up at f(x). Thus the phenomenon that we are about to describe is by no means pathological. It may be met with locally in diffeomorphisms of any manifold of dimension greater than two.

176

STABLE SYSTEMS

CH. 7

Consider f2{SlxB2). This is a thinner longer tube which winds twice around/(5 1 x B2). Similarly fiS1 xB2) is thinner and longer still, and winds twice around f2{S1xB2) (so eight times around SlxB2). Any point x of 1 2 r 1 S xB that is outside f (S xB2) for some r > 0 has a neighbourhood U (in S 1 X B2) that is also outside/'(S 1 x B 2 ) . T h u s f (£/) does not intersect U for s s= r, and hence x is wandering. Thus, if we are interested in the nonwandering set o f / i n S ' x B 2 , we need only look at A = Dr^ofiS1 xB2). If 1 2 we look at the intersections of f(S xB ) with a single cross section {[8]}xB2 (Figure 7.19), we notice a striking resemblance to the well-known

FIGURE 7.19

construction of the Cantor set from an interval by repeatedly deleting middle thirds. Thus it is no surprise to find that A n ({[#]} X.B2) is homeomorphic to the Cantor set. Technically speaking, A is a fibre bundle over S 1 with projection the product projection of S1xB2 and fibre the Cantor set. It is not hard to see that A is, in fact, the non-wandering set of / on 5 1 x B2. Let x G A, and let U be any neighbourhood of x. It is clear that any segment [a, b]xB2 of S1xB2, however short, will, under a sufficiently large iterate f, wrap once round S 1 x B2 (i.e. its image will intersect every disc {[9]} x B2). A similar remark holds for any of the image tubes f{SxxB2). We may choose a segment V of some such tube with i e ^ c [ / . B y the remark, for some 5 > 0, f (V) intersects every cross-sectional disc of V. Thus f(U)nU is non-empty, which proves that x is non-wandering. A similar argument shows that any open subset of A has its orbit dense in A, which implies that A is topologically transitive. Since A is locally like the product of a real interval and the Cantor set, we can talk of the 1-manifold part of A at x, meaning the

Ill

CHARACTERIZATION OF STRUCTURAL STABILITY

177

fibre of the projection onto the Cantor set. The hyperbolic structure of A is clear: the stable subspace at x = ([0], p) is the tangent to the cross-sectional 2-disc at x, and the unstable manifold is the tangent to the 1-manifold part at x. This expanding behaviour on the 1-manifold part is the reason we call the attractor expanding. We now describe our second exotic basic set, the Smale Horseshoe (Smale [5]). Again the example is for a diffeomorphism, but would yield a basic set of a flow by suspension. It is fundamentally a 2-dimensional phenomenon, but is, perhaps, rather harder to grasp than the expanding attractor. This time we take a solid 2-dimensional square S, and four subrectangles, two vertical, R0 and Ru and two horizontal, 0R and xR as shown (Figure 7.20). f{a)

fib)

f(c)

f(d')

f(d)

f(c)

Hi>)

nd)

FIGURE 7.20

We map the vertices of 0R to those of R0 as shown, and extend linearly for a map / of 0R to R0. Similarly for iR. Note the half turn in this second case. The map / so far defined is clearly hyperbolic. We now extend / to the whole square S, mapping it onto the shaded horseshoe shape in Figure 7.21 (whence the name of the example). We make sure that f(S) is contained in the interior of C = D 0 u S u D 1 , where D0 and Dx are half discs on the horizontal edges of S as shown. We then extend / over D0 and Du mapping them into int D0. We may actually arrange for f\D0 to be a contraction, so that it has a unique attracting fixed point p. At this stage, / has a basic set of the type that we wish to consider. Notice again that, since we could produce / by deforming some neighbourhood of C in the plane, we can consider it as embedded locally in any 2-manifold X, as part of a diffeomorphism f:X-* X. We suppose this done. We are interested in locating Cl(f) n S. We observe that if a point s e S is not in 0R or iR then it is not in H(/). This is because some neighbourhood U of it in 5 gets mapped into D0 or £>i by /, so into D0 by f, and so into D0 by

178

STABLE SYSTEMS

CH. 7

FIGURE 7.21

any higher power of /. Hence U nf(U) is empty for r > 0. Similarly if x is not in R0 u Ru then some neighbourhood of it came from outside C under/, and its iterates under / _ 1 are all outside C, since f(C) c C. Summing up, if x e ft(/) n S then x e ,i?y = ,i? n i?y, for some /, /. Let 5i = U.-R/We now make a similar observation for / 2 . Let /?,y=/(,J? ; ) and ,7i? = / \iRj) (see Figure 7.22). Note that Rt, <: /?, and ,7i? ' : jR. If x e 5 is outside

oiff_

>//MU///»MJ.>///////>//,

zzz^zzzzrzzzzz n f f ^ 'WWM///////H/////////7

1,0

7^

0"0

0"\

C7

C7

FIGURE 7.22

U nR, then so is some neighbourhood of x in S, / 2 takes the neighbourhood into D 0 n £>! and further iterates of / keep it there. Hence x£ 0 ( / ) . Similarly x e S\U#,7 is not in fl(/). Hence 0 ( / ) n 5 c UG/KflKw) = &. Now define ,i?;fc = ,i? n i?,fc and ,vi?fc = nR n i?fc. Notice that fdjRk) - iRjkDefine i;fci? =f~\iiRk) and i?,7fc = /(*»,*)• Notice that R^cR^ and ,•,*£ c

CHARACTERIZATION OF STRUCTURAL STABILITY

179

jkR.

Repeat the above argument, showing that Cl{f) nS<^ UdjkR <"> Rimn) = S 3 (see Figure 7.23). It is clear that, with a bit of care, we could define Sr

:

: S,

I I I I

I I I I

I 1 I •

•

1

•I

S2

5"3

FIGURE 7.23

inductively, for all r> 1, and, hence, Sx = (~]r»i Sr with the property that Cl(f) c Sao. In fact 5oo is homeomorphic to the Cantor set. There is a standard representation A of the Cantor set as infinite bisequences a — ... a-2a-ia0.aia2a3 • • .where each a, is 0 or 1. If U„ (a) is the set of all such bisequences that agree with a for n terms on either side of the decimal point then {Un(a): n 5= 1} is a basis of open neighbourhoods of a in A. We have a homeomorphism from A to Soo which sends the bisequence a to the (unique) point in f W o (a_n+1...aoR r\Rai...«„). Identifying Soo with A by this homeomorphism, it is clear that/|5oo is the well known shift automorphism of A, which moves the decimal point one place to the left (i.e. / ( a ) , = a,-_i for all i e Z). Notice that Un(a) contains many periodic points of/, for example the one whose entries are the 2n terms a _ n + i . . . an repeated as a block time after time. This shows that SKcPer/, and hence that Soo^Cl(f), so that Sx, — il(f) n S. Also there is a bisequence c which contains every possible finite sequence as a block of consecutive entries, so that, for every aeA and n >0,f(c)e Un(a) for some r e Z. Thus the orbit of c is dense in 5<x>. Finally note that the hyperbolic structure for S^ is induced from f\(oR u ii?); the stable summands are horizontal and the unstable summands vertical. Returning to the question of characterizing structural stability, and following Smale, we make another definition. We say that a system is AS if it satisfies both Axiom A and the strong transversality condition: for all x and y in the non-wandering set of the system, the stable manifold of the orbit of x intersects the stable manifold of the orbit of y transversally. This latter condition is, then, just the general version of the second condition in the definition of Morse-Smale systems. The best set of criteria of structural stability that has so far emerged is due to Robbin [1] (for C 2 diffeomorphisms) and, later, to Robinson [2, 3, 4] (for C 1 diffeomorphisms and flows).

180

STABLE SYSTEMS

CH. 7

They prove (7.24) Theorem. Any AS system is C 1 -structurally stable. It seems likely that the converse of this theorem is also true, which would characterize structural stability very satisfactorily. At the time of writing the nearest approach to this is due to Franks [3] (see also Robbin [2]). A diffeomorphism f:X^X is absolutely Cl-structurally stable if for some C -neighbourhood N of /, there is a map a (called a selector) associating with g e N a homeomorphism <x(g) of X such that (i) a(f) = idx, (ii) for all g e N, g = hfh~x, where h = a(g), (iii) a is Lipschitz at / with respect to the C u metric d (i.e. for some K > 0 and all geN, d(
IV. DENSITY It is in attempting to generalize the second part of Peixoto's theorem, which deals with the density of Morse-Smale systems, that things go disastrously wrong. Obviously Morse-Smale systems are not dense in higher dimensions, since there are, as we have seen, other types of structurally stable systems. One might, nevertheless, have hoped that structurally stable systems are dense. This is not the case in higher dimensions, and we are now in a position to see why. The two ingredients for a counterexample are an exotic basic set (because such a set contains arbitrarily close but topologically distinct orbits, viz. periodic and non-periodic orbits) and a failure of the strong transversality condition. The idea is to construct a system where the unstable manifold of a fixed point has a point of tangency with a member of the stable manifold system of an exotic basic set A. Roughly speaking, if the dimensions are right, nearby systems exhibit the same tangency, but the

181

DENSITY

IV

stable manifold in question neither consistently contains nor consistently avoids periodic points of A. More precisely, one ensures that the stable manifolds of periodic points of A are dense. Thus systems for which they are tangent to the unstable manifold of the fixed point are dense near the given system, whereas, by the Kupka-Smale theorem, the stable and unstable manifolds of periodic points intersect transversally for a structurally stable system. The above idea was originally due to Smale [4], who constructed examples for diffeomorphisms of compact 3-manifolds and flows on compact 4manifolds. Peixoto and Pugh [1], by similar methods, showed that structurally stable systems are not dense on any non-compact manifold of dimension 5=2. Finally Williams [2] completed the picture by reducing the dimensions of Smale's examples by 1. Thus, for instance, structurally stable diffeomorphisms are not dense on the two-dimensional torus T2. One starts with a toral automorphism/on T2, and modifies it in a neighbourhood of the fixed point 0 by composing with a diff eomorphism that is the identity outside some smaller neighbourhood, preserves the y-coordinate (the unstable manifold direction) but near 0 expands strongly in the x-direction (the stable manifold direction). This has the effect of turning 0 into a source and introducing two new saddle points, one each side of it. One might almost imagine that the old unstable manifold had been split in two lengthwise and the unstable manifold of a source grafted into the hole (see Figure 7.26).

_»

I

a

I

«

becomes

—*•

•

II ••-

•

\f/

*•

FIGURE 7.26

At this stage the system is called a DA (derived from Anosov) system. It has two basic sets, the source 0 and an exotic 1-dimensional expanding attractor A. The stable manifold system is still the stable manifold system of the original Anosov diffeomorphism (horizontal lines in Figure 7.26) except that one of them is broken in two at 0. We now remove a neighbourhood of the source at 0, and replace it by a plug consisting of two sources and a saddle point with the latter at 0 (see Figure 7.27). This modifies the stable manifold

182

STABLE SYSTEMS

CH. 7

becomes

FIGURE 7.27

system of A to give Figure 7.28. At the moment the unstable manifold of the saddle point 0 is horizontal. We now perturb the diffeomorphism (call it g now) slightly in the neighbourhood of some point p of the unstable manifold

FIGURE 7.28

to produce a kink in it at g(p) as shown in Figure 7.29. The diagram is oversimplified, as we do not attempt to reproduce the further kinks in Wu(0) (at g"(p),n>l) and in WS(A) (at and near g~n(p),n>0). However we now have the tangency that we are aiming for. There are two obvious courses of action open to us, as a response to the non-density of structural stability. We can either alter the equivalence relation on the space of all dynamical systems, in the hope that stability with respect to the new relation might be dense, or we can ask for something less than density in the given topology. We shall examine another equivalence relation in the next section. We finish this section by giving a positive result in the second direction.

IV

DENSITY

FIGURE

183

7.29

A natural question to ask is the following. Given an arbitrary dynamical system, can we deform it into a structurally stable system? If we can, how small a deformation do we need to make? We cannot make it arbitrarily C -small, since this would imply C 1 -density of structural stability. But we might be able to do it by an arbitrarily C°-small deformation. Notice that we are talking only about the size of the deformation needed to produce structural stability; the smoothness of the maps involved and the definition of structural stability are still as before. These questions are answered by the following theorem of Smale [9] and Shub [1] (see also Zeeman [1] and Franks [4]). (7.30) Theorem. Any Cr diffeomorphism (1 =£ r =£ oo) of a compact manifold is C isotopic to a C structurally stable system by an arbitrarily C°-small isotopyt. The idea behind the proof is to triangulate the manifold X (see Munkres [1]) and hence to get a handlebody decomposition (Smale [10]; see also Mazure [1], who calls them differentiable cell decompositions). The technique then adopted is illustrated in two dimensions by Figure 7.31. The shaded picture on the left, part of the handlebody decomposition of X is originally mapped into X by / as shown. We first isotope / to a map g with the property that the image of every /-handle is contained in a union of /-handles for various ;' =£ i. This effect may be produced on the whole of X, using an inductive argument. t Thus structural stability is dense in DifF X with respect to the C°-topology. Of course it is no longer open if we use this topology. The analogous theorem for flows has been announced by D'Oliviera [1],

184

STABLE SYSTEMS

CH. 7

FIGURE 7.31

We then isotope g to a new map h, such that (i) the 0-handles are contracted, (ii) the 2-handles are expanded at all points of h^(2-handles), and (iii) the 1-handles are mapped in a nice linear fashion, expanding lengthwise and contracting radially, at all points of h~l{l-handles). Notice the horseshoe that appears along the handle AC. With a little care, we have achieved a structurally stable map h, the fl-set consisting of periodic sources associated with 2-handles, periodic sinks associated with 0-handles and saddle-type basic sets associated with 1-handles which may very well be exotic, for example the above horseshoe. The size of the isotopy involved depends on thefinenessof the triangulation of X, and so can be made arbitrarily small, in the C°-sense, by taking a fine enough triangulation.

V

OMEGA STABILITY

185

V. OMEGA STABILITY The failure of structural stability to be dense in higher dimensions led to consideration of other equivalence relations on the pace of all systems. Of these, the one that has so far aroused the most interest is Vl-stability. We have already introduced the notion of fi-equivalence at the end of Chapter 2, and O-stability, is, of course, stability with respect to this equivalence relation. Thus, for example, a dhreomorphism f:X^> X is Cl-stable (in the Cr sense, 1 «£ r=soo) if, for any nearby g e DiflfX, there exists a topological conjugacy between /|fl(/) and g\£l(g). At first sight it may seem rather an extreme measure to ignore all behaviour off the fl-set. There are two answers to this criticism. Firstly, it is only in defining the equivalence relation that the non-fl behaviour is ignored; it is very important in determining stability with respect to the relation. Secondly, it is arguable that in applications of the theory to mathematical modelling the fi-sets are the only parts of the system with real physical significance. Certainly if a dissipative system in dynamics gives rise to a single dynamical system then only the attractors correspond to phenomena that are physically observable. In order to get some feeling for the way in which the non-wandering set of a system may alter when the system is perturbed, we examine a few simple examples. First consider a "one-way" zero of a flow on S 1 (Figure 7.32). By

FIGURE 7.32

making a small local perturbation (adding a small velocity in the one-way direction) we eliminate the zero, and, in the first case, introduce a periodic orbit. Thus we have exploded the fl-set from a single point to the whole circle. In the second case, we implode the fl-set from three points to two. If we have an isolated non-hyperbolic zero which is "two-way" then we cannot entirely eliminate it by a local perturbation. However, for any non-hyperbolic zero, we can, by a C^-small perturbation produce an interval, centred on the point, consisting entirely of zeros (the size of the interval varying with the size of the perturbation, of course), or introduce other more complicated behaviour on such an interval (Figure 7.33). Here the explosion in the fl-set is local; its size is limited by the size of the perturbation allowed, in contrast to the previous explosion.

186

STABLE SYSTEMS

CH. 7

FIGURE 7.33

A more spectacular explosion occurs if we start with the first picture in Figure 7.34, which represents the ft-set of a diffeomorphism / of S2, though, as usual, we draw flow-like orbits for easier visualization. The fi-set consists of two sources y and /, a sink z and a saddle point x, together with an invariant arc / joining* to itself as shown (so that / c W„(x) n Wu(x)). Notice that / cannot have a hyperbolic structure (Exercise: Why not?).

FIGURE 7.34

We perturb / on a small neighbourhood U of p, so that the resulting map g takes InU transversally across lng(U) at f(p) = g(p). The unstable manifold of x with respect to g, Wu(x), is the same as it was before the perturbation as one travels around it from x to p. However, after passing p moving to the left, it has a series of kinks which get larger and larger as they approach x. They press themselves up against Wu(x), and eventually spread out past p into the kinks already formed, and so on. Similarly as Ws (x) passes p going right, it develops a series of kinks which press themselves up against earlier parts of Ws(x). Of course we have just described again the transverse homoclinic point phenomenon of Chapter 6. The point is that Ws(x) now

V

187

OMEGA STABILITY

intersects Wu(x) in an extremely complicated and rather random way, and, by the sort of "cloud lemma" argument used in the proof of Theorem 7.14, any such intersection is in Cl(g). We leave the reader to prove the intuitively obvious fact that fl(/) is topologically different from il(g). At first site, it appears that fl-explosions are caused by a lack of hyperbolicity, so that we have, in Axiom A, the condition to eliminate them. This turns out to be true, in a sense, for local H-explosions, but Axiom A does not prevent global explosion. To see this consider the example (Smale [6]) of a diffeomorphism of S 2 with fl-set consisting of six hyperbolic fixed points, of which two are sinks, two sources and two saddles, laid out as in Figure 7.35. To see that the fl-set is as described note that any point outside the closed curve has a neighbourhood that either falls into the sink c under positive iterates of/, or into the source a under negative iterates. Similarly for points inside the closed curve. A small neighbourhood of a point such as p, under positive iterates of /, presses up against the underside of Wu(x) and the right-hand side of the portion of Wu (y) going to d. It does not come back and re-enter U, so p is wandering. Note that under negative iterates of /, U presses itself against the topside of Wu(x)n Ws(y) and so positive and negative iterates of U come arbitrarily close together. Thus if we, by local perturbation at q on Wu(x) n Ws{y), bring Wu(x) above Ws(y), we cause the positive and negative iterates f"(U) and f~"(U) to intersect for large n (Figure 7.35). Hence p becomes non-wandering, and similarly, any other

f before perturbation

f 0 f t e r perturbation FIGURE

7.35

point of Ws(x)nWu(y). On the other hand the points of Wu(x) n Ws(y) are wandering, just as they were before. Hence the new fl-set is the six points together with the arc Ws(x)r\ Wu(y). Note that, again, we cannot have a hyperbolic structure. If we make another perturbation so that Ws(x) and Wu(y) cross transversally, we cause another explosion. We have forced

188

STABLE SYSTEMS

CH. 7

intersections Ws{x)n Wu(x) and Ws{y)n Wu(y) to appear and these join the new versions of Ws(x)n Wu(y) and Ws(y)r\ Wu(x) and the six fixed points to give the O-set of the new map. The set is similar to, but even harder to visualize than the one in the last example. We now give an example of a system that is H-stable but not structurally stable. Consider the gradient flow of the height function on the torus T2 already described in Example 3.3 and illustrated again in Figure 7.36. A

FIGURE 7.36

slight perturbation of the vector field will make the separatrices beginning at q end at s instead of r. However for all C 1 -small perturbations the orbit structure is the same near the zeros, and elsewhere the orbits continue to move downwards, which precludes any global recurrence. Thus the fl-set is still the four zeros, and so the system is fl-stable. Since this last example does not have strong transversality, it is clear that this property is not a relevant consideration when trying to characterize fl-stability. We must examine the above examples to see what causes the explosions. We can distinguish two types, local explosions and global explosions. Local explosions are ones where the old and new fl-sets are close together, the degree of closeness depending on the size of the perturbation. Global explosions are ones where arbitrarily small perturbations produce H-sets that are not close to the original onest. The examples in Figures 7.33 and 7.34 are local, those in Figures 7.32 and 7.35 are global. As we have already hinted, local explosions are due to a lack of hyperbolicity. The basic sets of an Axiom A diffeomorphism / are locally stable. That is to say, given any basic set A, there is a neighbourhood U of A such that, for all g

t In some references (e.g. Shub [4]) the term "fl-explosion" is only applied to global explosions.

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OMEGA STABILITY

189

sufficiently C ! -near /, U nfl(g) has a subset A' such that (i) A' contains every g-invariant subset of U, and (ii) g|A' is topologically conjugate to /|A. Thus there may be other points of Cl(g) in U, but their orbits go outside U. This stability property is a close relative to the stability theorem for Anosov diffeomorphisms, and, not surprisingly its proof (see Smale [6] and Hirsch and Pugh [1]) has a lot in common with that of Theorem 7.9 sketched above. Several times in this chapter, in proving that a point is non-wandering, we have used arguments that involve open sets spreading out from basic set to basic set until they get back near where they started. Any two consecutive basic sets in the route have to be linked together, or the open set cannot make it across from the one to the other. The link consists of an unstable manifold of the one and a stable manifold of the other with non-empty intersection. If the manifolds meet transversally the open set can always get across; if not then it may still be lucky or it may not (see Figure 7.37).

Unlucky non-transverse intersection at p FIGURE

7.37

Another unlucky situation, illustrated in the first picture in Figure 7.35 occurs when the open set reaches the first basic set of a link but cannot get near the intersection point of the link. This can happen when, due to an

190

STABLE SYSTEMS

CH. 7

earlier lack of transversality, the open set does not intersect the stable manifold of the basic set (/"([/) does not intersect Ws(y) in the picture). These unlucky situations, since they are all due to non-transversality, may be eliminated by arbitrarily small perturbations producing transversality. Thus a recipe for a global fl-explosion of an Axiom A system is to start with a collection of basic sets, linked up to form a circular route, but with one or more unlucky situations incorporated (of course—we could not have separate basic sets otherwise). Then, by a small perturbation, we can remove all the bad luck situations, simultaneously if we do not insist on the perturbation being a local one, and we have our explosion. The above description motivates the following definition (Rosenberg [1]). An n-cycle of an Axiom A dynamical system is a sequence of basic sets fio, f l i , . . . , fl„, with fl 0 = fl„ and fl,•# fl, otherwise, and such that Wu(Ai-i) n Ws(Clj) is non-empty for all /, 1 =£ / =£ n. An Axiom A dynamical system satisfies the no-cyclic condition if it has no «-cycles for all n s* 1. The main theorem on fl-stability, due to Smale [6] in the diffeomorphism case, and to Pugh and Shub [1] in the flow case is: (7.38) Theorem. / / an Axiom A system on a compact manifold has the no-cycle property then it is il-stable. The idea of the proof is to show first, as indicated above, that Axiom A rules out local explosions. We then have a situation rather similar to that illustrated in Figure 7.36. If W„(fl,) n Ws(flj) is non-empty, we can think of fl* as being situated above fl, (the no-cycle condition ensures that we do not find, paradoxically, that fl, is above itself) and the movement between basic sets as being downwards. It seems quite clear that C1- small perturbations cannot produce an upwards motion and so global recurrence cannot occur in the perturbed system. Technically the proof uses a filtration (see Shub and Smale [1]). This is an increasing sequence of compact manifolds with boundary X\ <= • • • cXr=X, such that dim Xt = dim X, f(Xt) <= int Xt and each XAXi-i contains a single basic set. We picture a suitable filtration for the Figure 7.36 example in Figure 7.39. As with structural stability, it is conjectured that we have here a complete characterization; that fl-stability implies Axiom A and no cycles. Palis [1] has the partial result that Axiom A and fl-stability imply no cycles. There is, again, a notion of absolute stability, and this is characterized by Axiom A and no cycles (see Franks [3], Guckenheimer [1]). Unfortunately fl-stability is no more successful than structural stability as far as density is concerned. Abraham and Smale [1] took the product of a toral automorphism and a horseshoe diffeomorphism on 5 „ and by making the former DA over one of the two fixed points (... 00-00 . . . and . . . 11-11 . . . ) of the horseshoe, they constructed a

VI

BIFURCATION

191

XA--T2

FIGURE 7.39

diffeomorphism of T2xS2 that has a C 1 -neighbourhood all of whose members are neither Axiom A nor H-stable. Finally, Newhouse [1], by modifying the horseshoe diffeomorphism, constructed a counterexample to C 2 -density of Axiom A and of O-stability for diffeomorphisms of any surface.

VI. BIFURCATION Several other notions of stability have been put forward in the hope that they might prove to be generic, for example future stability (Shub and Williams [1]), tolerance stability (Takens [1], White [1]), finite stability (Robinson and Williams [1]) and topological il-stability (Hirsch, Pugh and Shub [1]). Some have survived longer than others; none has yet been wholly successful. In the course of time, a feeling has developed that perhaps it is too optimistic to expect to find a single natural equivalence relation with respect to which stability is dense (see, for example, Smale [8]). Recently, more attention has been focused on the interesting and important question of bifurcation of systems. Up to now we have not devoted much time to systems such as the one-way zero in Figure 7.32. Our feeling has been that the most important systems are ones which can be used to model the dynamics of real life situations. But no real life situation can ever be exactly duplicated, and we should expect this to lead to slight variations in the model system. Thus a theory making use of qualitative features of a dynamical system is not convincing unless the features are shared by nearby systems. That is to say, good models should possess some form of qualitative stability. Hence our contempt for the

192

CH. 7

STABLE SYSTEMS

one-way zero, which is extremely unstable. Now this argument has some force, but it is too simple-minded. Firstly, there may, in a physical situation, be factors present that rule out certain dynamical systems as models. Conservation laws have this effect, and so has symmetry in the physical situation. In this case, the subset of those dynamical systems that are allowable as models may be nowhere dense in the space of all systems, and so the stable systems that we have considered in this book are totally irrelevant. One has to consider afresh which properties are generic in the space of admissible systems. This has been done for Hamiltonian systems (see Robinson [1]). Secondly, even if the usual space of systems is the relevant one, the way in which a stable system loses its stability as it is gradually perturbed may be of importance, since the model for an event may consist not of a single system but of a whole family of systems. In his theory of morphogenesis, Thom [1] envisages a situation where the development of an organism, say, is governed by a collection of dynamical systems, one for each point of space time. The dynamical systems are themselves controlled by a number of parameters, which could for example be quantities like temperature and concentrations of chemicals in the neighbourhood of the point in question. A catastrophe is a point where the form of the organism changes discontinuously, and it corresponds to a topological change in the orbit structure of the dynamical systems. We say that the family of dynamical systems bifurcates there. We shall give some examples of bifurcations of flows. These particular ones are local changes that can take place on any manifold. By taking a suitable chart we may work in Euclidean space. First consider a vector field va on R given by va(x) = a + x . Here a is a single real parameter, so for each a e R we have a vector field on R. We are interested in how the orbit structure varies with a. The analysis is very easy, and we find that (i) for a > 0 we have no zeros, and the whole of R is an orbit, oriented positively, (ii) for a = 0 we have a single zero at x = 0, which is a one way zero in the positive direction, (iii) for a < 0, we have two zeros, a sink at — v — a and a source at v — a. Thus the bifurcation occurs at a = 0. Figure 7.40 illustrates the bifurcation. — » _ _ . . « — « — . . • • — • — — •

a<0

»

» — »

a=0 FIGURE 7.40

>

a>0

193

BIFURCATION

VI

If one takes the product of va with a fixed (i.e. independent of a) vector field on R " 1 having a hyperbolic fixed point at 0, one obtains a bifurcation of the resulting vector field on R". Of course we have essentially added nothing to the original bifurcation in doing this. All such bifurcations are known as saddle-node bifurcations, because of the picture that one gets for n = 2 (Figure 7.41). A saddle point and a node come together, amalgamate and cancel each other out.

o<0

a=0

a^O

FIGURE 7.41

Saddle-node bifurcations are important because (when properly defined) they are stable as bifurcations of one-parameter families. Roughly speaking, then, one-parameter families near to a family with a saddle-node bifurcation also exhibit something that is topologically like a saddle node bifurcation near (in the positions of the amalgamating zeros, for example) the original one. One speaks of them as being codimension one bifurcations; one is then visualizing the set of systems exhibiting zeros of the a = 0 type above as (in some sense) a submanifold of codimension one in Yr(X), and the oneparameter family as being given by an arc in Tr(X) crossing the submanifold transversally. Notice that the bifurcation illustrated in Figure 7.42 (a node bifurcating into two nodes and a saddle point) is not stable for oneparameter families. It can be perturbed slightly s,o that there is a saddle-node bifurcation near to but not at the original node. The saddle-node bifurcation is the typical bifurcation obtained when the sign of a real eigenvalue of (the differential at) a zero is changed by varying a single parameter governing the system. There is, similarly, a typical codimension one bifurcation which comes about when the sign of the real part of a complex conjugate pair of eigenvalues is changed by varying a single parameter. This is known as the Hopf bifurcation, since it was first described by E. Hopf [1]. Consider the vector field va on R 2 given by va(x, y) = (-y -x(a

+ x2 + y2), x - y(a + x2 + y2)).

194

STABLE SYSTEMS

CH. 7

FIGURE 7.42

Again a is a single real parameter. For all a e R, va has a zero at the origin, and the linear terms make this a spiral source for a < 0 and a spiral sink for a > 0. For a = 0 the linear terms would give a centre (recall that this is a zero surrounded by closed orbits), but the cubic terms make the orbits spiral weakly inwards. The interesting feature of the bifurcation is not, however, the zero, but the unique closed orbit that we get at x2 + y2= — a, for each a < 0 . The bifurcation is illustrated in Figure 7.43. The periodic attractor

F I G U R E 7.43

decreases in size until it amalgamates with the spiral source to form a spiral sink. The reverse bifurcation, in which a spiral sink splits into a spiral source and a periodic attractor, is intriguing because of the feeling one has that from something inert and dead (the source) one has created something pulsating and alive (the periodic orbit). This makes it very popular as a component in mathematical models. It is not possible at present to give a coherent account of bifurcation theory for dynamical systems, since the subject is still in an early stage of

VI

BIFURCATION

195

development. Papers of Sotomayor [1, 2, 3] give some general principles for tackling the problem and prove some generic properties of bifurcations of one parameter families of vector fields (see also Guckenheimer [3]). Newhouse and Palis [2, 3] have examined the situation after one passes the first bifurcation of a Morse-Smale vector field, and found that it may be very complicated. Newhouse and Peixoto have investigated conditions under which pairs of structurally stable vector fields may be connected by oneparameter families with only finitely many stable bifurcations (see Newhouse and Peixoto [1] and Newhouse [3]). Takens [2, 3, 4] has written papers on various aspects of bifurcations of singularities of dynamical systems. Guckenheimer [2] has shown that the analysis of bifurcation of functions on manifolds is fundamentally different from and simpler than the corresponding theory for gradient vector fields. The above discussion has mainly been for bifurcations in the context of topological equivalence of flows. The theory for topological conjugacy of diffeomorphisms is less well developed, and seems to differ in some respects. Also, of course, any other equivalence relation on dynamical systems has an associated bifurcation theory. Very little work has been done so far on such bifurcations. It seems likely, though, that for any interesting relation it will be a difficult task to establish a satisfactory general theory.

APPENDIX A

Theory of Manifolds

In this appendix we assemble most of the definitions and theorems about manifolds and maps that are needed in the course of the book. Our main purpose is to establish notations and terminology, and to collect useful examples. On the whole, therefore, arguments are omitted or summarized when they are already available in easily accessible textst. We do not make substantial use of infinite dimensional manifolds in the book, apart from the trivial example of an open subset of an infinite dimensional Banach space. Nevertheless, we shall give the infinite dimensional theory here, since it is useful for further reading in the subject, and the generalization requires only a modicum of care.

I. TOPOLOGICAL MANIFOLDS Let E be a real Banach space, finite or infinite dimensional. A topological manifold modelled on E is a topological space X such that, for each x e X, there is an open neighbourhood U of x and a homeomorphism £: U-* U' onto an open subset U' of E. It is usual to place further restrictions on the topology of X in order to avoid pathological examples. We shall always assume that X is Hausdorff with a countable basis of open sets. Together with the manifold hypothesis, these conditions imply that X is normal, metrizable and paracompact. In addition we shall suppose, unless otherwise stated, thatX is connected. tFor example, Hirsch [1] is an excellent reference for differential topology, backed up by Lang [1] for infinite dimensional manifolds and Helgason [1] for Riemannian metrics. There are also very readable introductions to differential topology by Guillemin and Pollack [1] and Chillingworth [1]. Chillingworth's book is particularly relevant since it contains a very nice account of dynamical systems theory. 196

I

TOPOLOGICAL MANIFOLDS

197

The homeomorphism f is called a chart at x. A set of charts whose domains cover X is called an atlas for X. If E is n-dimensional (n < oo) then X is said to have dimension n, and to be a (topological) n-manifold. Since two finite dimensional real normed vector spaces are homeomorphic if and only if their dimensions are the same (see, for example, § 1 of Chapter 4 of Hu [2]), the dimension of X is uniquely defined (with the trivial exception of X = empty set) and we may take E = R". If E is infinite dimensional, we say that X is infinite dimensional. The term Banach manifold commonly carries the connotation of infinite dimension.

Examples (A.O) The topological space consisting of a single point is a O-dimensional manifold. (A.l) Any connected open subset of E is a manifold modelled on E. The inclusion is a chart, and the single chart is an atlas. Any connected open subset V of a manifold X modelled on E is a manifold modelled on E. If £: U-> U' is a chart on X, £\U n V: U n V^ £{U n V) is a chart on V. (A.2) Let L(E, F) be the Banach space of continuous linear maps from the Banach space E to the Banach space F. We write L(E) for L(E, E), and denote by GL(E) the set of (topological) linear automorphisms of E. (Here topological means that the automorphisms and their inverses are continuous.) Then GL(E), which is called the general linear group of E, is open in L(E) (Lemma 7.6.1 of Dunford and Schwartz [1]), and hence is a (not necessarily connected) manifold modelled on L(E). For example, GL(R") has two components (automorphisms with positive determinant and automorphisms with negative determinant) which are » 2 -manifolds. (A.3) If p: E - » X is a covering map then, by definition, every xeX has an open neighbourhood U such that p~l(U) is a disjoint union of subsets U\ each of which is mapped homeomorphically onto U by p. Thus, for any /, (p\U'i ) _ 1 is a chart at x, and so X is a manifold modelled on E. In many simple applications, X is the quotient E / G of a discrete group G acting on E, and p is the quotient map. For example, the circle S1 is the quotient R / Z where the action is given by n . x = x + n. Points of S 1 are equivalence classes [x] under the equivalence relation on R given by x~x' if and only if x-x'eZ, and p is defined by p(x) = [x]. We shall give an alternative definition of S shortly. Some examples of 2-manifolds covered by R 2 are (i) the torus T = R 2 / Z 2 where (m, n). (x, y) = (x + m,y+n), (ii) the Klein bottle R 7 Z 2 where (m, n). {x, y) = ((—l)"x +m, y +n) and (Hi) the Mobius band R 2 / Z where n . (x, y) = (x + n, (-l)"y). See Figure A.3.

198

THEORY OF MANIFOLDS

APP. A

Similarly, if p:X-> Y is any covering of a space Y by a manifold X modelled on E, then Y is a manifold modelled on E. (A.4) The connected sum X # Y of manifolds X and Y modelled on E is obtained by removing a ball from each and gluing the remnants together along their spherical boundaries. More precisely, let Br(0) = {x e E: \x\ =s r} and5 r (0) = { x e E : \x\ = r). Take charts f: U-> U' on X and T?: V ^ y ' o n Y

(i) Torus

•This self-intersection is caused by our attempt to represent the manifold as a subset of R 3

(ii) Klein bottle

(iii) Mobius band

FIGURE A.3

I

199

TOPOLOGICAL MANIFOLDS

withfl r (0)<=C/'n V". Let X- = X\r\intBr/2(0))

and

Y~ = Y\v-\mt

Br/2(0))

and put X # Y = X~u

Y~l(f\x)

= v~\x)

• * e S,/2(0)).

The definition is, up to homeomorphism, independent of choice of charts. For example, if X = Y = T2 then X # Y is the pretzel, or sphere with two handles (see Figure A.4).

FIGURE A.4

(A.5) If X and Y are manifolds modelled on E and F respectively, then X x Y is a manifold modelled on E x F. If £ and 17 are charts on X and Y respectively, then £, x 17 is a chart on X x Y For example, the torus T as above defined may be identified with 5 x S1 in the obvious way. Precisely, there is a homeomorphism taking [(x, y)] to ([JC], [y]). We say that a linear subspace F of E splits E if it is closed in E, and, for some closed linear subspace G of E, E = F © G . We call dim G the codimension of F; it may be 00. Splitting is guaranteed to occur if F is finite dimensional. Let X be a manifold modelled on E and let Y be a connected subset of X. We say that Y is a (locally flat) submanifold of X if there is a subspace F of E that splits E and, for each y e Y, a chart £: U -* U' at y such that £ maps C/ n y onto U' n F (see Figure A. 6). The codimension of Y is the codimension of F. Then obviously the submanifold Y is (with the induced topology) a manifold modelled on F.

FIGURE A.6

200

THEORY OF MANIFOLDS

APP. A

(A.7) Submanifolds of E often arise in the following way. Suppose that V is an open subset of E and that / : V -» H is a C map (r s= 1) to a Banach space H. Fix on some value a e H and consider the level set f~l(a). If, for all x ef'1(a), the differential Df(x) e L(E, H) is surjective (split surjective in the infinite dimensional case, which means that ker Df(x) splits E) we say that a is a regular value of /, and, in this case, it is a consequence of the inverse mapping theorem that f~1(a) is a disjoint union of submanifolds of V (see Exercise C.13 of Appendix C). For example, if E = R" +1 , H = R and f(xi,... ,x„+i) = xi +• • -+x2n+i, then 1 is a regular value a n d / _ 1 ( l ) is the sphere x\ + • • -+xl+i = 1, the unit n-sphere in R" +1 , denoted 5". If n = 1, we get the unit circle S 1 , which is homeomorphic to the manifold defined in A.3. (A.8) Real n-dimensional projective space R P " is defined to be the set of lines through the origin in R" + 1 . The topology of the space is given by a metric, the distance between two such lines being the distance between their closest points of intersection with S". There is a map p:S" -+RP" which takes any point to the line joining it to the origin. Since this map is a (double) covering, R P " is a manifold, by A.3. Exercises (A.9) Let X" be the unit sphere {x e R" +1 : \\x\\ = 1} in R" +1 with respect to any norm || ||. Show that 2 " is a manifold homeomorphic to 5". (A.10) Show that when M and N are submanifolds of X neither M uN nor M r*N need be a submanifold of X. ( A . l l ) Show that if X and Y are manifolds then, for all y e Y, X x {y} is a submanifold of X x Y.

II. SMOOTH MANIFOLDS AND MAPS If U is an open subset of a Banach space E, and F is another Banach space, one has the notion of a C r ( = /• times continuously differentiate) map f:U->F. We shall assume familiarity with the basic theory of such maps: some definitions appear in Appendix B below. We shall not be concerned much with C° (= analytic) maps. The term smooth is used to mean C for some r with 1 =£ r =£ oo. We wish to extend the theory of differentiable maps from the context of Banach spaces to the context of manifolds. That is to say, we have to extend the notion of differentiability of a map from the local, chart level to the

II

SMOOTH MANIFOLDS AND MAPS

201

whole manifold. We do this by restricting ourselves to charts which are smoothly related where they overlap. By doing so we get what is known as a smooth structure on the manifold. Let si = {£•: i e /} be an atlas for a topological manifold X, where / is some indexing set and £, :[/;-» £/| c E are charts. We say that si is a C atlas if, for all i, jel, the coordinate change map gjgi1 :£,(£/,n Uj)->£,{Uir\ Uf) is C (see Figure A.12). A chart t- on X is C-compatible with si if si u {£} is a C r

FIGURE A. 12

atlas. The set of all charts C-compatible with si is the C structure generated by si. A Cr manifold is a topological manifold X together with a C structure si on X. Any chart in si is said to be an admissible chart on the C manifold. In the text of the book, and from now on in the appendices, "manifold" always means " C manifold for some r & 1" and "chart" means "admissible chart." We often use the symbol Xfor a C-manifold rather than the cumbersome (X, si). All the above examples and constructions for topological manifolds may be modified to fit into this new setting. For example the inclusion of an open set U of E in E is, by itself, a C°° atlas for U, since we have no overlaps to worry about beyond the trivial one of U with itself. If h: X -> Y is a homeomorphism from a Cr manifold AT to a topological space Y then h induces a Cr structure on Y, with a chart $h~l:h{U)-^U' corresponding to each chart £:U -*U' on X. This also works at a local level, so that any connected open subset of a Cr manifold has an induced Cr structure, and so does any space with covering a C manifold. We may define C submanifold by making the charts in the above definition of submanifold admissible charts of a Cr manifold. The level surfaces of the C r m a p / : V^>H of Example A.7 turn out to be unions of Cr submanifolds at regular values.

202

THEORY OF MANIFOLDS

APP. A

Thus, for example, S", and hence RP", are C°° manifolds. (The C°° structure of Sn may be denned, equivalently, by the two charts £:Sn\{N}->Rn and i?: 5"\{5} -» R", where N and 5 are the north and south poles and £ and 17 are stereographic projection from those poles.) Products of Cr manifolds are C manifolds. The only construction that needs some care is the connected sum construction; it takes a little ingenuity to make X # Y smooth at the seam. Now let X and Y be C manifolds and let f:X-> Y be any continuous map. We say that / is C if, for all admissible charts £:U^U'^EonX and 77: V -* V <= F on Y, the local representative is Cr (see Figure A. 13). Obviously it is sufficient to check this for all charts in a pair of admissible atlases, one for X and one for Y. If / is Cr and has a C inverse f~l:Y-*X, we call it a C diffeomorphism (r3=l). Any C map / : X -» R, where R has its standard C°° structure, is called a Crfunction on X.

FIGURE A. 13

Examples (A.14) For any C manifold X the identity function id:X^>X is a C diffeomorphism, and any constant map c : X - » Y onto a point of a C manifold Y is a C r map. (A.15) Any C r map / : U -> V of open subsets of Banach spaces is a C r map when U and V are given their standard C°° manifold structures (induced by inclusion). (A.16) Let X=YxY, where Y is a C manifold, and let 5: y ^ X , T : X ^ X and 7r:AT^ F be defined by <5(y) = (y, y), r(y,z) = (z,y) and 7r(y, z) = y. Then 5 and TT are C maps, and T is a Cr diffeomorphism.

II

SMOOTH MANIFOLDS AND MAPS

203

(A.17) If Y has the C structure induced by a homeomorphism h:X^> Y from a C structure on X, then h is a Cr diffeomorphism. (A.18) Any admissible chart £:U-+U' of a C manifold X is a C diffeomorphism, where U and U' inherit their structures from X and E via the inclusions. Exercises (A.19) Prove that the composite gf:X-*Z g: Y^Z is a C map.

of Cr maps f:X-*Y

and

(A.20) For any positive integer n, let Xn be the real line R with the C°° structure given by the chart £: R -» R defined by £(x) = x 2 " ~x. Show that the m a p / n m : X „ -»X m defined by fnm(x) = xp, where p = (2n — I)/(2m - 1 ) , is a C°° diffeomorphism, but that the identity map id: R -» R is a C diffeomorphism if and only \im = n. Any Cs atlas on a topological manifold is trivially a C atlas for all r with r s£ 5, and so any C s structure may be extended to a C r structure, by adding all Cr charts C r -compatible with it. Conversely, but non-trivially, (A.21) Theorem. Any C structure (r^\)ona finite dimensional topological manifold contains a Cs structure for all s with r =s s ^ oo. Any two such Cs structures are Cs-diffeomorphic. For a proof, see Theorem 2.2.9 of Hirsch [1]. Two Cs structures sd and 38 on a topological manifold X are Cs-diffeomorphic if there is a map f:X-*X that is a C ! diffeomorphism from (X, si) to (X, 31). Theorem A.21 encourages us to restrict our attention to C°° manifolds, and we do this in the text of the book. Note, however, that there are topological manifolds which admit no differentiable structure at all (Theorem A.20 needed rs=l). Kervaire [1] and Smale [1] discovered compact 8-dimensional examples. As we have seen in Exercise A.20, it is not hard to find different smooth structures on the same topological manifold; to say there that id:R-*R is not a diffeomorphism is equivalent to saying that the structures are different. However, it is very much harder, and correspondingly more interesting, to find non-diffeomorphic smooth structures on the same manifold. Nevertheless they do exist. For example there are 28 C°° structures on the topological 7-sphere S 7 , no two of which are C 1 -diffeomorphic (see Milnor [1]). If / : X -*• Y is a Cs map of Cs manifolds, and we extend the Cs structures of the manifolds to Cr structures, for r < s, then, of course, we decrease the degree of smoothness of / to Cr. If, conversely, we have a C map / : X -» Y of Cr manifolds, and we restrict the C structures to Cs structures (as we

204

THEORY OF MANIFOLDS

APP. A

may by Theorem A.21) we do not usually increase the smoothness of / to Cs; it would be a fluke if we did so. The best we can say about C maps f:X-> Y of Cs manifolds is that (in finite dimensions) they may be approximated arbitrarily Cs-closely by Cs maps. See Theorem 2.6 of Hirsch [1]; for an exact statement, one needs to discuss the topology of map spaces, as in Appendix B and Hirsch [1].

III. SMOOTH VECTOR BUNDLES This section is designed to provide a simple framework for the description of the tangent bundle of a smooth manifold and other associated concepts. The definition follows a pattern that can be used to specify many important structures on a manifold. The essential idea is to consider special types of chart, and to restrict the coordinate change maps to some particular type. We have already done this in defining smooth structures on a manifold. Let E and F be Banach spaces. Let U be open in E and let m: U x F -» U be projection onto the first factor (i.e. TT\{X, y) = x). We call TT\ (or sometimes U x F) a local vector bundle. To describe the coordinate change maps that interest us, we need the notion of a Cr local vector bundle map. This is a C map / : C / x F - * V x H (for V open in G, where G and H are Banach spaces) which (i) covers a Cr map f:U -*V, in the sense that the diagram L/XF

> VxH

U

*V

commutes, and (ii) is a C'-smoothly varying linear map of fibres. That is to say, for all

(x,y)eUx¥, f(x,y) = (f(x),Tx(y)), where Tx e L(F, H) and the map x >-» Tx from U to L(F, H) is Cr. Note that in finite dimensions fC automatically implies that x >-» Tx is C. Now let X be a Cr manifold modelled on E, let B be a topological space and let n: B -» X be a continuous surjection. A Cr vector bundle chart (or C

SMOOTH VECTOR BUNDLES

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205

local trivialization) on u is, for some open subset U of X a homeomorphism £ from U = ir~1(U) to a local vector bundle U'xF that covers some admissible chart £: £/-» U' on X. That is to say the diagram U

i

»[/'xF

«

^

1/ *U' commutes. A C vector bundle atlas for TT is, for some indexing set /, a set S& = {£r. i e 1} of C vector bundle charts £•: £/,-= 7r _ 1 (t/,)^ C/J x F satisfying (i) {£/,•: j e 7} is a covering of X, and (ii) for all /, jel

the map

is a C r local vector bundle map. We now proceed as when denning smooth structures on a manifold and say that a Cr vector bundle chart £ on TT is C-compatible with s& if ^ u { ^ } is a C vector bundle atlas. A Cr vector bundle structure on TT is a maximal Cr vector bundle atlas on TT, and a C vector bundle {TT, si) is TT together with a given C vector bundle structure si. As usual we abbreviate (TT, si) by TT if the structure si is unambiguous, and we refer to the elements of si as {admissible vector bundle) charts on TT. Moreover a further common abuse of language which we sometimes follow is to refer to "the vector bundle B" if the map IT: B -» X is unambiguous. We call 77, B, X and F the projection, total space, base (space) and fibre of the vector bundle, and say that TT is overX. (A.22) Example. If X is any C manifold and F is any Banach space, then 7 7 i : X x F - » X i s a C r vector bundle with base X, fibre F and an atlas given by charts of the form $ x id : U x F-> U' x F where £ e an atlas of X. Such a bundle is called the product or trivial bundle. (A.23) Example. We may regard any Banach space F as a smooth vector bundle with base {0} and fibre F. (A.24) Example. The Mobius band M = R 2 / Z defined in Example A.3 (iii) as a topological 2-manifold is the total space of a non-trivial C°° vector bundle with base S 1 and fibre R. The projection TT:M-* S 1 takes [{x, y)] to [x], where [ ] denotes the equivalence class under the relations defining M = R 2 / Z and S1 = R / Z . A C°° vector bundle atlas giving the structure consists of the two vector bundle charts £: U -> U' x R and rj: V -*• V x R,

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where £/' = ] 0 , l [ , V" = ] i f [ , U = .n~\U) = [ C / ' X R ] , V' = [ V " X R ] , and £ and 77 are the maps [(s, *)]**(*> 0- The coordinate change map is 0:i(W) + rj(W), where W=UnV, 6(s,t) = (s,t) for s e ] i 1[ and d(s, t) = (1 + 5, - / ) for s e ]0, | [ . It is clear that the bundle is not trivial, since M is not orientablet. (A.25) Remark. In the case r = 0, the condition that the first factor V of a local vector bundle is an open subset of a Banach space is inappropriate and unnecessarily restrictive. The definition works perfectly well with V any topological space. With this modification, we may define a C° vector bundle over any topological space as base. Such bundles occur naturally in the theory of dynamical systems (for example the tangent bundle over an exotic basic set X of a dynamical system on a manifold). Any Cr vector bundle structure M for IT : B -» X is certainly a C manifold atlas on B, and thus determines a C manifold structure on B. (As usual we restrict attention to B Hausdorff with a countable basis of open sets.) We always regard B as furnished with this structure. With respect to it, TT is a C map of manifolds. For all £ : U, ^ U\ x F in d, let Z, denote the subset £jl{U'y. {0}) of B. Since local vector bundle maps are linear on the second factor, it is clear that Z, n U,- = Z, n £7, for all /, / e /. Thus the union of all Zt, i e I, forms a Cr submanifold, Z say, of B which is termed, rather loosely, the zero section of B (see Figure A.26). Clearly TT maps ZC' diffeomorphically onto

FIGURE

A.26

the base X. For each point x e X, the subset Bx = TT~^(X) is a C submanifold of B, Cr diffeomorphic to F, called the fibre over x. It inherits a linear structure from F. That is to say, we may perform addition and multiplication by scalars in Bx by mapping to {x}xF by some chart £ performing the corresponding operations in {JC}XF (identified with F) and then mapping t A finite dimensional manifold is orientable if it has an atlas all of whose coordinate change maps have differentials with positive determinant at all points (see Hirsch [1]).

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back by £~\ By definition of local vector bundle maps the net result does not depend on f. Thus B is indeed "a bundle of vector spaces", whence, of course, the name. Notice that although each fibre Bx also inherits via charts a topology from that of F, it does not generally inherit a norm from F, since a coordinate change map is not generally an isometry of fibres. (A.27) Example. Let ir:B^>X and p: C-»X be Cr vector bundles over a common base X, with fibres F and G respectively. As we have seen, for each x e X the fibres Bx and Cx inherit topologies, and so we may consider the topological vector space L(BX, Cx) of continuous linear maps from Bx to Cx. Let L(B, C) = {Jx£xL(Bx, Cx). Then L(B, C) is the total space of a C vector bundle, which we denote L{TT, p), with fibre L(BX, Cx) over x, defined as follows. Let f : f / - » [ / ' x F and 17: V-* V'xG be charts on 77 and p covering the same chart £:U->U' (i.e. £ = 17, U = V, U' = V). We have an associated chart L(U' x F, U' x G) = U' x L(F, G) defined, for all TeL(Bx, Cx) and for all v e Bx, by £(T)(g(v)) = v{T(v)). We maytake C atlases {£•: / e 7} for 77 and {17,: / e 7} for p such that, for each i e 7, £ and rji have the above property.Tthen the associated atlas {£,: i e 1} is a Cr atlas for L(ir, p). We call L(TT, p) the linear map bundle from 7r to p. (A.28) Example. {Products and sums of vector bundles) Let TT.B^X and p:C^> Y be C r vector bundles with fibres E and F respectively. Then the product irXp:BxC^XxY has a C vector bundle structure, with fibre E x F, defined as follows. Let / : U ^ U' x E and ij: V -> V x F be C vector bundle charts for IT and p respectively. Then i x rj: U x V> ^ (U' x E) x ( V x F) = (U' x V ) x (E x F) is a vector bundle chart on TT X p, and the set of all such product maps ^ x rj forms a C r vector bundle atlas for ir x p. If X = y, and one restricts 7r x p to the fibres over the diagonal {(x, x): x eX}, one has a bundle over X (identified with the diagonal by x = (x, x)) called the (Whitney) sum n ©p of the bundles ir and p. Its fibre is still, of course, E x F, which is canonically isomorphic to E © F . Let ir.B^X and p:C-> Ybe C vector bundles. A m a p / : J3-» C is a CT vector bundle map if, for all admissible charts £ ; C / - » [ / ' x E on TT and 17: V-> V x F on p, the induced map (or local representative)

vft1 • iif'H

V)nU)^V'xF

is a C local vector bundle map. Thus / i s a CT map of C manifolds which maps fibres of IT linearly onto fibres of p. Note that/maps the zero section Z of 77 into the zero section T of p, and thus induces (via the diffeomorphisms TT\Z:Z^>X and p\T: T-> Y) a C map f:X^> Y such that the diagram over the page commutes. We say that f covers, or is over, f.

208

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7T

X

APP. A

p

*Y

r

\if:B-*C isabijective C vector bundle map, a n d / - 1 : C ->B is also a Cr vector bundle map, then / is said to be a C vector bundle isomorphism, and 77is Cr vector bundle isomorphic to p. (A.29) Exercise. Prove that the composite of two C vector bundle maps is another C vector bundle map. (A.30) Exercise. Prove that there are, up to C vector bundle isomorphism, precisely two C vector bundles with base S and fibre R" (n 3= 1).

IV. THE TANGENT BUNDLE Suppose that we are given a C manifold X (r s= 1). We shall associate with each point x of X a vector space TXX which we may think of as the set of all possible velocities at x of a particle moving on X (see the section on vector fields in Chapter 3). This gives us a vector bundle, with base space X, which is called the tangent bundle of X. There is more than one way of constructing TXX. We do it via charts. Any chart £ at x takes the path of a moving particle on X to the path of a moving particle in the model space E. We could define the velocity of the first particle at x to be the velocity of the second particle at £(x), but this latter depends on the chart £ We get round this problem by the ingenious use of an equivalence relation. Let sd = {£,: U,•-* U\: i e / } be the Cr structure of X. Consider the subset A of X x E x / given by A = {(x, p, /): x e [/;} and define an equivalence relation ~ on A putting (x, p, i) ~ (y, q, j) if and only if x = y and q = Dx(&(x))(p), where x'-&(Ui^n [/,-)-*fy(f/,-n Ut) is the coordinate change map, and Dx'-£i(Ui n £/,)-»L(E) is its derivative (see Appendix B), which is C _ 1 . Let TX = A/~, and denote the ~ class of (x, p, i) by [x, p, /]. Then there is a map TTX'TX^X given by irx([x,p, i]) = x. Let U{ = TT~X (Ut), and consider, for all / e /, the set of maps &: £/, -> U\ x E given by ii{[x,p, i)] = (ii(x),p). It follows trivially from the definition of ~ that {£,:/ e / } is a C_1 vector bundle atlas for TTX- This determines a C'~ vector bundle structure for TTX with fibre E. With this structure, vx (or TX) is called the tangent bundle of X. The fibre (TX)X is called the tangent space t o X a t x . It is usually written as TXX, or X x (by abuse of notation). Its points are called tangent vectors {to X)

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at x. It is sometimes useful to think of the base space X of trx as identified with the zero section of trx, the point x with the zero vector 0X of TXX. (A.31) Example. If U is an open subset of E, with its usual C°° manifold structure, then TU is C°° vector bundle isomorphic to the trivial bundle UxE. An explicit isomorphism is given as follows. Let £,: U-*E be the inclusion, where i e I. Then any point v e TU can be written uniquely as [x, p, /] where x e U andp e E. Define/: 7T/-> UxEbyfiv) = {x, /?).Then/ is a C°° vector bundle isomorphism. It is very common to identify TU with U x E by this isomorphism. (A.32) Exercise. Construct a C°° vector bundle isomorphism from TS to the trivial bundle S ' x R . (A.33) Exercise. Let X and Y be Cr manifolds modelled on E and F respectively. By Example A.28, TTX X TTY is a C _ 1 vector bundle over XxY. Construct a C r _ 1 vector bundle isomorphism from nx x TTY to -nxxy, the tangent bundle of the product manifold XxY. (A.34) Example. {Parallelizable manifolds) A Cr manifold X modelled on a Banach space E is said to be parallelizable if there is a Cr~ vector bundle isomorphism from TTX to the trivial bundle XxE. Such an isomorphism is called a trivialization. Thus any open subset of E is parallelizable, and so is 5 . The product of two parallelizable manifolds is parallelizable, by Exercise A.33. As we commented in the introduction to the book in connection with the spherical pendulum, 5 is not parallelizable. The concept of derivative, or linear approximation map, is basic to differential calculus, and, when we work on smooth manifolds, it makes its appearance as the tangent map. Let X and Y be Cr manifolds modelled on E and F respectively, and let / : X-» Y be a Cr map (r > 1). Let v £ TXX and let £•: U -* U'i and 17,: V, -» V) be charts at x and/(x) respectively. Then we have the local representative : W'-* Vy, where W = £(t/, n / ^ V , ) ) and &(x) = rijf(x) for all x e £7* (W). If u = [JC, p, /], we define an element Tf(v) of TY by r/(u) = [f{x), q,;'], where 4 = £><£(£(x))(p). One must check that Tfiv) is independent of choice of charts £ and 17,. This is a routine exercise, and we leave it to the reader, together with the proof of the following result: (A.35) Proposition. The map Tf: TX -* TY is a C'~ bundle morphism. If g: Y^Z is another Cr map of manifolds, then Tigf) = iTg)iTf). For any X, Tiidx) = idTx. • In the language of category theory, we have constructed a covariant functor from the category of C manifolds and C maps to the category of C'1 vector bundles and C~x vector bundle maps. We call T the tangent functor. We denote by Txf the restriction iTf)x : TXX ^ Tf(x) Y of Tf to a single fibre. It is a continuous linear map of topological vector spaces.

210

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APP. A

(A.36) Example. If X and Y are open subspaces of Banach spaces E and F then we may identify TX and TY with I x E and V x E as explained in Example A.31. The derivative Df:X-*L(E,F) and the tangent map Tf: TX-* TY are related by Tf(x, p) = (f(x), Df(x)(p)). The double tangent map T2f=T(Tf):T(TX)^T(TY) is given by T2f((x,p), (u, v)) = 2 ((f(x),Df(x)(p)),(Df(x)(u),D f(x)(p,u) + Df(x)(v))), where T(TX) is identified with ( X x E ) x ( E x E ) . If U is an open subspace of a C manifold X and t: U -> X is the inclusion, then the C " 1 vector bundle map Ti: TU-» TX maps TU bijectively onto the open subset TT~X (U) of TX. One customarily identifies TU with its image under Ti. If £ : [/,--» C/J is any C r admissible chart on X then the map T$t: TUi^TU'i = U\ x E is a C r " ' admissible chart on TX. Modulo the identifications, it is precisely the chart £, in the definition of TX. (A.37) Example. If I = [a, b ] is a real interval and y : I -> X is a smooth mapt, then y is called a cwrue on X The tangent space 77 is identified with / x R, and for all t e 7,1, denotes the element (f, 1) of TI. The vector Ty(l r ) in the tangent space Ty(,)X is called the velocity of y at (time) f. We usually abbreviate 7y(l,) to y'(t). If X = E and we identify Ty{t)X with E, this usage fits in with the standard notation of differential calculus. (A.38) Example. (The canonical involution) If ir.B-* Y is a C vector bundle and if f:B -»C is a bijection onto a set C, then / induces on C a C vector bundle structure such that / is a C vector bundle isomorphism. If C has from the outset its own C manifold structure (and in particular if C = B) then we are usually interested in maps / for which the induced C structure is the original one. Trivially this is so if and only if / is a Cr diffeomorphism with respect to the original structure on C. An example of this occurs when B = C = T2X = T(TX), for any C manifold X (/-3=2). In forming T{TX) from X, we, in effect, twice take tangents to X; firstly in forming TX, secondly in taking tangents to TX (for we envisage X as embedded in TX as the zero section). Thus T2X has a built in symmetry (to use the term rather loosely), and there is a C involution ( = diffeomorphism of period 2) that exchanges the two "tangent spaces to X" at each point x. This map is called the canonical involution of T X. We now give it a precise description. We continue with the notations in the definition of TX. Let 39 = {|,: / e 1} be the atlas of TX corresponding to the atlas si = {£•: i e / } of X. Points of T(TX) are of the form [[x, p, i], (q, r),;'], where/ e /, (q, r) e E 2 and [x, p, i] is a point of TX in the domain of £•. The outer brackets denote equivalence class with respect to ~ in the construction of T2X. We may write any such point as [[x, p, i], (u, v), i] for some (w, v) e E 2 , since certainly [x, p, i] e (/,-. t That is to say, y can be extended to a smooth map of some open interval / with I ^ J.

V

IMMERSIONS, EMBEDDINGS AND SUBMERSIONS

211

The canonical involution / takes this point to [[x, u, i], (p, v), /]. Checking that / is well defined and a C'~2 diffeomorphism is a useful exercise. Now recall that the C'1 map TTX takes [x, p, i] to x. Thus we have two important maps from T2X to TX, namely -rrTx and T(rrx)- The first takes [[x, p, i], (u, v), i] to [x, p, i], and the second, as may easily be verified, takes it to [x, u, /]. Thus the diagram

commutes. V. IMMERSIONS, EMBEDDINGS AND SUBMERSIONS Let f:X-> Y be a Cr map of C' manifolds (r3s 1). For each xeX, the tangent map TJ: TXX^ TyY, where y =f(x), is continuous linear. Its kernel ker TJ and its image im TJ are linear subspaces of TXX and Ty Y respectively. Ker TJ is automatically a closed subspace; im Txf is not necessarily closed unless Y is finite dimensional. We say t h a t / is immersive at x if TJ is injective and im TJ splits Ty Y. Dually, / is submersive a t * if 7^/issurjective and ker TJ splits TXX. We say that / is an immersion if it is immersive at x for all x eX, and a submersion if it is submersive at x for all x e X. (A.39) Note. If im Txf (resp. ker Txf) has finite dimension or finite codimension in TyY (resp. TXX) then splitting is automatic. This is because, firstly, any finite dimensional subspace or closed finite codimensional subspace splits any Banach space and, secondly, any finite codimensional image of a continuous linear map is closed (see Lang [2]). In particular, one may omit the splitting condition from the above definitions when either X or Y is finite dimensional. (A.40) Example. The C°° map / : S1 -> S1 given by f{[x]) = [nx] for all [x]eS = R / Z , where n is any integer, is an immersion (and also a submersion) if and only if n # 0. (A.41) Example. For any Cr vector bundle TT: B -» X, TT is a C submersion. In particular, if X is a C manifold, TTX is a C~l submersion for r3=2. The image of an injective C immersion is sometimes called a C immersed submanifold. This is an abuse of language, since it need not be a topological submanifold; think of the numeral 6 regarded as the image of a C°° immersion of R in R 2 . If / : X -» Y is an injective immersion, we say that

212

THEORY OF MANIFOLDS

APP. A

im TJ is the tangent space to the immersed submanifold at f(x). An injective C immersion f\X-*Y whose image is a Cr submanifold of Y is called a C r embedding. Any C immersion is locally a C embedding, by Exercise C.12 of Appendix C. (A.42) Example. In irrational flow on the torus (see Examples 1.25 and 2.9) the map: R -» T taking t to ([x +t], [y + 6t]) is an injective immersion but not an embedding. (A.43) Example. The inclusion of a C submanifold of X in X is a C embedding. (A.44) Example. If X and Y are C manifolds, and we define i:X -*Xx Y by i{x) = {x, y) for some given y e Y, then / is an embedding. Similarly the diagonal map x*-*(x,x) from X to X x X is an embedding. (A.45) Exercise. Prove that/: X -> Y is a C embedding (r 3= 1) if and only if it is a C immersion and a topological embedding (i.e. maps Xhomeomorphically onto f(X)). (A.46) Example. (Foliations and laminations) Let X be a C r manifold modelled on E and let E = F x G be a splitting of E. A C foliation is a disjoint decomposition of X into C r injectively immersed submanifolds, called leaves, satisfying the following condition: There is an admissible atlas of charts of the form ij: U - » F x G , called foliation boxes, such that, for all y e G , ^ ' ( F x {y}) is contained in a single leaf, and is the image of an open set under the injective immersion giving the leaf (see Figure A.46). The

FIGURE

A.46

dimensions of F and G are called respectively the dimension and codimen sion of the foliation. It is also possible to give an equivalent purely local definition of foliation by considering a maximal atlas of C admissible charts for which all coordinate change maps \ have the property ^ 2 (*, y) = Xiix', y') if and only if y = y', for all (JC, y) and (x', y') in F x G . Rational and irrational flows both give 1-dimensional C°° foliations of the torus (the leaves being the orbits of the flow). As we have seen in Chapter 6,

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213

the stable manifold of a hyperbolic closed orbit of a C flow is C foliated by the stable manifolds of the individual points of the orbit. This situation is not typical of hyperbolic sets in general. Usually the stable manifolds of the individual points only Cr laminate the stable manifold of the set. To define this notion, we regard E as a trivial vector bundle with base G and fibre F, and weaken the definition of Cr foliation by relaxing the condition that the foliation boxes £ are C admissible charts and insisting only that they are homeomorphisms with Fr inverses (see Appendix B for Fr maps). Of course they must still form a topological atlas for X and satisfy the foliation box condition. If / : X -» Y is a Cr map of manifolds, and, for some y e Y, f is submersive at every point of j^r(y), thenf1(y) is a disjoint union of C submanifolds of X, with dimension dim X - dim Y if this makes sense. This result generalizes the remarks of Example A.6 and is, again, a consequence of the inverse mapping theorem. More generally still, let W be a C r -immersed submanifold of Y. We say that / is transverse to W, written /if> W, if for all yef(X)n\V, Ty(Y)= Wy + im TJ, where y =f(x) and Wy is the tangent space to W at y, and (Txfy\Wy) splits TXX. In this case, f~\W) is a Cr submanifold of X whose codimension equals the codimension of W in Y. If V and W are two Cr immersed submanifolds of Y, we say that V is transverse to W, written V i\\ W, if some injective C immersion / with image V is transverse to W. It follows that VnW is a Cr immersed submanifold. VI. SECTIONS OF VECTOR BUNDLES Let 7r: B -> X be a C vector bundle. A map CT.X^B such that ira: X -* X is the identity on B is called a section of TT. Thus cr is a section if and only if it maps every point * of A" into the fibre Bx over x. Figure A.47 illustrates this

FIGURE

A.47

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THEORY OF MANIFOLDS

APP. A

idea in the case when B is the Mobius band (Example A.24). When ir is the tangent bundle TTX- TX-+X, sections are called vector fields on X. In the case of the trivial bundle TTI'.XXY^X, the sections of m are precisely all maps (id, f):X^>XxF, and, of course, the section (id, f) is Cr if and only if the map / : X -» F is Cr. The map / is called the principal part of the section. For a general vector bundle v. B -» X we may trivialize the situation locally, using an admissible chart. Let £: [/-» [/' x F be such a chart, so that the diagram U

U

(

—>-U'xF

*U'

commutes. Then the section a of TT, when restricted to U, induces a section, r say, of the trivial bundle vi, defined by commutativity of the diagram U

—*U'xF

t r


U

*U'

In this case r is said to be a local representative of a. (A.48) Exercise. Prove that any Cr section of a vector bundle is a C embedding. (A.49) Example. (The spherical pendulum) In the introduction to the book we discussed the spherical pendulum, and found that its motion could be modelled by a C°° vector field v on the C°° manifold X = TS2 of dimension 4. If U is the complement in S 2 of a single meridian of longitude, then there is an admissible chart g:U->U' on S 2 given by f (y) = (0, <j>) where #_and 0 are the Euler angles. Correspondingly there are admissible charts £:U ^> U'xR2 for X and | : U^(U'xR2)xR4 for 7X, where U = TT~HU) and C/ = 77-x1 (U). In terms of these local coordinates, the vector field v: X -* TX has local representative (0,<M,AO-»,((0><M,/4),P)

where (0,
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(A.50) Example. (Second order equations and sprays) Notice that, for the vector field v of Example A.49, the first two coordinates of the principal part of v(x) are the same as the last two coordinates of X, namely (A, /x). This came about because we originally converted a system of second order equations on U' into a system of first order equations on U' x R 2 by the substitution 6' = A, ' = /u. Since, as we commented in Note 3.17, this is a standard procedure, we ought to analyse the situation a bit further. Let X= TM, where M is a Cr manifold (/"3=2). A first order ordinary differential equation on M is just a vector field on M. A second order ordinary differential equation on M is a vector field u o n A ' satisfying T(TTM)V = idxThis is precisely the coordinate free generalization of the condition in the last paragraph. If £:£/-» U' is a chart on M, with corresponding charts i: U -» U' x E on X and f: U -* (U' x E) x (E x E) on TX, and if / : U' x E -* E x E is the principal part of the local representative of v, then the condition says that the first coordinate of f(x, y) is y. Now let y.I^X be any integral curve of v, and let 8:1 ^M be the projection of y onto M by -rrM- Thus S = TTMJ- We call S a solution of the second order equation. Differentiating at tel,we deduce that S'(t) = TnM(y'(t))

= TnM(vy(t))

= y(t).

That is to say, the velocity of the curve 8 at / is the value of the curve y at /. See Figure A.50. There is no reason why 8 should not have self intersections.

FIGURE

A.50

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APP. A

Uniquenesss of the integral curve of v at a given point of X corresponds to uniqueness of the solution 8 of the second order equation through a given point of M with a given velocity at t = 0. Concentrating on a particular point meM for a moment, we have infinitely many solutions 8X starting at m at t = 0, one for each possible velocity x at m. In fact we have infinitely many starting off in a given direction with various speeds, and there is no reason why these should be in any way related (see Figure A. 51(f)). However, given x e TmM and a e R, there is a very natural way of obtaining from 8X a curve with velocity ax at m at time t = 0, and that is by speeding Sx up by a factor a. That is to say, the curve 8ax defined by Sax(t) = 8x(at) has the required property (see Figure A.51(ii)). Note that for a # 0 , 8X and 8ax have the same image. It is a particularly nice situation when the solution curves 8 fit together in this way.

do

(i)

FIGURE

A.51

When they do so, we call the second order differential equation a spray, and the images of the solution curves geodesies of the spray. Let us find the condition on v that makes it a spray. We require that if 8'ax(Q) = a8'x{0) = ax, then 8ax(t) = Sx(at). Differentiating this latter relation gives 8'ax(t) = a8'x{t). Since v is a second order equation, we may restate this as yax(t) = ayx{at). To find the condition on v, we differentiate again. This not completely straightforward, because the outside a is scalar multiplication on the vector bundle TM; it affects the fibre direction but not the "zero section direction" (if we could make sense of this at points of TM which are not on the zero section). However, if we denote by ixa: TM -» TM scalar multiplication by a on the fibres, then we can write yax(t) = iAayx(at) and obtain y'ax(t) = Tfia{ay'x(at)). Putting t = 0, we get, for all x e TM and aeR, (A.52)

v(ax) =

aTfia(v(x)).

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VI

217

Conversely, if v is a C 1 second order differential equation on M satisfying (A.52) then, by the uniqueness theorem (Theorem 3.34) its integral curves satisfy yax = ayx(at), and hence the solution curves satisfy 8ax(t) = Sx(at) as required. Let y b e a C 1 spray on M. By (A.52), 0 m is a zero of v, for all meM. Thus, by Theorem 3.22, there is some neighbourhood of 0 m in TM such that, for all x in it, integral curves of v at x are defined on [0, 1]. Thus, for some neighbourhood N of the zero section of TM, there is a map &xp:N->M defined by exp x = Sx(l). This is called the exponential map of the spray. It is C when v is C. For all ra e Af, the restriction exp m of exp to N n TmM maps rays through the origin onto geodesies at m, since exp m (tx) = S,x(l) = Sx{t). Moreover, since d/dt(expm(tx)) = S'x(t) = x at t = 0, the derivative of exp m at 0 m is the identity map on TmM. More precisely, if we identify T( TmM) with TmMxTmM in the standard way, then T expm:T(TmM)-*TM satisfies T e x p m (0 m , x) = x for all xe TmM. It follows from the inverse function theorem that exp m maps some neighbourhood of 0 m in TmM diffeomorphically onto a neighbourhood of m in M. (A.53) Example. If U is an open subset of R" then the standard spray

v:UxRn

= TU^T2U

= (UxRn)x(RnxRn)

is given by v(m,x) =

(m,x,x,0). The integral curve of v at (m, x) is f<-»(m +tx, x) and the solution curve through m with velocity (m, x) is 11-» m + tx. The exponential map is given by exp (m, x) = m +x. (A.54) Example. The circle 5 = R / Z inherits its standard spray from the standard spray of R by the covering map m->[m]. Thus v([m], x) = ([m], x, x, 0), where we write TS1 as S ' x R . The exponential map is exp ([m], x) = [m +x]. In terms of the standard embedding of S 1 in R 2 , exp maps each tangent space round the circle, taking the tangent vector x at p to the point p e2mx. For higher dimensional spheres 5", the geodesies are great circle arcs which may be derived from the above example by taking 2-plane sections through 0 e R re+1 . A zero of a section a:X ->B is a point x e X such that a{x) is the zero 0X of the vector space Bx. The zero section is the section defined by cr(x) = 0X for all x e X (but, as we commented earlier, the phrase is sometimes used loosely for the image of the map). A section without zeros is said to be nowhere zero. For many vector bundles, nowhere zero sections do not exist. For example, the reader can soon verify experimentally that the Mobius band has no nowhere zero sections. Neither does the tangent bundle TS2", by the Poincare-Hopf index theorem (Theorem 5.79). The set C'{TT) of all Cr sections of IT has algebraic structures generated by the linear structure of the fibres. It is a real vector space, with structure

218

THEORY OF MANIFOLDS

APP. A

defined pointwise by (aa + br)(x) = aa{x) + br(x) for a, re C{TT), a, b e R and xeX. Moreover it is a module over the ring 3P'{X) of C functions on X, when we put {f.ci + g. r)(x)=f(x).

a(x) + g(x).

r(x)

r

for /, g G & (X). We shall discuss the space C(tr) further in Appendix B, and give it a natural Banach space structure in certain circumstances. When 77 is the tangent bundle of X, we denote Cr(ir) by Tr(X).

VII. TENSOR BUNDLES The total space of a Ck vector bundle rr: B -» X is partitioned into a set of fibres Bx. Each fibre Bx has an associated dual topological vector space B* = L(BX, R). If we cobble these dual spaces together in a natural way determined by the structure of IT, we get a new C vector bundle, called the dual 77* of 7T. Similarly we can, for each fibre Bx, form the space Lsr(Bx, R) of all real (r + s)-linear functions on (Bx)r x (B*)s, which we call tensors of type {r, s) and we can make these into a Ck vector bundle which we denote by 77-*. In particular, n* is 7Ti. We now describe the structure of vsr. We first observe that any (topological) linear automorphism / : F H > G of Banach spaces determines a linear automorphism/':Lr(F, R ) ^ L , ( G , R) by the formula fr(T)(pu...

,pr,qu..

.,qs)=T(r\pi),-

• • ,T1(Pr), qif, • • -,qsf),

where (pi,..., pr, qu ..., qs) e G r x (G*)s. For any subset U of X, let Lsr(U, R) denote the disjoint union of Lsr{Bx, R) for all fibres Bx of -n with xeU. Here, as usual, U = TT~\U). Let Trsr:Lsr(B, R)-*X send L\{B„ R) to x. Any admissible chart | : U-» U' x Fon -rr has the form |(M) = (g(x),fx(u)) where TT(U) = x, and fx:Bx -» F is a linear automorphism. We define a map £ : £ , ; ( # , R H I / ' x i l f F , R) by £ CO = (f (*), (/ X )J(T)), where x = TTJ (T). We may topologize Lsr(B, R) by insisting that for all admissible charts f on IT the maps £,sr are homeomorphisms, and it is easy to check that the £sr then form a Ck vector bundle atlas for nsr. We call n*, with the Ck vector bundle structure determined by this atlas, the bundle of tensors on ir of type (r, s) (or covariant of order r, contravariant oforder s). A section of nsr is called a tensor field of type (r, s) on rr. (A.55) Example. {The derivative of a smooth function) The dual ITX of the tangent bundle of a smooth manifold is called the cotangent bundle olX. The

VIII

RIEMANNIAN MANIFOLDS

219

total space is denoted by T*X, the fibre T*X (or X*) is called the cotangent space at x and its elements are called cotangent vectors. Sections of w% are called 1-forms on X. If f:X-*R is a C r map (r^V), then T,/ maps T^A" linearly to 7}(;t)R = {/(*)}xR. Thus if we identify {/(*)}xR with R in the obvious way, TJ becomes a cotangent vector at x. The C'~l map from X to T*A" taking x to TJ is a 1-form on X, and we call it the derivative Df of /. If X is an open subset of E, and we identify the tangent space TXX = {x}xE with E, then Df is the derivative in the usual sense (see Example A.36).

VIII. RIEMANNIAN MANIFOLDS As we commented earlier, a vector bundle does not usually come equipped with an inner product, or even a norm, on each individual fibre. However these may be given to the bundle as extra structure, and they then give rise to extra theory. In the case of the tangent bundle of a manifold, they enable us to define lengths of curves on the manifold and also to give the manifold a natural metric space structure. A C Riemannian structure, or Riemannian metric, on a vector bundle TT.B -*X is a C section p:X^L2{B, R) of the tensor bundle 77-° such that the bilinear form p(x) on Bx is symmetric and positive definite for all x e X. One usually writes p(x)(p, q) as (p, q)x when there is no doubt as to which metric p is in use. A CT Riemannian metric gives rise to a C" Finsler, which is a norm I \x on the fibre Bx that depends on x in a C fashion. Of course, 1 |x is defined by \p\x = V
220

THEORY OF MANIFOLDS

APP. A

(A.57) Example. (Gradient vector fields) A C Riemannian metric on a manifold X enables us to convert C 1-forms on X to C vector fields on X and vice versa. The former operation is called sharpening, the latter flatten ing. By the representation theorem for Hilbert spaces, given any element A of T*X, there exists a unique element p of TXX such that A (q) = (p, q)x for all q e TXX. Conversely any p e TXX gives rise to an element A of T*X denned by the given formula. We thus obtain inverse C vector bundle isomorphisms p#:T*X^>TX and p t : TX^> T*X, and, correspondingly, isomorphisms p#: ClrX -> TrX and fa: TrX -> ilrX, where ClrX is the Banach space of Cr 1-forms on X, and P#(HO = P#W,

pv(v)

= p\,v.

r+1

In particular, if / : X -* R is a C function on X, then the derivative Df is a C 1-form on X (see Example A.55) and /?#(£>/) is a C vector field on X called the gradient V/ of /. For more detail, see Example 3.3 of the text. (A.58) Exercise. Let f:X^R be Cr+1. Prove that V/ is orthogonal to the contours of /, in the sense that, for all x e X and p e ker TJ, {p, V / 0 ) ) x = 0. Let X be a Cr+ Riemannian manifold, for sufficiently large r (r s= 4 covers all eventualities). The Riemannian structure on X gives rise to a C'1 spray o- on X, called the Riemannian spray, defined as follows. Let £: U -» V be an admissible chart on AT, so that 7£: TU -* TV = V x E is an admissible chart on TX. We transfer the Riemannian structure to V by the chart. That is to say, we define a function . K ' l V x E x E - ^ R b y (A.59)

K{y, p, q) = {TC\y,

p), T£~\y,

q))e-\y)

so that, for fixed y € V, K(y,.,.) is a symmetric positive definite bilinear function on E x E. We wish to define a local representative T ol a with respect to the chart T2£: T2U •* T2 V = (V x E) x (E x E) on T 2 X We do so by the formula r(y,p) =

(y,p,p,v),

where v eE satisfies, for all w e E , the formula (A.60)

K(y, u, v) = {\TK(y, p, p, u, 0, 0) - TK(y, p, u, p, 0, 0))2,

the subscript denoting the second coordinate in TR = R x R. This, by the representation theorem for Hilbert spaces, gives rise to a uniquely defined element v eE, and it is clear from the formula that the second order equation r satisfies the condition (A. 5 2) for a spray, using the bilinearity of K(y,.,.). Obviously the left-hand side of (A.60) is connected with the derivative of (, )x with respect to x, but the geometrical motivation for its detailed expression is less clear. A very natural, but rather subtle, explanation of it appears in Lang [1]. However, this requires more of the machinery

221

RIEMANNIAN MANIFOLDS

VIII

of differential forms than we are prepared to introduce here. We content ourselves by verifying that a is well defined, in that it does not depend on the choice of chart £ Let i':U'^ V be another chart on X, and let x- i(U n U') -* £'(U n £/') be the coordinate change map. The map K of (A.59) and the corresponding map K':V'xExE->R induced by g' are related on the overlap by (A.61)

K(y,p,q)

=

K'(y',p',q')

where y' = x(y)> p' = Dx(y)(p) and q' = Dx(y)(q). We have to show that T2x takes r(y, p) to (y', p', p', v') where v' satisfies (A.60)', which is (A.60) with K and all the variables dashed. Now, by Example A.36, T2x(y, p, p, v) = (y\ p', p', D2x(y)(p,

p) + Dx{y){v)).

Moreover, differentiating (A.61), TK'(y', p', q\ u', 0, 0) = TK(y, p, q, u, -DX{yT1D2Xiy){p,

«),

-Dx(yr1D2x(y)(q,«)), where u' = Dx(y)(u).

Thus

\TK'{y', p', p\ u\ 0, 0) - TK'(y', p', u', p', 0, 0) = kTK(y,p,p, -TK(y,p,

u, -Dx(yy1D2x(y)(p, 1

u), -Dx{yY1D2x{y)(p,

2

1

u,p, -Dx(yV D x(y)(p,p),

-Dx(yy D x(y){u,p)).

By (A.60) and the bilinearity of K(y,.,.), expression reduces to K(y, u, v)-12K(y,p,DX(y)-1D2X(y)(p,

u)) 2

the second coordinate of this

u)) -\K{y,Dx{y)-lD2X{y){p,u),p)

+K(y, p, Dx{yYlD2X{y)(u, =

p))+K(y,

DX{yYlD2X(y)(p,

p), u)

K(y,u,v+DX(y)-1D2X(y)(p,p))

= K'(y',u',v') as required. By the theory of sprays (Example A.50) we now have a C~l exponential map exp :N^X where N is some neighbourhood of the zero section in TX. We call this the Riemannian exponential map. The Riemannian metric on X gives rise to a metric on X in the sense of metric space theory. We denote this by d, and always call it the Riemannian distance function, to avoid confusion. We define d(x, x') for x,x'eX as follows. Since X is connected, we may join x to x' by piecewise C1 curve y:I~*X (that is to say, / is a real interval [a, b], y is a continuous map and is C 1 on each subinterval [an ar+i] of some subdivision a= a0
•

-
222

THEORY OF MANIFOLDS

APP. A

Since |r'(0| Y (0 n a s a t worst a finite number of jump discontinuities on [a, b], the length of y, \a \ y'(t) \yW dt, exists finite, and we define d(x, x') to be the infimum of the lengths of y as y ranges over all piecewise C 1 curves joining x to x'. Trivially d is symmetric and satisfies the triangle inequality, and d(x, x') = 0. To complete a proof that (X, d) is a metric space, we need to show that d(x, x) = 0 implies that x = x'. This is an easy consequence of our final theorem, Theorem A.62 below, which says that, near x, distances from x in X correspond under the exponential map exp x to distances from 0X in TXX, and shortest paths from x correspond to rays through 0X (and hence are geodesies through x). Moreover, since small open balls with centre 0X correspond to open balls with centre x with respect to d, and since exp x is, near 0X, a homeomorphism with respect to the original topology on X, we deduce that the metric topology of d is the original topology. Theorem A.62 has other important corollaries. One, which we used in the proof of the generalized stable manifold theorem (Theorem 6.21), is that if A is a compact subset of X then we may choose a number a > 0 such that, for all x e A, exp* maps the ball in TxX with centre 0X and radius a with respect to | \x onto the ball in X with centre x and radius a with respect to d. (A.62) Theorem. Let x eX, and let B be any open ball with centre 0X in TXX small enough for expx to map B diffeomorphically onto its image. Then, for all y eB, d{x,exp y) = l v U Moreover \y\x is the length of the curve y:[0,1]-»X defined by y(t) = exp ty. Proof. We first comment that any C 3 diffeomorphism h: X -» Y from X to a smooth manifold Y induces a Riemannian structure on Y, defined for all p,qeTyY by (p,q)y=(Th~\p),Th~1(q))h-i(y). Correspondingly, the Riemannian spray a: TX'-* T2X induces a C 1 Riemannian spray T: TY-» T2 Y, and these are related by the commutative diagram T2X cr

TX

*T2Y T

^—*TY

Each solution curve e of T at y e Y is of the form hS where S is a solution curve of a at h'1{y). The length of a curve y in X is equal to the length of the curve hy in Y, and thus the distance from x to x' in X equals the distance from h(x) to h(x') in Y. We apply this idea with X = exp* B,Y = B and h the inverse of the restriction exp x : B -» exp x B. To prove the theorem, we have to show that,

VIII

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223

with respect to the distance function of the new Riemannian metric induced on B by h, the distance from 0X to y is \y\x, for all y eB, and that this is the length of the straight line segment y: [0,1] -> B joining 0X to y (i.e. y(t) = ty). We denote the new Riemannian metric on B by ({ »y and the associated Finsler by || ||y. Since TXX is a Banach space, we identify TB with B x TXX in the usual way, and write ip, q))y or K(y, p, q) for {{y, p), (y, q)))y and ||p ||y for ||(y, p)||y. Thus K is a real function o n 8 x TXX x TXX. Perhaps we ought to emphasize that B already has a Riemannian metric induced from the inner product ( )x on TXX. We denote this by ( )x as well; thus ((y> p)> (y> a)*- The associated Finsler is denoted by | \x. Now TXX may be identified with E (for example, by the isomorphism Tx% for any chart £ at x) and so we may think of h as being a chart. (In the literature h is usually called a normal chart at x.) Thus the Riemannian spray r : f i x E - > ( B x E ) x ( E x E ) is defined by the formulae for the local representative of the Riemannian spray a on X (see (A.60)). We know that solution curves of T come under h from solution curves of a, and thus are curves of the form 5p{t) = tp. Differentiating each curve <5P gives an integral curve of T of the form yp(t) = (tp, p). Differentiating again, we deduce that, for all p G B and for all t e R such that tp e B, r(tp, p) = {tp, p, p, 0). Thus, by (A.60). (A.63)

(\TK{tp, p, p, u, 0, 0) - TK(tp, p, u, p, 0, 0)) 2 = 0

for all u e E , and in particular (TK(tp, p,p, p, 0, 0))2 = 0. This says that \\p\\z is constant for z on the line joining 0X andp, and hence that ||p||,p = ||plk = I P U It is now clear that the line segment y joining 0X to y has length \y\x with respect to || ||. Note that we have, in fact, shown that the curve t >-» exp ty in X has constant speed |y|x at every point. The above property of \\p\\x holds, more generally, for ((p,q))x, for all q € E. We assume this for the time being: (A.64) Lemma. For all p,qeE

and for all teR

K(tp,p,q)

with tp e B,

= (p,q)x,

and complete the proof of the theorem. Let 8: [a, b]^B be any piecewise C 1 curve joining 0X to y. We wish to show that 8 has length 5= |y \x. We may assume that 8(t) # 0* for t > a, for otherwise we may shorten 8. Then, for all but finitely many values of t, the Schwarz inequality gives

ll«'(0ll«(.)*l««(0,«'(0»«(ol/ll«(0ll«(o and by Lemma A.64 the right-hand side equals \{8(t), 8'(t))x\/\8(t)\x. consider the curve F :[a, b]^> B defined by e(t) = \8(t)\xy/\y\x.

Now This

224

THEORY OF MANIFOLDS

APP. A

parametrizes the line segment [0X, y], possibly covering it more than once in places, and so its length is at least | y \x. But the formula for its length is \l Ik'(OIL.) dt, which reduces to ]"a (\(8{t), S'(t))x\/\S(t)\x) dt. Thus the length of 8 is s= the length of e. • Proof oi Lemma A.64. We first show that, for all u e E with (p, u)x=0, K(tp, p, w) = 0. We may assume that p and u have unit length with respect to | \x, so that, for all s eR, tp + su has length Vf2 + s 2 . For all / > 0 , we differentiate the relation K(tp + su, (tp+su)/Jt2

+ s2, {tp + su)Ht2 + s2) = 1

with respect to 5 at 5 = 0, and obtain (TK(tp, p, p, u, u/t, u/t))2 = 0 or, equivalently, (TK(tp, p, p, u, 0, 0))2 +(2/t)K(tp, using the symmetry and bilinearity of K(y,.,

p, u) = 0,

.)• But, by (A.63),

(TK(tp, p, p, u, 0, 0))2 = 2(TK(tp, p, u, p, 0, 0)) 2 . Thus if the real function ip is denned by t//(t) = K(tp, p, u) then it satisfies the differential equation ti//'(t) = -tf/(t) for t > 0. Hence tip(t) = constant. Since tp is continuous and
p)x)

= K(tp, p, q) - ((p, q)J{p, p)x)K{tp, p, p) =

K(tp,p,q)-(p,q)x.

•

APPENDIX B

Map Spaces In proving theorems about dynamical systems in the text of the book we have often applied the contraction mapping theorem to map spaces. Typically, we have had a collection of contractions indexed by some parameter. We have deduced smoothness results for the original dynamical systems by showing that the fixed point of the contraction depends smoothly on the parameter. This technique requires familiarity with the differential calculus in map spaces, and we outline some of the theory below. The proofs are rather straightforward and uninteresting, the main problem being how to organize the results in the least indigestible way. We give one approach; the reader may prefer a more economical treatment that appears in Franks [1]. The basic theorem is to the effect that if g: Y ->Z is a C + s map then the map g*: C(X, Y) -> C(X, Z) defined by g*(/) = gf is Cs, where Cr(X, Y) is the space of C maps from X to Y. For more intricate applications we need results of the form that the map comp: C(X, Y) x Cr+S( Y, Z) -» Cr(X, Z) sending (/, g) to gf is Cs. Unfortunately, it does not seem to be practicable to develop the theory in a sufficiently general context to cover all applications simultaneously. One needs, in the proofs, a certain element of uniformity in some of the maps concerned, and this can be introduced in various ways. For example, one can make certain spaces compact, or others finite dimensional, or one can restrict one's attention to spaces of uniform maps. Trying to cope with all these tactics simultaneously would further complicate what is already a not particularly appealing piece of theory. We prefer to stick to one approach in our exposition and to indicate possible modifications, some of which are actually applied in the text. We first deal with maps between Banach spaces, and later come on to the more general theory of sections of vector bundles. In the final section we define topologies for spaces of dynamical systems on a compact manifold. As we have previously commented, these spaces can actually be given Banach manifold structures, but we do not do so here. Throughout E, F, G, etc. are real Banach spaces, with norm written | |. 225

226

MAP SPACES

APP. B

I. SPACES OF SMOOTH MAPS Let X be any set. The set Cb(X, F) of all bounded maps from X to F inherits a vector space structure from F. That is to say, we define / + g and af by (f+g)(x)=f(x) + g(x), (af)(x) = af(x) where f,geCb

(X, F), x e X and a e R . We define a norm | |0 on Cb (X, F) by |/|o = sup{|fljc)|:*eXl

and this makes Cb(X, F) into a Banach space. If AT is a topological space, the subspace C (X, F) of all bounded continuous maps is closed in Cb (X, F) and hence is itself a Banach space. The vector space L(E, F) of all continuous linear maps from E to F has a Banach space structure with norm | | defined by \T\ = sup{\T(x)\:\x\^l}

=

sup{\T(x)\/\x\:x*0}.

If X is an open subset of E, we say that / : X -» F is (Frechet) differentiable at x e X if, for some map T e'L(E, F) \f(x + h)-f(x)-T(h)\

= o(\h\)

as h -» 0. If T exists it is unique, and we call it Df(x), the differential of / at x. If Df(x) exists for all x&X, we say that / is differentiable. The map Df:X->L(E, F) is called the derivative of /. Higher derivatives are defined inductively by Drf = D(Dr~lf). We say that / is C° if / is continuous, C if D'f is continuous and C°° if / is Cr for all r s= 0. Strictly speaking DJ is a map from X to L(E, L ( E , . . . L(E, F ) . . . ) ) , but, as usual, we identify the latter space with the space Lr(E, F) of all /--linear maps from E r to F, putting S = T when S(xi)(x2) • • • (xr) - T(x\, x2,. •., xr) for all (xi, x2, • • •, xr) e Er. Thus L r (E, F) has norm | | given by |T| = sup {|r(jci,..., xr)\: \xt\ ss 1,1 ss i *£ r}. If / is Cr, Drf(x) is a symmetric /--linear map. Note that continuous multilinear maps are themselves C°°. If X is any subset of E with X_<= intX, we shall say that f:X->¥ is Cr (0 =£ /-=soo) if it has a C extension / t o some open neighbourhood of X in E. In this case the differentials D'f(x), 1 *£ / =£ r at points x of the frontier dX of X are independent of the choice of the extension /, and we define D'f(x) to be D'f(x) for such x. We make the standing assumption that whenever a map f:X-* Yis said or implied to be differentiable, its domain Xalways satisfies XcintX If X is a subset of R, one commonly writes Df(x) or f'(x) instead of Df(x)(l). The context decides whether Df(x) is an element of F or L(R, F).

II

227

COMPOSITION THEOREMS

This remark also applies to partial derivatives Djf(x)(=df/dXi) when E is a product of Banach spaces Ei x • • • x E„ and the rth factor E, is R. Recall that the /th partial derivative of / at a = (ai,..., an)eX is denned to be the differential of the partial map Xj i - » / ( f l i , . . . , fly-i, Xj, aj+u

...,

an)

at ah It is an element of L(E ; , F). The Cr map f:X-*¥ is C"-bounded if the number

\f\r =

WQ{\Dlf(x)\:xeX,^i*zr)

exists and is finite. (B.l) Exercise. Prove that if / : X -» F and g: F -» G are maps with Df and Dg C r -bounded then D(gf) is C r -bounded. Prove that if, in addition g is C°-bounded then gf is C r + 1 -bounded. The set C(X, F) of all C r -bounded maps from X to F has the vector space structure of C°(X, F), and we give it the norm | |r. In some cases, C(X, F) inherits completeness from F; we need to know this in the following cases: (B.2) Exercise. Prove that if X is open in E then C(X, F) is a Banach space. Prove that if / is a compact real interval then Cl{I, F) is a Banach space. (This is basically a well known theorem on uniform convergence; see, for example (8.6.4) of Dieudonne [1]). When Y is a subset of F, C(X, Y) is (identified with) the subset of C(X, F) consisting of maps taking values in Y. If Y is closed in F then C(X, Y) is closed in C(X, F). If X is compact and Y is open in F, then Cr(X, Y) is open in C(X, F). If X = Y, we abbreviate Cr(X, X) to C(X). If Z is a subset of X, we say that / : X -> F is uniformly C at Z (r 3= 0) if given e > 0 there exists 5 > 0 such that, for all x e X and for all z eZ with \x-z\<8, sup {|£>'/(x) - D'f(z )|: 0 =£ / =£ r} < e. We say that / is uniformly Cr if it is uniformly C at X. We denote by UCr(X, F) the closed subspace of C(X, F) consisting of all uniformly C maps. (B.3) Exercise. Let f:X-> Y and g: Y^G be uniformly C maps. Prove that if Df and Dg are C r l - b o u n d e d then the composite gf is uniformly C. II. COMPOSITION THEOREMS Most of the following theorems are proved by induction. When the integer inducted upon is the degree of smoothness r of the map space C ( , ) concerned, the inductive step always depends upon the trivial relation

228

MAP SPACES

APP. B

l/lfc+i = max {\f\k, \Df\k} for 0 =s k < r. Since these proofs run to a pattern, we tend to cut down on the detail. Throughout Y is a subspace of F, and X is a topological space if r = 0 and a subset of E if r > 0. The first two lemmas are special cases of a general result about left composition with a continuous multilinear map. (B.4) Lemma. / / T e L ( F , G ) then, for all fe Cr(X, F), \Tf\r^\T\\f\r. Thus T%(f) = Tf defines a continuous linear map T* : Cr(X, F) -» C(X, G). Proof. By induction on r. r = 0.

| Tf\0*z | r | | / | 0 , trivially.

Inductive

step. Assume the inequality holds for r = k. Let be the continuous linear map S^TS. Then D(Tf) = rDf, and so, by hypothesis, \D{Tf)\k^\r\\Df\k = \T\\Df\k. Combined with T:L(E,F)-»Z,(E,G)

|r/|fc=s|r||/|k,thisgives|7y|k+1^|r||/|fc+1.

•

(B.5) Lemma. IfB : F X G H » H / S continuous bilinear, then for all f e C(X, F) and g e C ' ( I , G ) the map £*(/, g) e C(AT, H) defined by £*(/, g)(x) = B(f(x), g(x)) satisfies \B*(f,g)\r^2r\B\\f\r\g\r. Thus B*: C(X, F) x Cr(X, G) -*• Cr(X, H) is continuous bilinear. Proof. By induction on r. Inductive step. This uses \D(B*(f, g)\k = |/3*(/, Dg) + y^(Df, g)\k where /3 : F x L(E, G) -> L(E, H) and y : L(E, F) x G -» L(E, H) are the continuous bilinear maps fi(y,T) = (x>-^B(y,T(x)) and y(S, z) = (x>-^B(S(x), z)). Notethat |iS| = | r | = | B | . D We are particularly concerned with the case when B is the composition m a p 5 : L ( F , G)xL(G, H)-*L(F, H) taking (5, T) to TS. Note that \B\ = 1. In this case we simplify notation by writing g. / for the so-called compositional product B*(/,g) of /eC r (AT,L(F, G)) and geCr(X, L(G,H)). Thus Lemma B.5 becomes (B.6)

\g.f\r^T\f\r\g\r.

The space L(R, G) is commonly identified with G, equating maps with their value at 1, and in this case B becomes the evaluation map (x, g)>-*g(x). (B.7) Lemma. For all r5?0, there is a constant A {independent of X, Y, G) such that, for all f:X^Y with Df C'1-bounded and for all g e C(Y, G), \gf\r^A\g\M(f), where Mrf = max {1, (|£>/| r -i) r }.

229

COMPOSITION THEOREMS

II

Proof. By induction on r. ' = 0. | g / | o « | g | o . Inductive step. \D(gf)\k = \(Dg)f. Df\k « 2k\(Dg)f\k\Df\k by (B.6).

•

r

If X is compact and g: Y -> G is C then g induces thete/fcomposition map g*: C(.Y, y ) -» C(X, G) defined by g*(f) = gf. This map may also exist in other circumstances, for example if g is C r -bounded (by Lemma B.7). (B.8) Lemma. Let X be compact and g: Y->G be Cr. Then g*: C(X, Y)-* Cr(X,G) is continuous. If g is uniformly continuous then g*:C (X, Y)-> C°(X, G) is uniformly continuous. Proof. By induction on r. r = 0. Let / 0 e C°{X, Y). Then g is uniformly continuous at the compact subset f0(X). Thus \g*(f) - g*(/o)|o is small for / C°-near f0. Moreover if g is uniformly continuous, |g*(/)-g*(/o)|o is uniformly small ( a s / 0 varies). Inductive step. For all /, f0 e Ck+\X,

Y), where 0 =£ k =£ r,

\D(g*(f) - g,(/o))| t = \(DgUf).

Df- (DgUfo)

• Dfolk

k

^2 (\(Dg)M)-(DgUf0)\k\Df\k + \(DgUf0)\k\Df-Df0\k).

O

(B.9) Exercise. Prove that if g is uniformly Cr and Dg is C r _ 1 -bounded then g*: C r (X, y ) -» C(AT, G) exists and is uniformly continuous at subsets & of C r (X, y ) such that sup {|Z>/|r_! :fe &}
If g is uniformly C , and r = 0, then g* is uniformly Cs. Proof. We first prove that if s 5= 1 g* is differentiable, by induction on r. We may assume that Y is open (otherwise g has a C r + 1 extension to an open neighbourhood Y' of Y, inducing an extension of g* to C(X, Y')).

230

MAP SPACES

APP. B

r = 0. Let/o e C°(X, Y). For all sufficiently C°-small TJ e C°(X, F) (i.e. such that, for all xeX, the line segment Lx = [fo(x), f0(x) + r\ (x)~\ is in Y), |g*(/o + 17)-g*(/o) - (£>g)/o • Vlo = sup{|g(/o(jc) + Tj(x))-g/b(x)-D g (/o(Jc))(T?U))|:jeeA'} «|T ? | 0 sup{|Dg(y)-Dg(/o(x))|: y 6 L , , x e X } by the mean value theorem (Corollary 2 in § 4 of Chapter 5 of Lang [2]). Since Dg is uniformly continuous atf0(X), the right-hand side is 0(|T?| 0 ) as

hlo^o. Inductive step. Let f0eCk+1(X, k+1 (X,V), VeC \D(g*(fo+v)-g*(fo)-(Dg)f0.V)\k

Y), for 0=sfc
= \Dg(f0 + r,). (Dfo + DV)-

(Dg)f0 . Df0

2

- (D g)f0 .r,. Df0- (Dg)f0 ^2k(\(DgUf0 2fc

.DV\k

+ v)-(DgUfo)-(Dig)f0.

r,| k |D/o + £)T,|t

2

+ 2 |r> g)/oU|T7|fc|DT,|,). The right-hand side is 0(|T?|fc+i) as |rj|fc+i-»0. The remaining results follow by induction on s. The case s = 0 is Lemma B.8. If the theorem holds for s = k and g is cr+k+1 then Dg* is clearly Ck since it is (Dg)* followed by the continuous linear map A from Cr(X, L(F, G)) to L{C'{X, F), Cr(X, G)) that takes (to (17 -* £. 77). Similarly for the uniformity result. D ( B . l l ) Corollary. Given r s= 0 and ss?l, there exists a constant A {independent ofX, Y and G) such that if X and Yare as above, f e C(X, Y) and g is cr+s with Dg C ' + , _ 1 -bounded, then \D'gM^A\Dg\r+.-lMr(f), where, again, Mr(f) = max {1, (|£)/| r _i) r }. Proof. By induction on s. s = l.

sup{\(Dg)f.r1\r/v\r:v^OBCr(X,F)}

\Dg*(f)\ =

n2rB\Dg\rMr(f) Inductive step.

\D

k+1

by Lemma B.7. k

g*(f)\ = \D (\(Dg)*)(f)\

(A as in Theorem B. 10)

^\\\\Dk(DgUf)\ ^2k\Dk(DgUf)\.

a

231

COMPOSITION THEOREMS

II

(B.12) Corollary. Let X and Y be as above. For all Cs maps g with Dg C*'1-bound the map g* : C°(X, Y) -* C°(X, G) is Cs-bounded. Moreover the map from C\Y, G) to CS(C°(X, Y), C°(X, G)) taking g to g* is continuous linear. D The above theory is sufficient for most applications. However, we now move on to smoothness of the map comp: Cr(X, Y)x C+S(Y, G)-» C(X, G) denned by comp(/, g) = gf, and we approach this via its partial derivatives. We have already dealt with differentiability of the partial map g*. We also have to consider right composition maps of the form /* : Cr+S(Y, G)-» Cr(X, G), where / e C(X, Y) and f*(g) = gf. In fact, such maps are denned for all C maps / such that Df is Cr~ -bounded, by Lemma B.7. Since they are trivially linear, Lemma B.7 also gives: (B.13) Lemma. The maps f* are continuous linear.

•

However, this remark is not sufficient for our purposes. We now know that Df*: Cr+S(Y, G)^L(Cr+s(Y, G), C(X, G)) is the constant map with value (£>-»£/). We need to know how smoothly this value depends on /. The necessary result is: (B.14) Lemma. Let X be compact and Y be open or a closed ball. Then for sssl, the map d:Cr(X, Y)^L(Cr+s(Y,G), Cr(X,G)) defined by 8(f) =

U~Zf)isC'-\ Proof. By induction on s. We may assume that Y is open in F, and so Cr(X, Y) is open. s = l. Let /oe C{X, Y). For all sufficiently C r -small

v

e C(X, F),

\O(fo + v)-0(fo)\ = \£»(£(fo + v)-tfo)\ = sup{\Ufo

+

v)-Ufo)U\(\r+i:(*OeCr+\Y,G)}

« s u p { | U ^ ( / o + hj)lhlr/Klr + i:0«r=s l , f * 0 } we have by the mean value theorem applied to £%. By Corollary B . l l , \D^(fo + t7})\^A\C\r+iMr(fo + tri), and since Mr(f0 + tr}) is bounded for C r -smallij, 6(fo + v)^0(fo)as TJ-»0. Inductive step. We assume that the theorem holds for 5 = k s* 1 and prove it for s = k + l. Let f0eC(X, Y) and veC(X,F). We assert that 6 is differentiable with derivative given by D0(fo)(r}) = (£-+(D£)f0 .17). This is because, for sufficiently C r -small 77,

\Hfo +

v)-0(fo)-{£~(D£)fo.v)\

= \£~(Mfo

+

r,)-Ufo)-DMfo)(v))\

232

MAP SPACES

= mp{\U(fo

+

^s\ip{\D(t(fo

v)-U(fo)-DU(fo)(v)l/U\r+k+i:£*0}

+

2

^sup{\D U(fo

APP. B

tr,)-DU(fo)Hr/\C\r+k+i:0^t^l,^0} utr,)\t(\v\r)2/\{\r+k+1:0<st^l,0^u^l,(^0}

+

by two applications of the mean value theorem, and this is o( |TJ | r ) as | TJ | r -> 0, using Corollary B . l l again. Thus D6 is the composite of d:Cr(X, Y)-> L((Cr+k(Y, L(¥, G)), C(X, L(F, G)), which is Ck~l by hypothesis, and the continuous linear map from the target of 6 to L(Cr(X, F), L(Cr+k+1(Y,

G), C(X,

G)))

defined by <MT) = (T,-»(£-» r ( D f ) . i j ) ) .

D

We can now prove: (B.15) Theorem. Let X be compact and Y be open or a closed ball. Then comp: Cr(X, Y) x C,+S(Y, G)^Cr(X, G) is Cs. Proof. By induction on s. 5 = 0. This uses the inequality |comp (/, g) - comp (/„, g0)\r «\(g - g0)f\r + \gof~gofo\r «£A|g-go| P M,(/) + |go*(/)-go*(/o)l and Lemma B.8. Induction step. We know that for s = k +1 > 1 comp has partial derivatives given by Dx comp(/, g) = (-n~(Dg)f.

r

v)eL(C r+s

£>? comp (f, g) = ( f - > # ) e L(C

(Y,

(X,F),

C'{X, G)) r

G), C (X, G)).

k

Thus D2 Comp is C by Lemma B.14. We break down L>i Comp as a composite of three maps (/, g) ~ ( / , £>g) ~(£>g)/ ~ ( T ? ^(£>g)/- T,). The first and third are continuous comp:Cr{X,Y)xCr+k(Y,L(¥,G)^Cr(X,L(F,G)), inductive hypothesis.

linear,

and the second is which is Ck by D

By putting X = a singleton {x} and identifying C°(X, F) with Y(f /(*)) we have:

with

(B.16) Corollary. / / Y is open or a closed ball, then the evaluation map ev: YxCs(Y, G ) H > G defined by ev(y, g) = g(y) is Cs.

II

COMPOSITION THEOREMS

233

We also need the following result, obtained similarly from Lemma B.14. (B.17) Corollary. / / Y is open or a closed ball, the map ev' from Y to L(CS(Y,G)), G) defined by eV(y) = evy = (g-»g(y)) isCs~\ • This is as far as we wish to take the theory with X compact. Let us just recall how that hypothesis was used. Firstly, we needed it in the case s s= 1, together with Y open or a closed ball, in order to ensure Cr(X, y)<= int C'{X, Y). Secondly, we used it in Lemma B.8 and Theorem B.10 to infer that the continuous maps g and Dg were uniformly continuous at f0(X). How can we dispense with compactness of XI If s s= 1, we can put Y = F so that C(X, Y) becomes the whole space C(X, F). If F is finite dimensional, we can enclose fo(X) in a compact set K (since f0 is C°-bounded) and the proof proceeds as before. Actually, if r > 0, we also need E finite dimensional, so that L(E, F) is finite dimensional, or the induction will not work. We leave it to the reader to reformulate the main theorems in this finite dimensional case. Another possible course of action is to build in the uniformity by restricting attention to maps g that, together with their derivatives, are uniformly continuous. We have used theorems of this sort in the text, so we state them below, leaving proofs to the reader. We are able to consider a slightly more general situation than before. Let f0:X^>¥ be a fixed C map with C r _ 1 -bounded derivative Df0. (B.18) Theorem. Let g e UCr+s(F, G). The map from C'(X, F) to C\X, G) taking f to g(f + f0) is Cs {and uniformly continuous if s = 0). For s>0 its differential at f is (17 *-^>Dg(f + f0) • rj). • (B.19) Theorem. The map from Cr(X, F) x UCr+s(F, G) to C\X, G) taking (f,g)tog(f+f0)isCs. • In the text, in the course of proving the stable manifold theorem, we made use of spaces of sequences. These may be interpreted as map spaces, and so the above theory may be invoked. Perhaps we should be more explicit. Let N = {0,1,2, 3,...} and let YN be the set of all sequences in Y. As before Y is a subset of the Banach space F, so Y N is a subset of the vector space F N . We denote by 38(F) the Banach space C°(N, F) of bounded sequences in F, and by 5^(F) the closedsubspace consistingof convergent sequences. Notice that £P(F) may be identified with C°(N, F), where N is the one point compactification of N. Let 38 (Y) = C°(N, Y) and $f(Y) = C°(N, Y). Any bounded map (or any Lipschitz map) g : y - » G induces a map g%: S3 (Y) -> Sfc (G) taking y to gy. Similarly, any continuous map g: Y'-»G induces g^-.J/'iY)->5^(G). We obtain results about these spaces by putting r = 0, X = N or, when possible, N in the above composition theorems. In the text, it was convenient to use Sf0(G), which is the closed subspace of 5^(G)

234

MAP SPACES

APP. B

consisting of series converging to 0 in G. The relevant theorems are: (B.20) Theorem. / / Y is open or a closed ball and if g: Y -» G is C\ with g(0) = 0, then g*: Sr0{ Y) -> Sf0(G) is Cs. If, further, g is uniformly Cs then so is g*D (B.21) Theorem. If Y is open or a closed ball then the map comp from Sf0(Y) xCs0(Y, G) to Sf0(G) is C5, where comp (/, g) = gfand CS0{Y, G) is the subspace of CS(Y, G) consisting of maps g with g(0) = 0. • Finally in Exercise 6.11 of the text, and its applications to hyperbolic closed orbits, we need the following generalization of the previous pair of theorems: (B.22) Exercise. Let a be a positive real number, and let a : F N -» F N be the isomorphism taking y to (n*-*any(n)). Let J | denote the norm induced by a on a(39(F)) from that of 33(F) (so a\S\ =sup {a~"\S(n)\: n e N}). Suppose that Y is a ball with centre 0 in F and that g: Y -» G is a Lipschitz map with g(0) = 0. Prove that g induces a Lipschitz map g * : a ( ^ 0 ( y ) ) - » « ( ^ 0 ( G ) ) (defined by gHs(y) = gy). Now suppose that a =£ 1. Show that g* is C when g is C s (5 & 1) and uniformly Cs when g is uniformly C s . Prove also that the map c o m p : a ( 5 ^ 0 ( y ) ) x C s ( y , G)-^a(^ 0 (G)) is Cs. Investigate the situation when a is greater than 1.

III. SPACES OF SECTIONS Let ir:B^>X be a C vector bundle with fibre F. We suppose throughout for simplicity thatXis compact. Of course for r > 0 X is a Cr manifold. Recall (Appendix A) that C(ir) (or, sometimes, C(B)) denotes the vector space of all C sections of ir. We wish to give C(TT) a norm. We say that an admissible chart £: U -» U' x F on TT is a norm cnart if it is the restriction of an admissible chart 77: V -» V x F where U'^Kc V' for some compact K A norm af/as is a finite atlas of norm charts. Let si = {£;: 1 =e i« n} be such an atlas for -77, and let
Ill

SPACES OF SECTIONS

235

Moreover if SM is another norm atlas for v then | U>r and | \^r are equivalent norms. We suppose from now on that some atlas si has been chosen, and we abbreviate | |^>r to | |r. (B.23) Exercise. Show that norm atlases exist, and fill in the details of the previous paragraph. Notice that there is a canonical norm-preserving isomorphism between Cr(U'i,F) and Cr(in), where ir,:C/J x F - > t / J is the trivial bundle. This points to the way in which spaces of sections generalize the map spaces of the preceding sections. The generalization is a substantial one for, whereas above we composed elements of the map space with Cr+S maps of F, we now compose with maps of the total space of the vector bundle satisfying a weaker differentiability condition. Clearly we have to introduce some new condition, because it does not usually make sense to talk of a C+s map of the total space of a C vector bundle. The condition that we introduce occurs naturally because of the fundamental distinction between base and fibre coordinates in vector bundles. Let TT:B-*X and p:C^>X be C vector bundles, and let g : B ^ C b e a fibre preserving map (over the identity). Thus g maps each fibre Bx to Cx, not necessarily linearly. Recall (Example A.27 of Appendix A) that the linear map bundle L(ir, p) is a vector bundle with base space X and fibre L(BX, Cx) over x. Its total space is denoted by L(B, C). We define inductively Ln(v,p) = L(-n;(Ln~1(ir,p)). Suppose that, for all xeX, the restriction gx:Bx->Cx of g is n times differentiable. We say that g is n times fibre differentiable and define its nth fibre derivative F"g:B ->Ln(B, C) by Fng(v) = D"gx(v) where v € Bx. We say that g is of class rFs if F"g exists and is C for 0 =s n =£ s. We write Fs for Fs. Thus rFs is a stronger condition than C but weaker than C+s. In terms of local coordinates, rFs means that all partial derivatives up to order 5 in the fibre direction exist and are Cr (as functions of all coordinates together, both fibre and base). We may also use this local criterion to define what we mean by a map g: B -» Y being rFs, where B is the total space of a vector bundle and Y is any smooth manifold. In the text we used the following results, which may be given proofs closely resembling those of analogous theorems in the last section (see Eliasson [1] and Foster [1] for details). (B.24) Theorem. Let TT.B^Xand p:C^Xbe Cr vector bundles and let r s g.B^Cbe F . Then g*: C'(ir)•* C'(p) defined by g*(o-) = gcr is Cs. For

236

MAP SPACES

APP. B

s s= 1 its derivative Dg% is defined by (B.25)

DgM)(r)

= (Fg)cr.r

r

for all a and T in C (ir), where, as in the last section,. denotes compositional product. • (B.26) Remarks. The above theorem may be modified in various ways. For example, the domain of g may be an open neighbourhood N of the image of some given section of IT, rather than the whole of B. Of course g* is then only defined on sections taking values in N. Again, we may (and in fact, in the proof of the generalized stable manifold theorem, do) replace C°(TT) by Cb(n), the space of bounded but not necessarily continuous sections. We may in this case prove that gFs implies g* C \ with Dg* as in (B.25), provided that the fibres of tr and p are finite dimensional (this condition being required for uniformity arguments in the proof). Finally, if one gives the space of rFs maps from B t o C a natural Banach space structure, one may prove an analogue of Theorem B.15 above (see Theorem 15 of Foster [1]). Theorem B.24 has a partial converse. This is very useful when, as in the proof of the generalized stable manifold theorem, one attempts to establish results about a subset of a manifold by applying differential calculus to the space of sections of the tangent bundle of the manifold over the subset. In this situation, one needs a tool for transferring results from the space of sections back down to the manifold again, and the following theorem (due to Foster [2]) serves this purpose: (B.27) Theorem. Let TT.B^Xand p.C^Xbe C° vector bundles and let g-.B^C be a fibre preserving map. If g% : C°(TT) -> C°(p) is Cs then g is Fs. Before proving this theorem, we establish two preliminary results. The first tells us that small open sets in the total space of a vector bundle may be embedded nicely in the space of sections. (B.28) Lemma. Let v.B^Xbe a C° vector bundle, and let x0eX. There exists an open neighbourhood U of x0 in X and a continuous map o-. U-*C°(TT) (where U = 7r _1 (£/)) with value atp denotedap such that (i) for all xeU, a. \BX is continuous linear, (ii) for all peU, o-p takes the value p at ir(p), (iii) for all p e U, \ap | 0 = \p | where | | is the norm induced on the fibres of U by some chart of si. Proof. Let £: W -> W x F be a norm chart in si with x0 e W. Choose open neighbourhoods U and V of x0 in X with U^V and Vc W. Let A: W^> [0, 1] be a continuous function which takes the value 1 on U and 0 on W\ V

237

SPACES OF DYNAMICAL SYSTEMS

IV

(A exists by Urysohn's theorem). For all peU,

define ap e C (ir) by

(0,

if

lC\£(x),Hx)l-(p)2)

if

xeX\V xeV.

Then a. = (p>-*crp) has the required properties.

•

(B.29) Lemma. The formula tp{y){a) = y.cr defines a continuous linear map ip: C°{L{TT, p))-»L(C°(7r), C°{p)) which is injective and has a closed image. Proof. We take local representatives with respect to norm charts for TT and p and the associated chart for L{rr, p) (see Example A.27). In terms of these charts, ip sends {id, 8) to {{id,r)^{id,S.T))t where SeC°{U',L{E,F)), T e C°{U', E) and E, F are the fibres of ir, p. It is now clear that ip is well defined and continuous linear. For injectivity we need to show that if y.o- = 0eC°{p) for all aeC°{ir) then y = 0 e C°{L{ir, p)). This follows providing that for any p e B there is a C° section a e C°{ir) taking the value p at ir{p), and this is so by Lemma B.28. Similarly, closure of Im ip amounts to showing that if sup{|"y m .o--"y n .<x|o:|o"|o s £ 1} is arbitrarily small for sufficiently large m, n then (y„) is a Cauchy sequence, and this again uses Lemma B.28. • Proof of Theorem B.27. We prove the theorem by induction on s. Let a. : U^> C°{ir) be as in Lemma B.28. We write g on U as the composite *

U

(IT, a-.)

r,

*• U x C°(TT)

idxg.

n

1* U x C°{p)

ev

-» C.

Thus continuity of g* implies continuity of g (the evaluation map ev is continuous, essentially by Corollary B. 16). If we restrict g to a single fibre Bx of U we have the composite evx g*o-., so, since ev* and a. are continuous linear, differentiability of g* implies the existence of Fg. From the definition of derivatives, we deduce that Fg satisfies the relation Dg% = tp{Fg)*, where \p is the map of Lemma B.29. Now assume that the theorem holds for s = ?3=0 and that g* is C' + 1 . Then Dg% is C, and hence, by the above relation and Lemma B.29, {Fg)* is C'. Thus by the inductive hypothesis Fg is F'. We deduce that g is F'+1. •

IV. SPACES OF DYNAMICAL SYSTEMS Let X be a compact differentiable manifold. We wish to describe a topology for the set of all Cr dynamical systems on X which takes into account all derivatives of the system up to order r{r s= 1). For vector fields on

238

MAP SPACES

APP. B

X we already have such a topology. A C vector field on X is just a C section of the tangent bundle TTX of X, and we have seen in the last section how to give the space C(irx) a Cr norm | |„ As we commented above, the norms corresponding to any two norm atlases are equivalent, so we may sensibly talk of the C topology on the space. We usually write V(X) for

CVx). The situation for the set Diff' (X) of all C diffeomorphisms of X is rather more complicated. The easiest way to link it with what we have done so far is to note that if X has a Riemannian metric then right composition with the exponential map exp: TX-*X takes QeTr(X) to the identity map idx€ DhT (X), and induces a bijection from a small enough neighbourhood U of 0 in Tr(X) to a subset V of Diff (X) containing idx. We define a basic system of neighbourhoods of idx in Diffr (X) to be the subset V as U ranges over all discs in T r (X) with centre 0 and with radius (small) e > 0 . We may now obtain a basic system of neighbourhoods at any / e Diff' (X) by composition (either right or left) with /. An equivalent and more straightforward way to define a basic system of neighbourhoods at / is as follows: Take any pair of finite atlases si = {&• : Ui -> U't} and $ = {77,: V, -» V-} with the property that, for all i, /(Ud c Vm for some /(/), and define We (/) to be the set of all g e Diff (X) such that, for all i,

and

sup I vmgtT1 - vmfti11 < ?. i

Then {WE(f): e > 0 } is a basic system of neighbourhoods of / in the C topology. Yet another description of the C topology is in terms of jet bundles (see, for example, § 2.1 of Hirsch [1]). We may easily define topologies for r°°(X) and Diff°° {X) by taking as a basis the Cr topologies for all finite r. For example U is open in Diff0 (X) if and only if for all fe U there is an open subset V of Diff (X), for some r,

with fe

V^U.

One may now consider the case when X is finite dimensional but non-compact. There is more than one candidate for the C topology. See § 2.1 of Hirsch [1] for a good discussion of the weak and strong {Whitney) topologies. One may also extend these ideas to X with boundary, and even to infinite dimensional X.

APPENDIX C

The Contraction Mapping Theorem

Let X and Y be non-empty metric spaces, with distance function denoted by d. Let K be any positive number. A map f:X^>Y is Lipschitz {with constant K) if, for all x and x' eX, d(f(x),f(x'))^Kd(x,x'). The chords of the graph of / have slope ««• (see Figure C.l). Clearly any

^-^^

graph f

-^JS-""

d{xtxf)

FIGURE C.l

Lipschitz map is continuous (in fact, uniformly continuous). An invertible Lipschitz map with Lipschitz inverse is sometimes called a Lipeomorphism. A map / is locally Lipschitz if every x e X has a neighbourhood on which / is Lipschitz. (C.2) Proposition. Let X and Ybe subsets ofBanach spaces and letf: X -» Y be a map. If f is C with |.D/(JC)| bounded by K and if X is convex, then f is Lipschitz with constant K. In particular, any C1 map is locally Lipschitz. 239

240

THE CONTRACTION MAPPING THEOREM

APP. C

Conversely, if f is Lipschitz with constant K and f is differentiable at x then \Df{x)\^K. Proof. These are immediate consequences of the mean value theorem (see § 4 of Chapter 5 of Lang [2]) and the definition of differentiability. D (C.3) (i) (ii) (iii) (iv)

Exercise. Which of the following maps are Lipschitz? / : R -> R defined by f(x) = sin2 x, / : R ^ R defined by f(x) = xu\ / : R 2 -> R defined by f(x, y) = x2 + y2, / : E-» R defined by f(x) = \x\, for any norm | | on a vector space E.

We say that / is a (metric) contraction if it is Lipschitz with constant K < 1. If / is invertible and fl is a contraction we call / an expansion. When X nY isnon-empty,afixedpointoffisanyx eX n Y such that f{x) = x. One of the simplest and yet most widely used of all fixed point theorems is due to Banach and Cacciopoli. The idea is as follows. Suppose that ^:A"-»A" is a contraction, with Lipschitz constant K < 1. Let x0 e X, and choose a number r with r(l-t<)>d(xo, x(*o)) • If Br(x) denotes the closed ball with centre x and radius r in X, then BKr(x(x0)) is contained in Br(x0) (see Figure C.4). Since x

FIGURE C.4

maps B =Br(x0) into BKr(x(x0)), the iterates x"(B) for n = 0 , 1 , 2 , . . . f o r m a nested sequence, as in Figure C.4. Since x decreases diameters by a factor K, it is intuitively obvious that there is single point at the core of the sequence, and this must be a fixed point of /. It is almost as quick to give a proper proof: (C.5) Theorem. {Contraction mapping theorem) A contraction x'- X'-> Yhas at most one fixed point. IfX=Y and X is complete then x has a fixed point.

THE CONTRACTION MAPPING THEOREM

241

Proof. Let x and x' be fixed points of \- Then d(x, x') = d(x(x), x(x'))=£icd(x,

x')

where K < 1 is the Lipschitz constant of x- Thus d(x, x') = 0, and so x = x'. Suppose that X = Y is complete. Let x e X. Consider the sequence (x"(x))n»o- For all integers « 5s m 3= 0 d(x"(x),xm(x))^Kmd(xn'm(x),x) n—m—1

^xm

d(xr+\x),xr(x))

1 r= 0 n—m—1

^*m

E

Krd(x(x),x)

r=0

^ Km(1 - K)'1 d{x(x), x), which tends to 0 as m -» oo. Thus the sequence is a Cauchy sequence. Since X is complete, the sequence converges to some limit, / say, in x. By the continuity of x, Xd) = x( Hm xn(x))

= lim *(*"(*)) = /•

•

We often find in applications that there is a variable parameter present, and that we need to know how the fixed point depends on this parameter. Let us be more precise. A map ^ : X x Y-*Z is uniformly Lipschitz on the first factor if for some constant K > 0 and all y e Y the map Xy '• X -» Z taking x to x(x,y) is Lipschitz with constant K. Similarly for the second factor. Clearly x is Lipschitz if and only if it is uniformly Lipschitz on both factors. It is a uniform contraction on either factor if it is uniformly Lipschitz on that factor with constant < 1 . When x is a uniform contraction on the second factor, say, and Y = Z is complete, each map x* has a unique fixed point, which we denote by g(x). This defines the fixed point map g:X^ Y of xSuch a map, satisfying for all x e X (C.6)

x(x,g(x))

= g(x)

may, of course, exist even when the above Lipschitz conditions do not hold. We now investigate the extent to which properties of x influence properties of g. (C.7) Theorem. Let x'XxY^>Z be a uniform contraction on the second factor, and letg:X^> Ysatisfy (C.6). Ifx is continuous then g is continuous. If X is Lipschitz then g is Lipschitz. If, further, X is a subset of a Banach space E, Y and Z are subsets of a Banach space F and x is Cr then g is Cr (r s= 1). If, further, Dx is C'~ -bounded then Dg is C -bounded.

242

THE CONTRACTION MAPPING THEOREM

APP. C

Proof. We denote the distance from p to p' by \p —p'\. Let K > 1 be a Lipschitz constant for x on the second factor. Then, for all x and x'eX, \g(x)-g(x')\

=

\x(x,g(x))-X(x',g(x'))\

^\x(x,g(x))-x(x',g(x))\ +

\x(x',g(x))-X(x',g(x'))\

=£\x(x, g(x))-x(x',

g(x))\ +

K\g(x)-g(x%

and so \g(x)-g(x')\^(l-K)-l\x(x,g(x))-x(x',g(x))\. Thus g is continuous when x is continuous, and Lipschitz when x is Lipschitz. Now suppose I c E , F u Z ^ F and that x is Lipschitz and C ( r ? l ) . Then for all (x, y) e X x Y, \D2x(x, y)\=£ K, and thus /rf -£> 2 *(*, y) is a linear homeomorphism of F. We first show that g is differentiate at x e X, with (C.8)

Dg(x)=T(x)DlX(x,g(x)) gix)))"1. For all sufficiently small f e E ,

where T{x) = (id-D2x(x, \g(x +{)-g(x)-

T(x)DlX(x,

g(x))(0

« | T(x)\\g(x + i) -g(x)-D2x(x,

g(x))(g(x

+&-g(x))

-D1x(x,g(x))(i)\ = \T{x)\\x{x+&g(x -DX(x,

+

€))-X(x,g{x))

g(x))((x + £ g(x + £)) - (x, g(x)))\.

By the differentiability of x, this expression is o(\(£, g(x +£) — g(x))\) as l(£ g(* + l)~g(*))l _ ) , 0, whence o ( | £ | ) a s | f | - » 0 (since g is Lipschitz). This gives differentiability of g. The proof that g is C is by induction on r{ s= 0). The case r = 0 is trivial, since g is Lipschitz. The inductive step is clear, since (C.8) expresses Dg as a composite (C.9)

X

>XxY

—

^L(E,F)x5

idxp

*

comp

> L(E, F) x L(F, F)

> L(E, F),

where B is the ball with centre 0 and radius K in L(F, F), and p: B ->• L(F, F) is the C°°-bounded uniformly C°° map sending X1 to (id-T)~l. Note that comp is here continuous bilinear. The last part comes, similarly, by induction using Lemma B.7. •

243

THE CONTRACTION MAPPING THEOREM

Notice that the proof of continuity of g works in principle when X is merely a topological space. Note also that continuity of x is implied by continuity of the maps Xy f ° r y e V. We now take the theory one stage further. Our attitude is that results in the text such as Theorem 3.45 (relating a change in a vector field to the corresponding change in its integral curves) should be immediate applications of theorems in this section. To achieve this, we introduce a further parameter, taking values in a topological space A. The spaces X, Y and Z are as in Theorem C.7. We are now, however, given a map ^ : A x X x F ^ Z such that, for each a e A, xa-Xx Y^Z is a uniform contraction on the second factor with constant K < 1. We also have, for each a e A, a fixed point map ga:X^Y satisfying ga(x) = x"(x, g"(x)) for all xeX. (CIO) Theorem.Leta 0 €.A. Suppose that, forallaeA,x" is C (r 2= 0) and, ifr>0, Lipschitz. Suppose also thatDx" is Cr~ -bounded and thatx" ~Xa° is C°-bounded. Then ga - g"° is C-bounded. If, further, Dx"° is uniformly C'1 and the map a: A -» C(X x Y, Z) taking a to x" ~x"a is continuous at a0, then the map /3: A -» Cr(X, Y) taking a to ga — ga° is continuous at a0. Proof. By Theorem C.7 Dga is C r _ 1 -bounded for all aeA. a e A and xeX, \ga{x)-ga°(x)\

= \xa(x, ga(x))-xa°(x,

Also, for all

ga°(x))\

^\xa(x,ga(x))-xa°(x,ga(x))\ + ^\xa-Xa°\o

\xa°(x,ga(x))-xa°(x,ga°(x))\ +

«\ga(x)-ga°(x)\,

and \ga(x)-ga°(x)\^(l-K)-1\xa-xa°\o. This completes the proof that ga - ga° is Cr-bounded. It also gives continuity of (3 at a0, when a is continuous at a0, in the r = 0 case. We complete the proof by induction. Suppose that fi:A^> Ck(X, Y) is continuous at a0. To perform the inductive step, we show that the map y: A^-Ck(X,L(E, F)) taking a to Dg" is continuous at a 0 . First note that, by hypothesis, the map from A to Ck(X, XxY) taking a to (0, ga-g"°) is continuous at a0. So is the map (a>->Dxa) from A to Ck(XxY,L(ExF,F)). Now (id, g"0) has a C fc_1 -bounded derivative, a and Dx ° is uniformly Ck. We may apply Theorem B.18 and the s = 0 argument from Theorem B.15 to show that the composition map from Ck{X,XxY)x Ck(Xx Y, L ( E x F , F)) to Ck(X, L ( E x F , F)) taking (6, ) a to (4>(0 + {id, g °) is continuous at ((0, 0), Dx"°). Thus the map A from A to

244

THE CONTRACTION MAPPING THEOREM

APP. C

Ck(X, L(E x F, F)) taking a to Dx"(id, ga) is continuous at a0. We identify Ck(X, I ( E x F , F)) with Ck(X, L(E, F)) x Cfc(AT, L(F, F)) by the canonical isomorphism. The second component of A takes values in Ck(X, B), where B is as in the proof of Theorem C.7. We now describe a decomposition of the map y. One first applies A. Then one operates on the second factor by p*, where p is as in the proof of Theorem C.7. Finally one takes the compositional product of the two factors (continuous bilinear, by Lemma B.5). Since A is continuous at a0, and the maps that follow are continuous, y is continuous at a0D ( C . l l ) Exercise. (Lipschitz inverse mapping theorem) Let B be the closed ball with centre 0 and radius b (possibly b=00) in a Banach space E. Let J T : E - » E be a (topological) linear automorphism, and let rf.B^IL be Lipschitz with constant K<\T~1\~1 and such that rj(0) = 0. Let C be the closed ball with centre 0 and radius Z>(|r _ 1 | _ 1 -K) in E. Prove that, for all y e C, there is a unique x e B such that (T + rj){x) = y. (Hint: rewrite this as x = T~1(y - TJ(JC)).) Hence, if D = int C and we write x = g(y), then g{D) is an open neighbourhood of 0 in B and the map g: D -» g(D) is inverse to the restriction T + -q:g{D)-*D. Prove that g is Lipschitz, and Cr (r 2= 1) when -q is C. Deduce the following local form: If / is a Cr (r s= 1) map of some open subset of E into E and if Df(x0) is an automorphism then there exist open neighbourhoods U of x0 and V of f{x0) such that the restriction / : £/-» V is a Cr diffeomorphism. (C.12) Exercise. {Immersive mapping theorem) Prove that if / : X -» Y is a C map of manifolds (rs* 1) and / is immersive at x0 then / restricts to a C embedding of some neighbourhood of x0. (Hint: Assume that X and Y are open in Banach spaces E and F, x0 = f(x0) = 0, F = E x G and Df(0) = (id, 0). Apply the inverse mapping theorem to the map ^ : X x G - > F denned by (x,z)=f(x) + (0,z).) (C.13) Exercise. (Submersive mapping theorem) Prove that if / : X-* Y is a Cr map of manifolds (r 3= 1) and / is submersive at x0 then some neighbourhood of x0 in f~1(f(x0)) is a C submanifold of X modelled on ker Df(x0). (Hint: Assume that X and Y are open in Banach spaces E and F, xo = f(x0) = 0, E = F x ker D/(0) and Df(0) is projection to the first factor. Apply the inverse mapping theorem to the map 4>: X -»E defined by 4>(x) = (xu x2) = (fix), x2).) (C.14) Exercise. {Implicit mapping theorem) The implicit mapping theorem is, basically, concerned with solving the equation (C.15)

T(y) + v(x,y) = 0

THE CONTRACTION MAPPING THEOREM

245

for y in terms of x, where T is an automorphism of a Banach space F, x takes values in a topological space X and 17 is Lipschitz on the second factor. The theorem is usually presented in a local form, where we are given a single solution y = b when x = a, and have to show the existence of a unique continuous map x>-+g{x) defined on some neighbourhood of a in X such that g(a) = b and (C.16)

T{g(x)) + r,(x,g(x)) = 0

for all x in the neighbourhood. We can always modify 17 so that a = b = 0. Let B be the closed ball in F with centre 0 and radius b (possibly 6=00). Suppose that x = 0, y = 0 satisfies (C.15) and let 17: X x B -» F be uniformly Lipschitz on the second factor with constant K <\T~ \~ . Suppose that, for all xeX, \r}(x, 0 ) | = S | T _ 1 | _ 1 - K . Prove that there is a unique map g:X-*B satisfying (C.16) for all x e X. {Hint: Rewrite (C.15) as y = - T " 1 T 7 ( X , y) = 0.) Prove that g is continuous if 17 is continuous. Prove that if X is open in a Banach space E then g is Lipschitz if 17 is Lipschitz, and Cr (r 2* 1) if 17 is C'. Deduce the following local form: Let X and Y be open subsets of Banach spaces E and F respectively, and let / : X x Y -»F be C (r 2= 1) with f(a, b) = 0 and D2f(a, b) an automorphism, for some (a, b) e X x y. Prove that there exist neighbourhoods £/ of a in X and V of b in Y such that there is a unique map g: U -* V satisfying f(x, g(x)) = 0 for all xeU. Moreover the map g is C.

BIBLIOGRAPHY As a reading of Chapter 7 should make clear, many recent advances in dynamical systems research stem from ideas of Stephen Smale, and his 1967 survey article (Smale [5]) has been profoundly influential. Some other surveys that may help to guide further reading on the subject are Nemitskii [1], Palis [2], Robbin [3], Shub [2, 3,4, 5] and Smale [7, 8]. In the list of references that follows, two conference proceedings appear so often that I have abbreviated them. "Global Analysis" denotes "Global Analysis, Proc. Symp. in Pure Math. 14 (Providence, Rhode Island: American Math. Soc, 1970)", and "Dynamical Systems" denotes "Dynamical Systems (Ed. Peixoto, M. M.) (New York: Academic Press, 1973). A B R A H A M , R. (assisted by M A R S D E N , J. E.)

1. "Foundations of Mechanics", Benjamin, New York, 1967. A B R A H A M , R. and R O B B I N , J.

1. "Transversal Mappings and Flows". Benjamin, New York, 1967. A B R A H A M , R. and S M A L E , S.

1. Non-genericity of fl-stability. Global Analysis, 5-8. A N D R O N O V , A. and P O N T R J A G I N , L.

1. Systemes grossiers. Dokl. Akad. Nauk SSSR 14 (1937), 247-251. A N O S O V , D.

1. Roughness of geodesic flows on compact Riemannian manifolds of negative curvature. Dokl. Akad. Nauk SSSR 145 (1962), 707-709 (English: Soviet Math. Dokl. 3 (1962), 1068-1070.) 2. Geodesic flows on compact Riemannian manifolds of negative curvature. Trudy Mat. Inst. Steklov 90 (1967) (English: Amer. Math. Soc. translation (1969).) A N T O S I E W I C Z , H. A.

1. A survey of Lyapunov's second method. "Contributions to the Theory of Non-Linear Oscillations". (Ed. Lefschetz, S.), Vol. 4, pp. 141-166. Annals of Math. Study 41, Princeton University Press, Princeton, 1958. A R N O L D , V.

I.

1. "Ordinary Differential Equations". MIT Press, Cambridge, Massachusetts, 1973. BIRKHOFF, G. D .

1. "Dynamical Systems". Amer. Math. Soc. Colloquium Pub. 9 American Math. Soc, New York, 1927. B O U R B A K I , N.

1. "General Topology, Part II". Herman and Addison-Wesley, Reading, Mass., 1966. B R I C K E L L , F. and C L A R K , R.

S.

1. "Differentiable Manifolds". Van Nostrand Reinhold, London, 1970. 246

BIBLIOGRAPHY

247

C H I L L I N G W O R T H , D. R. J.

1. "Differential Topology with a View to Applications". Pitman, London, 1976. C O D D I N G T O N , E. A. and L E V I N S O N , N.

1. "Theory of Ordinary Differential Equations". McGraw-Hill, New York, 1955. DANKNER,

A.

1. On Smale's Axiom A dynamical systems. Asterisque 49 (1977), 19-22. D E B A G G I S , H.

F.

1. Dynamical systems with stable structures. "Contributions to the Theory of Non-Linear Oscillations". (Ed. Lefschetz, S), Vol. 2, pp. 37-59. Annals of Math. Study 29, Princeton University Press, Princeton, 1952. DENJOY,

A.

1. Sur les courbes definies par les equations differentielles a la surface du tore. /. de Math. Pure et Appl. 11 (1932), 333-375. D E O L I V I E R A , M.

1. C°-density of structurally stable vector fields. Bull. Amer. Math. Soc. 82 (1976). DIEUDONNE, J.

1. "Foundations of Modern Analysis". Academic Press, New York and London, 1960. D U N F O R D , N. and

S C H W A R T Z , J. T.

1. "Linear Operators" Part 1. Interscience, New York, 1958. E L I A S S O N , H.

I.

1. Geometry of manifolds of maps. /. Differential Geometry 1 (1967), 169-194. F O S T E R , M.

J.

1. Calculus on vector bundles. /. London Math. Soc. (2) 11 (1975), 65-73. 2. Fibre derivatives and stable manifolds: a note. Bull. London Math. Soc. 8 (1976), 286-288. F R A N K S , J.

1. "Manifolds of C r -mappings". 2. Differentiably H-stable diffeomorphisms. Topology 11 (1972), 107-114. 3. Absolutely structurally stable diffeomorphisms. Proc. Amer. Math. Soc. 37 (1973), 293-296. 4. Constructing structurally stable diffeomorphisms. Annals of Math. 105 (1977), 343-359. G A N T M A C H E R , F.

R.

1. "The Theory of Matrices" Vol. 1. Chelsea, New York, 1959. G R E E N B E R G , M.

J.

1. "Lectures on Algebraic Topology". Benjamin, New York, 1967. G R O B M A N , D.

M.

1. Homeomorphisms of systems of differential equations. Dokl. Akad. Nauk SSSR 128 (1959), 880-881. 2. Topological classification of the neighbourhood of a singular point in ndimensional space. Mat. Sb. (N.S) 56 (98) (1962), 77-94. GUCKENHEIMER, J.

1. Absolutely fl-stable diffeomorphisms. Topology 11 (1972), 195-197. 2. Bifurcation and catastrophe. Dynamical Systems, 95-110. 3. One parameter families of vector fields on two-manifolds: another nondensity theorem. Dynamical Systems, 111-127.

248

BIBLIOGRAPHY

G U I L L E M I N , V. and

P O L L A C K , A.

1. "Differential topology". Prentice-Hall, Englewood Cliffs, New Jersey, 1974. GUTIERREZ,

C.

1. Structural stability for flows on the torus with a cross-cap. Trans. Amer. Math. Soc. 241 (1978), 311-320. H A L E , J.

K.

1. "Ordinary Differential Equations". Wiley-Interscience, New York, 1969. H A R T M A N , P.

1. "Ordinary Differential Equations". Wiley, New York, 1964. H E L G A S O N , S.

1. "Differential Geometry and Symmetric Spaces". Academic Press, New York and London, 1953. HiRSCH, M. W. 1. "Differential Topology". Springer-Verlag, New York, 1976. H I R S C H , M.. W. and

P U G H , C.

C.

1. Stable manifolds and hyperbolic sets. Global Analysis, 133-163. H I R S C H , M. W. and

S M A L E , S.

1. "Differential Equations, Dynamical Systems, and Linear Academic Press, New York and London, 1974. H I R S C H , M. W., P U G H , C. C. and

SHUB,

Algebra".

M.

1. "Invariant Manifolds". Springer Lecture Notes No. 583 (1977). H O L M E S , R.

B.

1. A formula for the spectral radius of an operator. Amer. Math. Monthly 75 (1968), 163-166. HOPF, E.

1. Anzweigung einer periodischen losung von einer stationaren losung eines differentialsystems. Ber. Verh. Sachs, Akad. Wiss. Leipzig Math. Phys. 95 (1943), 3-22. H u , S.-T. 1. "Introduction to general topology'*. Holden-Day, San Francisco, 1966. 2. "Homology theory". Holden-Day, San Francisco, 1966. HUREWICZ,

W.

1. "Lectures on Ordinary Differential Equations". MIT Press, Cambridge, Mass., 1958. I R W I N , M.

C.

1. Transformation groups with a common orbit. Bull. London Math. Soc. 5 (1973), 164-168. 2. A new proof of the pseudo-stable manifold theorem. /. London Math. Soc. (to appear). KERVAIRE,

M.

1. A manifold which does not admit any differentiable structure. Comm. Math. Helv. 34(1960), 304-312. K U I P E R , N.

H.

1. The homotopy type of the unitary group of Hilbert space. Topology 3 (1965), 19-30. 2. The topology of the solutions of a linear differential equation on R". Manifolds-Tokyo 1973, pp. 195-203. Univ. Tokyo Press, Tokyo, 1975. K U I P E R , N. H. and

R O B B I N , J.

W.

1. Topological classification of linear endomorphisms. Invent. Math. 19 (1973), 83-106.

BIBLIOGRAPHY

249

KUPKA, I.

1. Contribution a la theorie des champs generiques. "Contributions to Differential Equations" Vol. 2 (1963), pp. 457-484: Vol. 3 (1964), pp. 411-420. Wiley-Interscience, New York. 2. On two notions of structural stability. /. Differential Geometry 9 (1974), 639-644. LANG, S. 1. "Differential manifolds". Addison-Wesley, Reading, Mass., 1972. 2. "Analysis II". Addison-Wesley, Reading, Mass., 1969. L E S L I E , J.

1. On a differential structure for the group of diffeomorphisms. Topology 6 (1967), 263-271. L E F S C H E T Z , S.

1. "Differential Equations, Geometric Theory". Wiley-Interscience, New York, 1957. L I A P U N O V , A.

M.

1. "Probleme general de la stabilite du mouvement." Annals of Math. Study 17. Princeton University Press, Princeton, 1947. MANNING,

A.

1. There are no new Anosov diffeomorphisms on tori. Amer. Jour. Math. 96 (1974), 422-429. M A U N D E R , C. R.

F.

1. "Algebraic Topology". Van Nostrand Reinhold, London, 1970. M A R K L E Y , N.

1. Homeomorphisms of the circle without periodic points. Proc. London Math. Soc. (3) 20 (1970), 688-698. M A T H E R , J.

1. Characterization of Anosov diffeomorphisms. Nederl. Akad. van Wetensch. Proc. Ser. A, Amsterdam 71 = Indag. Math. 30 (1968), 479-483. M A Z U R , B.

1. Pub. Math. I.H.E.S. 15 (1963) (also 22 (1964), 81-92). M I L N O R , J.

1. On manifolds homeomorphic to the 7-sphere. Annals of Math. 64 (1956), 399-405. 2. Topology from the Differentiable Viewpoint". University of Virginia Press, Charlottesville, 1966. 3. "Morse Theory". Annals of Math. Study 51 Princeton University Press, Princeton, 1963. M O S E R , J.

1. On a theorem of Anosov. /. Differential Equations 5 (1969), 411-440. MUNKRES, J.

1. "Elementary differential topology". Annals of Math Study 54, Princeton University Press, Princeton, 1963. NELSON,

E.

1. "Topics in Dynamics I: Flows". Princeton University Press, Princeton, 1969. N E M I T S K I I , V.

V.

1. Some modern problems in the qualitative theory of ordinary differential equations. Russian Math. Surveys 20 (1965), 1-34. N E M I T S K I I , V. V. and

S T E P A N O V , V.

V.

1. "Qualitative Theory of Ordinary Differential.Equations". Princeton University Press, Princeton, 1960.

250

BIBLIOGRAPHY

NEWHOUSE, S. E .

1. Nondensity of Axiom A(a) on S2. Global Analysis, 191-202. 2. Hyperbolic limit sets. Trans. Amer. Math. Soc. 167 (1972), 125-150. 3. On simple arcs between structurally stable flows. "Dynamical Systems— Warwick 1974". Springer Lecture Notes No. 468, pp. 209-233. N E W H O U S E , S. E. and P A L I S , J.

1. Hyperbolic nonwandering sets on two-dimension manifolds. Dynamical Systems, 293-301. 2. Bifurcations of Morse-Smale dynamical systems. Dynamical Systems, 3 0 3 366. 3. Cycles and bifurcation theory. Asterisque 31 (1976), 43-140. N E W H O U S E , S. E. and P E I X O T O ,

M.

1. There is a simple arc joining any two Morse-Smale flows. Asterisque 31 (1976), 15-41. N I T E C K I , Z.

1. "Differentiable dynamics". MIT Press, Cambridge, Mass., 1971. P A L I S , J.

1. A note on fl-stability. Global Analysis, 221-222. 2. Some developments on stability and bifurcation of dynamical systems. "Geometry and Topology". Springer Lecture Notes No. 597, pp. 495-509 (1977). 3. On Morse-Smale dynamical systems, Topology 8 (1969), 385^105. P A L I S , J. and

S M A L E , S.

1. Structural stability theorems. Global Analysis, 223-231. P E I X O T O , M.

M.

1. On an approximation theorem of Kupka and Smale. / . Differential Equations 3 (1966), 214-227. 2. Structural stability on two dimensional manifolds. Topology 1 (1962), 1 0 1 120. 3. On the classification of flows on 2-manifolds. Dynamical Systems, 389-420. P E I X O T O , M. and

P U G H , C.

C.

1. Structurally stable systems on open manifolds are never dense. Annals of Math. 87 (1968), 423-430. POINCARE, H .

1. Memoire sur les courbes definies par une equation differentielle. / . de Math. 7 (1881) 375-422 and 8 (1882), 251-296. Sur les courbes definies par les equations differentielles. /. deMath. Pure etAppl. 1 (1885) 167-244. (These are all in Vol. 1 of "Oeuvres de Henri Poincare". Gauthier-Villars, Paris, 1951. PUGH, C. C. 1. The closing lemma. Amer. J. Math. 89 (1967), 956-1009. 2. An improved closing lemma and a general density theorem. Amer. J. Math. 89 (1967), 1010-1021. 3. On a theorem of P. Hartman. Amer. J. Math. 91 (1969), 363-367. P U G H , C. C. and

SHUB,

M.

1. H-stability for flows. Inventiones Math. 11 (1970), 150-158. R E N Z , P.

L.

1. Equivalent flows on Banach manifolds. Indiana Univ. Math. J. 20 (1971), 695-698. R O B B I N , J.

W.

1. A structural stability theorem. Annals of Math. 94 (1971), 447-493.

BIBLIOGRAPHY

251

2. Topological conjugacy and structural stability for discrete dynamical systems. Bull. Amer. Math. Soc. 78 (1972), 923-952. R O B I N S O N , C.

1. Generic properties of conservative systems. Amer. J. Math. 92 (1970), 562-603 and 897-906. 2. Structural stability of C'-flows. "Dynamical Systems—Warwick 1974". Springer Lecture Notes No. 468, pp. 262-277. 3. Structural stability of C'-diffeomorphisms. /. Differential Equations, 22 (1976), 28-73. 4. The geometry of the structural stability proof using unstable discs. Bol. Soc. Bras. Mat. 6 (1975), 129-144. R O B I N S O N , C. and W I L L I A M S , R. F.

1. Finite stability is not generic. Dynamical Systems, 451-462. R O S E N B E R G , H.

1. A generalization of Morse-Smale inequalities. Bull. Amer. Math. Soc. 70 (1974), 422-427. S H U B , M.

1. Structurally stable systems are dense. Bull. Amer. Math. Soc. 78 (1972), 817-818. 2. Stability and genericity for diffeomorphisms. Dynamical Systems, 4 9 3 514. 3. Dynamical systems, filtrations and entropy. Bull. Amer. Math. Soc. 80 (1974), 27-41. 4. Stabilite globale des systemes dynamiques. Asterisque 56 (1978). 5. The Lefschetz fixed point formula: smoothness and stability. "Dynamical Systems" (Ed. Cesaro, L., Hale, J. K. and LaSalle, J. P.) pp. 13-28. Academic Press, New York, 1976. S H U B , M. and S M A L E , S.

1. Beyond hyperbolicity. Annals of Math. 96 (1972), 587-591. S H U B , M. and W I L L I A M S , R. F.

1. Future stability is not generic. Proc. Amer. Math. Soc. 22 (1969), 483-484. S I M M O N S , G. F.

1. "Differential Equations with Applications and Historical Notes". McGrawHill, New York, 1972. S M A L E , S.

1. On gradient dynamical systems. Annals of Math. 74 (1961), 199-206. 2. Generalized Poincare's conjecture in dimensions greater than four. Annals of Math. 74 (1961), 391-406. 3. Stable manifolds for differential equations and diffeomorphisms. Ann. Scuola Normale Superiore Pisa 18 (1963), 97-116. 4. Structurally stable systems are not dense. Amer. J. Math. 88 (1966), 4 9 1 496. 5. Differentiable dynamical systems. Bull. Amer. Math. Soc. 73 (1967), 7 4 7 817. 6. The O-stability theorem. Global Analysis, 289-297. 7. Notes on differentiable dynamical systems. Global Analysis, 211-287. 8. Stability and genericity in dynamical systems. "Seminaire Bourbaki 19691970". Springer Lecture Notes No. 180, pp. 177-185 (1971). 9. Stability and isotopy in discrete dynamical systems. Dynamical Systems, 527-531. 10. On the structure of manifolds. Amer. J. Math. 84 (1962), 387-399.

252

BIBLIOGRAPHY

SOTOMAYOR, J.

1. Structural stability and bifurcation theory. Dynamical Systems, 549-560. 2. Generic bifurcations of dynamical systems. Dynamical Systems, 561-582. 3. Generic one-parameter families of vector fields on two-dimensional manifolds. Publ. Math. IHES 43 (1974), 5-46. S P A N I E R , E.

H.

1. "Algebraic topology". McGraw-Hill, New York, 1966. S U L L I V A N , D. and

W I L L I A M S , R.

F.

1. On the homology of attractors. Topology 15 (1976), 259-262. S T E R N B E R G , S.

1. Local contractions and a theorem of Poincare. Amer. J. Math. 79 (1957), 809-824. 2. On the structure of local homeomorphisms of euclidean n-space: II. Amer. J. Math. 80(1958), 623-631. T A K E N S , F.

1. On Zeeman's tolerance stability conjecture. "Manifolds Amsterdam 1970". Springer Lecture Notes No. 197, pp. 209-219 (1971). 2. Tolerance stability. "Dynamical Systems—Warwick 1974". Springer Lecture Notes No. 468, pp. 293-304 (1975). 3. Integral curves near mildly degenerate singular points of vector fields. Dynamical Systems, 599-617. 4. Singularities of vector fields. Publ. Math. IHES 43 (1974), 47-100. THOM,

R.

1. "Stabilite Structurelle et Morphogenese". Addison-Wesley, Reading, Mass., 1973. (English translation by D. H. Fowler, 1975). W A R N E R , F.

W.

1. "Foundations of Differentiable Foresman, Glenview, 111., 1971. W E L L S , J.

Manifolds and Lie Groups". Scott,

C.

1. Invariant manifolds of nonlinear operators. Pacific J. Math. 62 (1976), 285-293. WHITE,

W.

1. On the tolerance stability conjecture. Dynamical Systems, 663-665. W I L L I A M S , R.

F.

1. One dimensional non wandering sets. Topology 6 (1967), 473-487. 2. The " D A " maps of Smale and structural stability. Global Analysis, 329-339. 3. Expanding attractors. Publ. Math. IHES 43 (1974), 169-203. Z E E M A N , E.

C.

1. Morse inequalities for diffeomorphisms with shoes and flows with solenoids. "Dynamical Systems—Warwick 1974". Springer Lecture Notes No. 468, pp. 44-47 (1975).

Index

A Absolute fl-stability, 190 Absolute structural stability, 180 Action, topological group, 26 Adapted metric, 155 Admissible chart, 201 Admissible vector bundle chart, 205 a-limit set, 42, 46 Angular variation, 133 Anosov diffeomorphism, 168 Anosov vector field, 168 Anosov's theorem, 168 Arbitrary constant, 64 AS system, 179 Asymptotic stability, 6, 119, 129 Atlas, 197 Attractor, 175 Autonomous system, 64 Auto-oscillatory system, 66 Axiom A, 171 weak, 180 B Baire category theorem, 50 Baire space, 50 Baire subset, 161 Banach manifold, 197 Base of a vector bundle, 205 Basic set, 171, 175 Bifurcation, 192 Hopf, 193 saddle-node, 193 Brouwer's theorem, 137

C C° stability theorem, 183 C function, 202 C map, 226 C topology, 161, 238 C"-bounded map, 227 C r -continuous variation, 156 Canonical involution, 210 Cantor set, 179 Catastrophe, 192 Characteristic multiplier, 122 Chart, 197 norm,234 vector bundle, 204 Circle, 197 Circle orbit, 24 Classification, 28 of hyperbolic closed orbits, 125 of hyperbolic fixed points, 116 of hyperbolic linear automorphisms c R", 95 of hyperbolic linear vector fields o R", 99 of hyperbolic zeros, 118 Closed orbit, 24, 25 Cloud lemma, 172 Codimension, of a bifurcation, 193 of a foliation, 212 of a submanifold, 199 of a subspace, 199 Commuting flows, 21 Commuting homeomorphisms, 21 253

254 INDEX Compatible chart, 201 Dunford's spectral mapping theorem, Compatible vector bundle chart, 205 106 Complexification, Dynamical system, 10, 12 of a vector space, 106 of a linear map, 107 E Composition map, 225, 229, 231 Elementary fixed point, 110 Compositional product, 228 Elementary linear vector field, 86, 97 Configuration space, 4 Embedding, Connected sum, 198 C, 212 Conservation of energy, 79 of a homeomorphism in a flow, 18 Conservative system, 5, 79 topological, 212 Contraction, 240 Energy, 5, 66 linear, 85, 90, 94, 105 Energy function, 133 Contraction mapping theorem, 240 Equations of motion, 1 Contravariant tensor, 218 Equilibrium position, 133 Coordinate change map, 201 Equivalent norms, 92, 99 Cotangent bundle, 218 Evaluation map, 228, 232 Cotangent space, 219 Eventually hyperbolic subset, 155 Cotangent vector, 219 Exotic basic set, 175 Covariant tensor, 218 Expanding attractor, 175 Critical point, 164 Expansion, 240 Critical value, 164 linear, 90 Cross section, 120 Expansive map, 169 Curve, 210 Explosion, 185 Cycle (orbit), 24 Exponential of a linear endomorphism, Cycle (n-), 190 82,97 Exponential map, D of a spray, 217 DA system, 181 Riemannian, 221 Decomplexification, 107 Decreasing sequence of subsets, 50 F Degree of a map, 136 Dense property, 164 F" map, 235 Derivative, 219, 226 Fibre of a vector bundle, 205, 206 Fibre derivative, 235 directional, 78 Fibre differentiable map, 235 Deterministic process, 1 Filtration, 190 Diffeomorphism, 202 Finite stability, 191 Diffeomorphic structures, 203 Differentiable conjugacy, 29 Finsler, 219 Differentiable map, 226 First integral, 78 Differential, 226 First order differential equation, 215 Dimension, First return function, 120 Fixed point, 24 of a foliation, 212 Fixed point map, 241 of a manifold, 197 Fixed point set, 24 Direct method of Liapunov, 132 Flattened vector field, 220 Directional derivative, 78 Flow, 10, 13 Discrete dynamical system, 11, 13 Flow equivalence, 32, 75 Discrete flow, 11, 13 Dissipative system, 6 Flow map, 31 Distance function, 221 Focus, 87 Double tangent map, 210 Foliation, 212

255 Immersion, 211 Immersive map, 211 Immersive mapping theorem, 244 Implicit mapping theorem, 244 Implosion, 185 Increasing sequence of subsets, 50 Indecomposable set, 171 Index, of a fixed point, 135, 137, 140, 141 Gradient vector field, 59, 220 of a map, 136, 138 General solution, 64 of a submanifold, 133, 137 Generalized stable manifold theorem, Index sum, 141 156 Induced norm, 80 Generic property, 164 Induced system, 20 Geodesic of a spray, 216 Initial conditions, 64 Global O-explosion, 188 In-set, 144 Global stable manifold theorem, Integrable vector field, 61 for diffeomorphisms, 150 Integral curve, 3, 57, 62 for flows, 152 Integral flow, 10, 61, 74 Grobman's theorem, 117 Integrating the inequality, 70 Invariant, 28 Invariant set, 43 H Inverse mapping theorem, 244 Hamiltonian function, 66, 79, 133 Involution, 210 Hamiltonian system, 5, 192 Irrational flow, 20, 33 Hamilton's equations, 66 Irrational rotation, 30 Hartman's theorem Isomorphic hyperbolic linear automorfor diffeomorphisms, 114 phisms, 93 for flows, 117 Isomorphic hyperbolic linear vector Height function, 60 fields, 99 Hessian vector field, 111 Isomorphism of vector bundles, 208 Heteroclinic point, 145 Isotropy subgroup, 23 Homoclinic point, 144 Hopf bifurcation, 193 Hopf index theorem, 141 Horseshoe, 177 Jordan canonical form, 83 Holder continuous function, 127 Hyperbolic closed orbit, 120 K Hyperbolic fixed point, 110 Hyperbolic linear automorphism, 81, Kinetic energy, 5, 65 Klein bottle, 197 88,91 Kupka-Smale theorem, 165 Hyperbolic linear flow, 86 Hyperbolic linear vector field, 81, 86, 97 Hyperbolic periodic point, 151 Hyperbolic structure, 155 Lagrange's equations, 65 Hyperbolic subset, 155 Lamination, 213 Hyperbolic toral automorphism, 22 Lefschetz number, 141 Left composition map, 229 I Length of a curve, 222 Level surface, 129 Immediately hyperbolic subset, 155 Liapunov function, 131 Immersed sybmanifold, 211 INDEX

Form (1-), 219 Frechet differentiable map, 226 Free action, 53 Fundamental domain, 93 Fundamental theorem of algebra, 136 Future stability, 191

256 INDEX Liapunov stability, 128 Mobius band, 198, 205 Liapunov's direct method, 132 Morphogenesis, 192 Morse-Smale system 162, 165 Liapunov's instability theorem, 132 Liapunov's theorem, 131 Lie group, 27 N Lie group action, 27 n-manifold, 197 Lie transformation group, 27 Negative definite function, 131 Lifting, 31 Negative semi-definite function, 131 Limit set, 42, 46 Linear approximation, 81, 110, 115, Node, 87 No-cycle condition, 190 118,124 Non-autonomous system, 64 Linear automorphisms, Non-wandering point, 47 of R, 88 Non-wandering set, 47 of R 2 , 88 Norm atlas, 204 Linear conjugacy, 83, 88 Norm chart, 204 Linear contraction, 85, 90, 94, 105 Normal chart on a Riemannian maniLinear equivalence, 82 fold, 223 Linear expansion, 90 Nostalgic point, 47 Linear flow, 82 on R, 86 on R2, 86 O Linear map bundle, 207 (o- limit set, 42, 46 Linear vector field, 82 fl-conjugacy, 48 Lipeomorphism, 239 fi-decomposition, 171 Lipschitz condition, 67, 239 A-equivalence, 48 Lipschitz constant, 239 Lipschitz inverse mapping theorem, 244 fl-explosion, 185 A-implosion, 185 Lipschitz map, 239 A-set, 47 Local first integral, 79 fl-stability, 185 Local integral, 62 absolute, 190 Local fl-explosion, 188 topological, 191 Local product structure, 172 fl-stability theorem, 190 Local representative, 202, 207, 214 One-parameter group of homeomorLocal stable manifold theorem, 146 phisms, 13 Local stable set, 144 One-point compactification, 54 Local topological conjugacy, 38 Open property, 164 Local topological equivalence, 41 Orbit, 14 Local trivialization, 205 Orbit component, 40 Local vector bundle, 204 Orbit space, 14 Local vector bundle map, 204 Ordinary differential equation, 64, 215 Locally Lipschitz map, 239 Lucky non-transverse intersection, 189 Orientation of an orbit, 14, 31, 32, 51 Orientation preserving map, 137 Orientation reversing map, 137 M Out-set, 144 Manifold, Overlap map, 201 Banach, 197 C, 201 P Riemannian, 219 topological, 196 Parallelizable manifold, 209 Manifold of maps, 155 Partial derivative, 227 Minimal set, 45 Partial flow, 10, 39

INDEX

Peixoto's theorem, 167 Pendulum simple, 2 spherical, 7, 214 Period, 24 Periodic orbit, 24 Periodic point, 22, 24 Periodic point set, 25 Periodic rotation conjecture, 89 Perturbation, 76, 126, 160 Phase portrait, 3, 15 Phase space, 13 Picard's theorem, 57, 68 Piecewise C 1 curve, 221 Poincare-Hopf theorem, 141 Poincave map, 120 Positive definite function, 131 Positive orbit, 153 Positive semi-definite function, 131 Potential energy, 5, 65 Pretzel, 199 Principal part, 59, 214 Product bundle, 205, 207 Product of dynamical systems, 17 Projection, 91 stereographic, 54 vector bundle, 205 Projective space, 200 Pseudo-hyperbolic map, 149 Pseudo-stable manifold theorem, 150

257

Riemannian exponential map, 221 Riemannian manifold, 219 Riemannian metric, 219 standard (on R"), 219 Riemannian spray, 220 Riemannian structure, 219 Right composition map, 231 Robbin's theorem, 180 Robinson's theorem, 180 Rotation, 15 Rotation flow, 15 Rotation number, 31

S

Saddle point, 15, 119 Saddle-node bifurcation, 193 Sard's theorem, 164 Second countable space, 53 Second order ordinary differential equation, 215 Section of a vector bundle, 213 Selector, 180 Separatrix, 167 Sequence space, 233 Sharpened 1-form, 220 Shift automorphism, 179 Simple pendulum, 2 Singular point, 73 Sink, 119 Q Size of a stable set, 156 Skewness, 92 Quotient system, 21 Smale's expanding attractor, 175 Smale's horseshoe, 177 R Smale's fl-stability theorem, 190 Rational flow, 19, 33 Smale's spectral decomposition Rational rotation, 30 theorem, 172 Real Jordan form, 83 Smooth map, 13, 200 Real projective space, 200 Solution of a second order equation, 215 Real spectral decomposition theorem, Source, 119 108 Spectral decomposition of an fl-set, 171 Rectification theorem, 112 Spectral decomposition theorem, Recurrent point, 47 for dynamical systems, 172 Regular point, 38, 111 for linear endomorphisms, 106, 108 Regular value, 200 Spectral mapping theorem, 106 Residual subset, 161 Spectrum, 90, 103 Resolvent, 103 Speed of a rotation flow, 15 Restriction of a flow, 40 Speeding up factor, 21 Reverse flow, 20 Spherical pendulum, 7, 214 Riemannian distance function, 221 Split surjective map, 200

258

INDEX

Splitting of a Banach space, 197, 211 Spray, 215 Stability, absolute n , 190 absolute structural, 180 asymptotic, 6, 119, 129 finite, 191 future, 191 n , 185 structural, 6, 162 tolerance, 191 topological ft, 191 Stabilizer, 23 Stable equilibrium position, 133 Stable manifold, 143 of a compact hyperbolic subset, 155 of a hyperbolic closed orbit, 153 of a hyperbolic fixed point, 144, 152 of a hyperbolic linear automorphism, 92 of a hyperbolic linear vector field, 98 of a hyperbolic periodic point, 151 Stable manifold theorem for diffeomorphisms, 146, 150, 151 for flows, 152, 154 generalized, 156 Stable set, of an equivalence relation, 81 of a hyperbolic fixed point, 144, 152 Stable summand, 92, 98 Standard Riemannian metric on R", 219 Standard spray on R", 217 State space, 1 Sternberg's theorem, 126 Strange basic set, 175 Strong Liapunov function, 131 Strong topology, 238 Strong transversality condition, 179 Structural stability, 6, 161-162 absolute, 180 Structure, C manifold, 201 Riemannian, 219 vector bundle, 205 Submanifold, 199 C, 201 immersed, 211 Submersion, 211 Submersive map, 211 Submersive mapping theorem, 244 Sum of vector bundles, 207

Suspension, of a homeomorphism, 19 of a local homeomorphism, 122 Symmetry, 192 T Tangent bundle, 208 Tangent map, 209 double, 210 Tangent space, 208 Tangent vector, 208 Tensor bundle, 218 Tensor field, 218 Thorn's theory, 192 Time t map, 13 Tolerance stability, 191 Topological conjugacy, 28, 123 Topological dynamics, 12, 26 Topological embedding, 212 Topological equivalence, 32, 75 Topological linear automorphism, 197 Topological manifold, 196 Topological fl-stability, 191 Topological transformation group, 26 Topological transitivity, 171 Toral automorphism, 22 Torus, 197 Total space of a vector bundle, 205 Transitivity, of group action, 53 topological, 171 Translation, 80 Transversality, 164 strong, 179 Transversally heteroclinic point, 145 Transversally homoclinic point, 144 Transverse disc, 120 Transverse intersection, 165 Transverse map, 213 Transverse submanifold, 213 Trivial bundle, 205 Trivial dynamical system, 13 Trivialization, 209 U Uniform contraction, 241 Uniformly C map, 227 Uniformly Lipschitz map, 241 Unit circle, 200 Unit sphere, 200

259

INDEX

Unit vector field, 60 Unlucky non-transverse intersection, 189 Unstable fixed point, 129 Unstable manifold, 92, 98, 143 Unstable manifold theorem, 149 Unstable set, 144, 152 Unstable summand, 92, 98

Van der Pol's equation, 66 Vector bundle, 205 Vector bundle isomorphism, 208 Vector bundle map, 207

Vector field, 58, 214 Velocity, 57, 60, 210 Velocity vector field, 61 W Weak Axiom A, 180 Weak topology, 238 Whitney sum, 207 Whitney topology, 238

Zero of a vector field, 73 Zero section of a vector bundle, 206, 217