Lecture Notes in Mathematics Editors: A. Dold, Heidelberg B. Eckmann, Ztirich F. Takens, Groningen Subseries: Fondazione C.I.M.E., Firenze Advisor: Roberto Conti
1589
J. Bellissard M. Degli Esposti G. Forni S. Graffi S. Isola J.N. Mather
Transition to Chaos In Classical and Quantum Mechanics Lectures given at the 3rd Session of the Centro Internazionale Matematico Estivo (C.I.M.E.) held in Montecatini Terme, Italy, July 6-13, 1991 Editor: S. Graffi
Fondazione
C.I.M.E.
Springer-Verlag Berlin Heidelberg NewYork London Paris Tokyo Hong Kong Barcelona Budapest
Authors
Editor
Jean Bellissard Laboratoire de Physique Quantique Universit6 Paul Sabatier 118, route de Narbonne F-31062 Toulouse Cedex, France
Sandro Graffi Dipartimento di Matematica Universith degli Study di Bologna Piazza di Porta S. Donato, 5 1-40127 Bologna, Italy
Mirko Degli Esposti Giovanni Forni Sandro Graffi Stefano Isola Dipartimento di Matematica Universith degli di Studi di Bologna Piazza di Porta S. Donato, 5 1-40127 Bologna, Italy John N. Mather Department of Mathematics Princeton University Princeton, NJ 08544, USA
Mathematics Subject Classification (1991): 58F, 58F05, 58F15, 58F36, 81Q, 81Q05, 81Q20, 81Q50, 81S, 81S05, 81S10, 81S30
ISBN 3-540-58416-1 Springer-Verlag Berlin Heidelberg New York
CIP-Data applied for This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. 9 Springer-Verlag Berlin Heidelberg 1994 Printed in Germany Typesetting: Camera ready by author SPIN: 10130140 46/3140-543210 - Printed on acid-free paper
FORE'WORD
This volume collects the texts of two series of 8 lectures, and the expanded version of a seminar, given at thelC.I.M.E. Session on "Transition to Chaos in Classical and Quantum Systems", which took place at the Villa "La Querceta" in Montecatini, Italy, from July 6 to July 13, 1991. The purpose of the Session was to give a broad survey of the mathematical problems and techniques, as well as of some of the most relevant physical motivations, which arise in the study of the stochastic behaviour, if any, of deterministic dynamical systems both in classical and quantum mechanics. The transition to chaos in the most relevant and widely studied examples of classical dynamical systems, the area preserving maps, is thoroughly covered in the first series of lectures, delivered by Professor John Mather and written in collaboration with Dr. Giovanni Forni. In particular the reader can find in this text an up-to-date version of the well known Aubry-Mather theory. The lectures of Professor Jean Bellissard cover in turn, in addition to his algebraic approach to the classical limit, the behaviour of the quantum counterpart of the above systems, with particular emphasis on localization, and on qualitative as well as quantitative properties of the spectra of the relevant SchrSdinger operators in classically chaotic regions. They can be therefore considered an exhaustive introduction to the mathematical aspects of the so-called "quantum chaos". The third series of lectures, delivered by Professor Anatole Katok, covered the basic stochastic properties of classical dynamical systems and some of tlheir most recent developments. Unfortunately Professor Katok could not find the time to write up the text of his course. A very prominent role in describing tlhe chaotic behaviour of classical dynamical systems is played, as discussed also in Professor's Katok lectures, by the proliferation and equidistribution of the unstable periodic orbits of increasing period. An overview of recent results in this direction, and of their intimate connection to the problem of the classical limit of the quantized toral symplectomorphism~, is contained in an outgrowth of a seminar held by M.Degli Esposti, written in collaboration with S.Isola and the Editor.
Bologna, April 1994
Sandro Graffi
TABLE
J. BELLISSARD,
M.
G.
DEGLI
FORNI,
Non Commutative Methods in Semiclassical Analysis ...................
ESPOSTI,
J.N.
OF CONTENTS
S. G R A F F I , S. I S O L A , Equidistribution of P e r i o d i c O r b i t s : a n o v e r v i e w of classical vs quantum results ............
MATHER,
A c t i o n m i n i m i z i n g o r b i t s in H a m i l t o n i a n Systems .................................
1
65
92
Non Commutative Methods in Semiclassical Analysis Jean Bellissard Laboratoire 118, route
de Physique de Narbonne
Quantique
Universit6
F-31062 Toulouse
Paul Sabatier
Cedex, France
Contents 1
The kicked rotor problem
2
The 2.1 2.2 2.3 2.4
4
2
Rotation Algebra The Polynomial Algebra 5oi . . . . . . . . . . . . . . . . . . . . . . . . Canonical calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Rotation Algebra ,4i . . . . . . . . . . . . . . . . . . . . . . . . . Smooth functions in .41 . . . . . . . . . . . . . . . . . . . . . . . . . .
6 6 7 9 10
Continuity with respect to Planck's constant 3.1 Mean values of observables . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 The time evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 The spectrum of observables . . . . . . . . . . . . . . . . . . . . . . . .
11 12 13 14
Structure
15
o f t h e R o t a t i o n A l g e b r a `4/
S e m i c l a s s i c a l a s y m p t o t i c s for t h e s p e c t r u m 5.1 2D lattice electrons in a magnetic field . . . . . . . . . . . . . . . . . . 5.2 Low field expansion . . . . . . . . . . . . . . . . . . . . 5.3 Qualitative analysis of the spectrum . . . . . . . . . . . . . . . . . . . .
.........
17 18 19 23
Elementary Properties of the Kicked Rotor 6.1 The Furstenberg Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Calculus on BI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Representations and structure of 131 . . . . . . . . . . . . . . . . . . . . 6.4 Algebraic Properties of the Kicked Rotor . . . . . . . . . . . . . . . . .
26 26 27
Localization and Dynamical Localization 7.1 Anderson's Localization . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 The Observable Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Localization Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Localization in the Kicked Rotor . . . . . . . . . . . . . . . . . . . . .
34 34 38 39 43
28
30
1
The kicked rotor problem
One considers a spinning particle submitted to rotate around a fixed axis. Let 0 E T = R/27rZ be its angle of rotation, L c R its angular momentum, I its moment of inertia, # its magnetic moment, and B a uniform magnetic field parallel to the axis of rotation. Its kinetic energy is given by : L2
~t0 ~ + ~ B L ,
(1)
We assume that this system is kicked periodically in time according to the following Hamiltonian : L2 Tl = ~-[ + p B L + k cos(0) ~ 5(t - n T ) . (2) nEZ
where T is the period of the kicks, and k is a coupling constant representing the kicks strength. Here 5 is the Dirac measure. Classically the motion is provided by the solution of the Hamilton-Jacobi equations : dO cOTl dt - cOL
dL dt -
07-I 08 "
(3)
Between two kicks, 0~-//c90 = 0, so that L is constant whereas 0 varies linearly in time. When the kick is applied, L changes suddenly according to L ( n T + O) = L ( n T - O) + k sin(0). If we set : A~=T(L(nT-O)
+#B)
O~=O(nT-O),
(4)
On + An+l mod 27r ,
(5)
the equation of motion can be expressed as : A.+I = As + K sin(0.)
0.+1
=
where K is the dimensionless coupling strength namely : K-
kT I
(6)
The phase space is the cylinder C = T x R, if A is considered as a real number. If we set
f(0, A) = (0', A')
8' = 0 + A + K sin(P)
A' = A + K sin(P) ,
(7)
the solution of the equation of motion can be written as : (0,~+l,An+l) = f(On, An) 9
(s)
f is an analytic diffeomorphism of the cylinder C, which is area preserving, namely dO~ A d A ' = dO A d A , and a twist map, namely cOO~/OA > 0, which preserves the ends (see the course of John Mather in this issue). We remark that f also commutes with the translation A ~-~ A + 2~r of the action variable A in such a way that it also defines a map of the 2-torus T 2.
The orthodox way of quantizing this model consists in choosing the Hilbert space /C = L2(T, dO/27r) as the state space, and replacing L and 0 by operators as follows : L-
hO
P = multiplication by V(0),
i 00
(9)
whenever )2 is a continuous 27r -periodic function of the variable 0 . Q u a n t u m Mechanics requires using a new parameter h, the Planck constant which gives rise to a new dimensionless parameter : h T _ 47r b'QM
"Y- I
(10)
,
//ca
where vCL = 1 / T is the kicks frequency, whereas/]QM is the eigenfrequency of the free quantum rotor in a zero magnetic field. To compute the motion, we need to solve SchrSdinger's equation, namely, we look for a path t ~ R ~ r c tC such that : L2
iliCt = H ( t ) r
H ( t ) = - ~ + # B L + k cos(0) ~
5(t - n T ) .
(11)
nEZ The &kicks may create a technical difficulty. To overcome it let us consider a smooth approximation 5~ of 5 given by a non negative Ll-function on R supported by [0, e], with integral equal to 1. The solution can be given in term of a convergent Dyson expansion. Then letting e converge to zero, we get the following result (see Appendix
1): T h e o r e m 1 The solution of (11) is given by the following evolution equation : CT-0
=
F-I'r
-
F -1
=
e-iA2/2"re-iKc~176165 i9
(12)
~)=(#B)2--~.
(13)
where A=T(L+#B)
,
Let us also introduce the dimensionless magnetic field x : X2
x = -#BT
=>
/) = ~--~.
(14)
The operators of the form )2 whenever 1~(0) is a continuous 2rr -periodic function of the variable 0, can be obtained as the norm limit of polynomials in the operator U = e/~ .
(15)
In much the same way, one can quantize the action in the torus geometry by considering the operator : V = e -iA . (16) U and V are two unitary operators satisfying the following commutation rule : U V = ei~VU .
(17)
The C*-algebra generated by these two operators is the non commutative analog of the space of continuous functions on the 2-torus. By analogy with the commutative case, this algebra will be seen as the space of continuous functions on a virtual space, the "quantal phase space". Any such function will be the norm limit of polynomials of the form : a = ~ a ( m ) U m l V ' ~ 2 e -'~'~''~2/2 , (18) mEZ2,]m[_
where the a ( m ) ' s are complex numbers. We denote by .A~ the norm closure of this algebra. Whenever 7 = 0, this algebra coincides with the space C(T 2) of continuous functions on the 2-torus. One remarks that cos(0) E J~, but there is no way of writing F0 = exp ( i A 2 / 7 ) as an element of A~ since it is not periodic with respect to A. Therefore F0 ~ JLy in general. However, the following properties hold : (i)FoVFo ~ = V
(19)
(ii)FoUFo 1 = UV-le ~/2 ,
so that, setting D0(a) = F o a F o 1 for a C .A~, ~0 defines an automorphism of .A~, which coincides for 7 = 0 with the free rotation f0 in T 2, namely : fo(0, A) = (0 + A, A) ,
(20)
In particular if V ~ 0, a E J[~, we get : ~ ( a ) = F a F -1 = eiK c~176 Do(a)e-~K c~176
e .A~ ,
(21)
which means that D is an automorphism of .A~. At last, D admits a classical limit as 7 ~ 0, namely the automorphism of C(T 2) corresponding to the standard map (see section 3 below). For if V = K cos(0), let us denote by/2~ the "Liouville operator" defined by : ))a - a)) s
the limit of s be written as :
- -
-
(22)
,
iv
as V ~-+ 0 coincides with the Poisson bracket of V with a, and ~ can (23)
= e -c~ o Do 9
To summarize, we have obtained an algebraic framework describing the quantal observables which is completely analogous to the classical description of the system, and which converges to the classical analog as V ~ 0. In this framework, (i) the observable algebra .A~ is the non commutative analog of the space C(T 2) of continuous functions on the classical phase space T 2. (ii) the quantal evolution is described through the automorphism ~ of ~ which admits the standard map as a classical limit. Before leaving this section, let us describe the complementary point of view, given in wave Mechanics by the Feynman path integral, which happens to be exact and finite dimensional in this case. L e m m a 1 I r e E C~162 (e-~A'/~r
t h e n the f o l l o w i n g f o r m u l a holds :
(u) = e -'~/4 / + =
du'
e,(, . . . . . ) ~ / 2 ~ e - i ~ ' / ~ r
.
(24)
P r o o f : From (9)&(13), we get A = - i 7 0 / 0 0 Fourier series, so t h a t :
(0)-- E
9
nEZ
. "~ E
- x. If r E C ~ ( T ) , let (r
f L '~ dO'ei,~(o-o')-,(~,~-~)212~r
its
)
nEZ
To compute the distribution kernel coming into this sum, we use the Poisson summation formula : C '~(~176 nEZ
~ ~
2 ~ i(o-o'+~+2~l)~/2,~ IEZ
Now we perform the change of variables u' = 0' + 2~rl, u = 0, and the sum over l E Z will give rise to an integral over R with respect to u', leading to (24). Using (12)&(24), we immediately get the following Feynman p a t h integral representation : C o r o l l a r y 1 For any t ~ N and r E C~176
get :
(25) where Uo = u, and the right-hand-side defines a convergent oscillatory integral which is periodic of period 27r with respect to u. Remark
: The expression contained in the phase factor
s(~l...,~,;~o,x)=]<s
(26)
is nothing but the "Percival" Lagrangean or the "Frenkel-Kontorova" energy functional used by A u b r y and Mather to describe the trajectories of the s t a n d a r d map. For indeed the s t a t i o n n a r y points of such a Lagrangean are finite sequences (Us)l<s
1) , u t -
ut-1 - x + K s i n ( u t ) = 0 .
In particular if we set ps = us - "//.s-1 (for 1 < s < t) we get Us+l = us + Ps+l for 0 < s < t - 1, and ps+l = ps + K s i n ( u s ) for 1 < s < t - 1, x = pt + K s i n ( u t ) , namely we recover the s t a n d a r d map (5) in R 2 now instead of T ~, for a t r a j e c t o r y (00, A 0 ) , ' " , (0,, At) such that 0o = uo mod 27r, and At+l = x m o d 27r.
2 2.1
The Rotation Algebra The
Polynomial
Algebra
7)s
In this section we define properly the algebra .h v and we will describe without proof its most i m p o r t a n t propelties. We refer the reader to [BaBeF1] for more details. Actually given an interval I of R, we will rather consider the algebra .AI which is
roughly speaking the set of continuous sections of the continuous field 7 E I ~-* .Av. The semiclassical limit will be included whenever I contains 7 = 0. Let I be a compact subset of R. The polynomial algebra Pl is defined as follows : the elements of P1 are the sequences (a(m))meZ2 with finite support, where for each m = (ml, m2) e Z 2, a ( m ) : 7 c I ~ a(m, 7) E C is a complex continuous function on I. Px admits a natural structure of C(I)-module by setting, for a, b E Pl, and l E C ( I ) -
-
(a + b) (m) = a(m) + b(m)
ha(m; 7) = )~(7)a(m; 7) 9
(27)
- any element a c 7)i admits an adjoint a* defined by : a*(m; 7) = a ( - m ; 7) ,
(28)
where ~ denotes the complex conjugate of z in C. -if a, b E 7:'I, their product is defined by : (ab) ( m ; 7 ) =
~
a(m'; 7)b(m - m'; 7 ) e i ' r m ' ^ ( m - m ' ) ,
(29)
m~EZ 2
where we have set if m/, m " E Z 2 : m ' A m " = m l,m ,,2 - m 21m ,,1 .
(30)
- the topology on 79I, is the direct sum topology obtained from the uniform norm on
c(i). Denoting by 79~ the algebra 791 whenever I = {7} it follows that 79~ = 797+4.. Moreover setting c~(a) = ((-)ml'~2a(m))m~z~ , c~ defines a *-isomorphism between 797 and 79~+2~. Thus, as far as 79~ is coneerned, one will consider that 7 is defined rood. 27r. The same definition holds if we replace I by the toms T namely the continuous functions on I by the continuous 27r-periodic functions on R. We will denote by 79 the corresponding algebra. The following elements in 791 are remarkable : I(m;7)
=
~m,O
U(m; 7) = 6m,(1.o)
V(m;7)
=
~m,(o,1)
9
(31)
For indeed, I is the identity of 79I whereas U, V, are unitaries namely U U * = U * U = V V * = V * V = I, and obey to the commutation rules (17). Moreover, 79I is algebraically generated by U, V as a C(I)-algebra, namely if a c 79I, it can be written as a=
~
a ( m ) W ' ~ l V ' ~ 2 e -i~'~1m2/2 .
mtEZ 2
It will be convenient to introduce the "Weyl operators" as follows : W ( m ) = U TM v m 2 e
-iTralra2/2
.
(32)
From the interpretation given in the previous section, it follows that 79I is the set of trigonometric polynomials over the "non-commutative" 2-torus. In particular if I = {0}, we recover the convolution algebra, which by Fourier transform is exactly the algebra of usual trigonometric polynomials. The "evaluation" homomorphism r/~ is defined as the map from 791 into 79~ by : ~7~(a) = (a(m;7))mEZ 2
.
(33)
It is immediate to check that ~ is a *-homomorphism, namely it is linear, and preserves the product and the adjoint.
2.2
Canonical
calculus
Using the analogy with the space of trigonometric polynomials on the 2-torus, we now define some rules for the differential calculus. The integral is given by the trace defined by : ~(a) : a(O) c C ( 5 9
(34)
We will denote by 7~(a) the value of T(a) at V. The trace ~- is a linear module map from 7'i into C(I) satisfying : (i) positivity : T(a*a) = ~m,eZ 2 la(rn)l 2 > 0, a E ~ / , (ii) normalization : T(I) = 1, (iii) trace property : 1-(ab) = r(ba), a, b E Pl. We remark that the value of ~-(a) at V = 0 is the 0 th Fourier coefficient of rl0(a), namely the integral of its Fourier transform : dOdA
-(a)l~:o = fT~ -~-~--2ad(0, A)
.
(35)
where ad is the Fourier transform of r/0(a). The angle average, is defined by the element (a / in PI given by : (a)(m) = 5m,,oa(O, m 2 ) .
(36)
The m a p a ~-* Ca) is a module-map taking values in the commutative subalgebra Z)I generated by V as a g(I)-module. The usual Fourier transform permits to associate with any element b of Z)i a continuous function of (7, A) E I • T denoted by bay as follows : bav(3',A)= ~ b(O, m 2 ; v ) e - ~ 2 A 9 (37) m,EZ ~
The mapping b E Z)i ~-~ b~ E C(I x T), is a .-homomorphism, namely (bc)a~ = b~ca~ and (b*)~v = b**. We will say that b E 7Pi is positive whenever ba~ is positive. Using these definitions, the angle averaging satisfies : (i) positivity property : (ii) projection property: (iii) normalization : (iv) conditional expectation :
(a'a) > 0 , a E P l (Ca)) = (a) , (I) = 1 , (ab) = (a)b , (ba) = b{a) , if b E D1 , a E P l (38) A differential structure is defined on P / t h r o u g h the data of two ,-derivations 00 and OA given by : (Ooa) (m) = i m l a ( m )
(OAa) (m) = i m 2 a ( m ) .
(39)
These two derivations 0" (if # = 0, A) actually commute and satisfy : (i) (ii) (iii)
(iv)
they are C ( I ) - l i n e a r c%(a*) = (Oga)* Ot,(ab ) = (Ot,a ) b + a (Ogb) o o g = i u , o o v = o , o A u = o , OAK = - i v .
a E "~I , a, b E "PI ,
(40)
8
Moreover one can exponentiate them, namely defining by 2-parameter group of *-automorphisms given by : po,A(a)(m) = e~('~'~
{PO,A;(0, A)
C T 2} as the
(41)
,
we get : = \
0#
/#=A=0
#=0, A.
(42)
Actually PO,A is a module-*-homomorphism such that (0, A) E T 2 ~-~ po,A(a) E PZ is continuous and : PO,A o PO',A' = PO+O',A+A' , (43) If a, b E 7)z their Poisson (or Moyal [Bou]) bracket {a, b} is defined as follows : {a,b}(m;7)=
~]
a(m';7)b(m-m';7)2sin(-~m'A(m-m'))
,
(44)
mtEZ 2
where we set ( s i n x ) / x = 1 for x = 0. In particular that for "~ = 0, it coincides with the usual Poisson bracket, namely : {a, b}~l = {a~l, b~l} = OoaclOAbr -- OAaclOobr ,
(45)
From (44), the right-hand-side defines a continuous function of "y on I, so that the Poisson bracket {a, b} still belongs to T'x. The "Liouville operator" associated to w E "Pl is the module map defined by : L~(a) = {w, a} , a E T'I.
(46)
The properties of this operator are the following : (i) (ii) (iii) (iv)
L~ is C ( I ) - l i n e a r L~(a*) = gw. (a)* L~(ab) = n~.(a)b + aL~.(b) [L~,,L~,] = L{~.,~,} (Jacobi's identity)
w, a C P l , w, a, b E Pz , w,w' E Pi 9
(47)
e T 2,
(48)
We also remark that T(po,a(a))=~'(a)
r({a,b})=0
a, b E P I , ( O , A )
which is equivalent to the "integration by parts formula" : "r(O~,a. b) = - ' r ( a . O~,b)
2.3
The
Rotation
Algebra
7- (L,~(a). b) = --T (a. L~(b)) ,
(49)
Az
In order to get all continuous functions on our non commutative torus, we ought to define the non commutative analog of the uniform topology on Pz. This can be done by remarking that in the commutative case, the uniform topology is defined through a C*-norm, namely a norm on the algebra which satisfies : Ilabl] _< Ilall[Ibll
[[a'all--Ilall
2 9
(5o)
The importance of this relation comes from the fact that such a norm is actually entirely defined by the algebraic structure, namely it is given by the spectral radius of a*a. Therefore, the algebraic structure is sufficient and the uniform topology becomes natural. To construct such a norm, one uses the representations of 79I. A "representation" of PI is a pair (:r, 7-/~), where 7-/~ is a separable Hilbert space, and 7r is a ,-homomorphism from 79I into the algebra B(7-/~) of bounded linear operators on 7-/~. The formulae (17)&(18) give an example of representation for which ~ = L2(T, dO/27r). In particular 7r(U), 7r(V) will be unitary operators on 7-/~ so that if a 9 79I, one gets (if HfllI denotes the sup norm in C(I)) :
Jbr(a)ll_< ~
lla(m)Ib< oc.
(51)
mEZ 2
Two representations (Tr, 7-/~) and (C, 7-/w) are equivalent whenever there is a unitary operator S from 7-(~ into 7"/~, such that for every a 9 791 : S~(a)S -l : r'(a)
(52)
9
Up to unitary equivalence, one can always assume that 7-/~ ~2(N), so that the family of all equivalence classes of representations of 79I is a set denoted by Rep(791). We remark that tile norm 117r(a)II depends only upon the equivalence class of 7r. We then define a seminorm on 791 by : =
Ilalb = sup{ II=(a)II; ~ 9 mep(79J}.
(53)
This notation agrees with the sup-norm on C(I) if a 9 C(I). Then one has [BaBeF1] : Proposition
1 The mapping a 9 79I ~-~ ]la[ll 9 I%+ is a C*-norm.
R e m a r k : The only non trivial fact in this statement is that it is a norm, namely that Ilallz : 0 implies a : 0. D e f i n i t i o n 1 The algebra .,4i (resp. .A) is the completion of 79i (resp. 79) under the norm I1" I]x (resp. I]" liT). ,4 is called the "universal rotation algebra". P r o p o s i t i o n 2 1)-Any representation of 79i extends in a unique way to a representation of A I 2)-If B is any C*-algebra, and/9 is a *-homomorphism from 79I to B, then/9 extends in a unique way as a *-homomorphism from A~ to 13. 3)-Any pointwise continuous group of *-automorphisms of 791 extends in a unique way as a norm pointwise continuous group of *-automorphisms of,41. 4)-The trace r and the angle average (.) satisfy :
IlT(a)llI < Ilalb
II(a)lb < Ila11I
a
9 791 ,
(54)
and therefore they extend uniquely to At. 5)-The norm I1" I[I satisfies : Ila11i = supll~(a)ll
a 9 79I .
(55)
10
In practice the explicit computation of the norm does not require the knowledge of every representation. It is enough to have a faithfull family, namely a family {Trj}jeg where J is a set of indices, such that ~rj(a) = 0 for all j ' s implies a = 0. In other words njegKer(Trj) = {0}. We recall that the spectrum Sp(a) of an element a of a C*-algebra with unit A, is the set of complex numbers z such that zI - a is non invertible in A. Proposition 3
Let (Trj)je J be a faithfull family of representations of the C*-algebra
A, then : ][a[[l = sup H~rj(a)[[
Sp(a) = closure{tJj~gSp(Trj(a))} .
(56)
jEJ
In particular if 7r is faithfuU (namely if J contains only one point), [[a[[i = lit(a)[[ and Sp(a) = Sp0r(a)). 2.4
Smooth
functions
i n .AI
Beside Pl, one can define many dense subalgebras of AI playing the role of various subspaces of smooth functions. (i) For N E N, the algebra CN(AI) of N-times differentiable elements of 1)1 is the completion of .AI under the norm : 1 1
~
Ilallc"/=
,~,
n! n'! IIO;OA (a)ll' "
(57)
O~_n,n ~; n T n ~~ N
(ii) Coo(.Az) = NN>OC~V(.Ax). It coincides with the set of elements a = (a(rn))meZ2 with rapidly decreasing Fourier coefficients. It is a nuclear space, similar to the Schwartz space on the torus. Its dual space S(AI) is a space of non commutative tempered distributions which can be very useful in investigating unbounded elements. (iii) 7-/s(Ar) is the Sobolev space, namely the completion of Ps under the Sobolev norm : Ilall
.,1
(T(a*a)+r(a*(--A)8/2a)) '/2
A=
2+
2
(58)
where --A is the Laplacean on the non commutative torus. The imbedding 7-/8'(.As) ~-* 7C(Ax) is compact if s ~ > s and Coo(A1) = A~>07"/~(Ax), showing that Coo(Ai) is a nuclear space. (iv) An element of AI is holomorphic in some domain D of ( T + i R ) 2 if the continuous mapping (0, A) e T 2 ~-~ Po,A(a) C At, can be extended as a holomorphic function on D. A special interesting case consists in considering the algebra .Al(r) for r > 0, obtained by completing 7)i with the norm :
miami,,, -- sup
~
]a(m;7)[e rl'~ll ,
(59)
~'EI m E Z 2
where [m[1 = Iml[ + ]m21. Then .41(r) becomes a Banach ,-algebra of holomorphic elements in the strip D(r) = {[Im0] < r , limA[ < r}. (v) Let us consider now the case for which I is an open interval, and let ~o~ be the subalgebra of ~~ the elements of which have Fourier coefficients given by C~176 on I. Let us define the operator 07 on T'~~ by :
Ova
( Oa(m) / =
)m
(60) Z2
"
11 Then az obeys the following rules (Ito's derivative) : (i)
it is linear
(ii)
cO,~(a*) = (O,~a)*
a E "R~ ,
(61)
(iii) d T ( a ) / d ' , / = "r(O.ya) a c 7:'f ~ , (iv) O~(ab) = (c9~a)b + a(O~b) + 89(OoaCgAb - cOAaOob) a, b e 7 ~ . One can extend 0r to the dense subalgebra CN'L(AI), obtained by completing P T with respect to the norm : []a[[c~,L,1 =- MaxI<_L ]]O~a[[cN,I .
Let [[. [] be an algebraic *-norm. Then the following norm is also an algebraic ,-norm. [[a[[c1 = I]a[[1 + [[c9oa[[ + HOAaI[ + HcO~a[[ 9
(62)
By recursion we will set II' [[cN = (]1" HcN-')c1 with ]]. ]]co = H" H. It defines then an algebraic *-norm on gN'N(AI) equivalent to [[" []C~,N
3
Continuity
with respect to Planck's constant
Since the effective Planck constant 7 is a tunable physical parameter in many examples, one can wonder whether the various quantities of interest such as mean values of observables, or the evolution, or the spectrum of observables, are continuous functions of % The main difficulty in dealing with this problem is that the family of algebras V ~ -s even though continuous in the sense of Tomiyama [Tom], is not locally trivial. Indeed, A z is isomorphic to A~, if and only if 7 = 4-7~ rood 27r [Rie, PiVo]. Therefore such continuity properties must be carefully studied. We will give in this section and again without proofs, three kinds of continuity properties. The first concerns the mean value of observables, namely the function -r(a) if a E Az. One important consequence is the Weyl formula for the semiclassical limit of the density of states. The second type of result concerns the continuity of the evolution. It requires the use of a non commutative analog of the Canchy-Kovaleskaya theorem. In particular, the semiclassical limit of any time correlation function at fixed time, is equal to the corresponding classical expression. The last type of result is the continuity of the gap edges of the spectrum of any observable. This fact will permit to compute the spectrum numerically (see section 4). It is to be noticed that the algebra A~ can be constructed from the algebra of pseudodifferential operators of order zero acting on the unit circle. These results are well known in the context of pseudodifferential calculus. However, it turns out that all proofs given here are purely algebraic, and do not require any explicite reference to the form of the symbol. In particular they are valid for any element in the norm closure. But the closure contains much more than pseudodifferential operators, it also contains Fourier integral operators, and elements with no special behaviour.
3.1
Mean values of observables
Our first result is elementary in view of the definition of the algebra .At.
12
P r o p o s i t i o n 4 If a E ,41, the mapping 7 E I ~ ~'7(a) is continuous. P r o o f : If a E 7~i the result comes from the definition of 7~1.If a E ,41, given c > 0, there is a r 7~i such that ]]a-adix < ~, and therefore by (54), supTei ]TT(a)-~'7(ae)l <_ e. Thus "r(a) is a uniform limit of continuous function on I namely it is continuous. Let us now consider a self adjoint element H = H* in `4I and let ~ be its spectrum. Let f be a continuous function on Z. Then the map f E C(Z) ~ TT(f(H)) E C is linear positive and bounded. Therefore there is a Radon measure jV'.yon R supported by Z such that :
v~(f(H)) = fI~ dAf~(E)f(E) ,
(63)
This measure is called the "density of states" of H. The "integrated density of states" (IDS) is :
NT(E) = fu<E dAfT(E').
(64)
It is a non decreasing function of E E R. From the proposition 4, we get [BaBeF1] : P r o p o s i t i o n 5 If H = H* E `41 letJV'~ be the integrated density of states of H. Then
if E is a point of continuity of Aft, we get : lim Af~,(E) = Aft(E) ,
(65)
If I contains 3' = 0, since .4o = C(T 2) it is easy to check that :
Afo(E)=fEdAfo(E')=fHc,(O,A)<E d O d A _ _ 47r2
(66)
where H a is the Fourier transform of ~/o(H). Thus N0(E) is the area of the set H ~ l ( - c ~ , E) in the 2-torus. A consequence of the Proposition 4 is the following : C o r o l l a r y 2 If I contains "7 = 0 and H = H* E Az let Af~ be the integrated density of states of H. Then if E is a real number such that the level set H ~ I ( E ) has zero
Lebesgue measure in the 2-torus, we get : limAf~(E) = N 0 ( E ) , (Weyl's formula).
(67')
Let us also mention the following non trivial result [BaBeF1] : P r o p o s i t i o n 6 If H = H* E Pl, then its integrated density of states Af~ is continuous
with respect to E for any "7 E I.
13 3.2
The
time
evolution
Our next result concerns the continuity of the evolution with respect to 7. Let w E ~i:~I, and let us consider the automorphism of A z (for 3' ~ 0 ) given by : (68)
~ ( a ) = e-m~(w)/~aem~(~)/~ ,
Is it possible to prove that ~ can be continued at 7 = 0 in such a way as to define an automorphism of .4i ? To show that it is actually possible, let us consider the algebra At(r) introduced in 2.4 with the norm defined by (58). Then we get [BeVi]: T h e o r e m 2 Let w = w* be an element of ~4i(r) where I is an interval containing 7 = O. Then f o r any p such that 0 < p < r, (i) the Liouville operator L~ associated to w is well defined as a bounded linear operator f r o m .Ai(r) into .Al(r - p), (ii) f o r t small enough, exp(tL~) defines a linear bounded operator f r o m A l ( r ) into A z ( r - p), (iii) exp(tLw) can be extended as a *-automorphism of A I f o r any t 9 R , in such a way as to satisfy : de tn~ (a) _ etL~ (L~(a) ) (69) dt f o r any a E A i ( r ) .
To prove this result, we will proceed in several steps. First of all : L e m m a 2 I f w 9 .Al(ro) and a 9 A1(r), (r < ro), then f o r any p such that O < p < r one has :
[l{w, a} IIr_p <
211wllr~
- e~p(ro-r-p)
(70) "
P r o o f : From (44), using the inequalities ]sin(x)l < Ixl, Iml < Im'l + Im - m'l whenever m, m ' 9 Z 2 and Im' A m " l < Im~ IIm"21+ Im~ Hm " l l w e g e t :
II{w, a}ll,,-pll -< sup~el Em,,,,v,r z ~ er~
~/)le'm"la( m", 7)
(71) 9"" e -(~~247
(Im~llm~l + Im~llm~l) 9
The inequality (70) will be obtained by using the estimate sup,~ez Inle -vl'~l = 1/ep. L e m m a 3 I f w 9 A 1 ( r ) f o r any r such that 0 < p < r and any n 9 N one has :
P r o o f : One can write
_<
=
for any family (Pk)l
14
....
L~
p
1(2) -< ~ j
n Ilwll~"
~__l~1 1 Pk(Pk - Pk-i)
"
Let us choose Pk = pk/n; since n'~e -~ < n!, we immediately get the result.
Proof of theorem 2 : the point (i) is exactly the content of the L e m m a 2. To prove (ii), it follows from the Lemma 3 that if It] < p2/(2Ilw]lr ) = T, the expansion for exp(tL~) in ~owers of t converges in norm as an operator from .Ai(r) into .Ai(r - p) and in addition : Ile'L~ll ....
~ -< 1 -
Proving (iii) is more subtle : if we set /3t *-derivation, by (47), then, if a, b e .Ai(r) :
1
(73)
2ltliiwllr/p =
(i) ~t(ab) = ~t(a)/3t(b) (ii) /3t(a*) = ~t(a)* (iii) ~t+~(a) = ~t (~,(a)) (iv) d~t(a)/dt = ~t(L~(a))
~ "
e tLw, we observe that since L~ is a
for for for for
Itl < Itl < Itl + It[ <
T, T, Is] < T , T.
(74)
Therefore given any representation 7r of "/ol, 7r can be extended as a representation of .Al thus of Az(r). In particular, 7r o fit gives also a representation of A i ( r ) , so that by the same type of argument used in 2.3 (see (51)&(53)), one gets 1lTro/3t(a)] b < ]]all1, and since 7r is arbitrary :
II~(a)lb ~ Ilalb, for Itl < T.
(75)
In particular,/3t can be extended by continuity to .As and the extension still satisfies (74). Now if t C R, let n be a positive integer such that It/n I < T. Then we set (/3t/,~) "~. Thanks to (74) (iii), it is standard to check that this definition does not depend upon the choice of n. Moreover, (74) continues to hold at any value of t : this is obvious for (i), (ii), (iii). (iv) also holds once one notices that L~ commutes with /3t. Therefore (/3t)teR defines a 1-parameter group of *-automorphisms of .AI. At last it is norm-pointwise continuous, namely : lira IIA(a) t~-*O
alli
=
o
(76)
For indeed, by a 3e-argument, it is enough to check it for a E .Ai(r), which is simply a consequence of the Lemma 3.
3.3
The spectrum of observables
Our last result concerns the continuity of the spectrum with respect to 7. Let (E(t))teR be a family of compact subsets of a topological space X. This family is continuous at t = to if the two following properties hold : (i) it is continuous from the outside, namely given any dosed set F in X, such that E(to) N F = 0, there i 5 > 0, such that if It - to[ _< 5, then E(t) N F = O.
15 (ii) it is continuous from inside, namely given any open set O in X, such that E(t0) M O ~ 0, there is 5 > 0, such that if It - t01 _< 5, then E(t) M O ~ 9. If X = R a gap of E(t) is a connected component of R - E(t). One can check that this definition is equivalent to the continuity of the gap edges of E(t) at to. For a E .AI we set E(7) = Sp(r/7(a)), whenever "y 6 I. The main result of this subsection is [BaBeF1] : T h e o r e m 3 For any normal element a E ,4i, (namely such that aa* = a'a), the family (E(7))~e I is continuous at every point of I. The proof of this theorem will not be given here. It can be found in [BaBeFl]. However, it is of very high importance in view of the numerical computation of the spectrum. For we will see in the next section that the spectrum can be easily computed on a computer for rational values of 7/27r. The continuity of the gap edges everywhere on I implies that this type of computation is sufficient to get an idea of the spectrum for irrational values of 3,/2rr. Actually for smooth self adjoint elements of .AI one gets a better result [BaBeF1], namely : T h e o r e m 4 For any self adjoint element H E C3'IAj, the gap edges of any open gap of (E(7))~el are Lipshitz continuous at every point of I. Similar but weaker results have already been obtained previously by Choi et al. [ChE1Yu], and by Avron et al. [AvSi] on the almost Mathieu model. They found HSlder continuity only. Here we get a stronger result. However the Lipshitz constant depends explicitely of the width of the gap considered, and it diverges whenever the width tends to zero. As we will see in section 4 below, there is no chance to get a better result because the gap edges have discontinuous derivative at each rational value of ~//27r. On the other hand, if a gap closes for some value of "y then generically with respect to H E C3,1jtI, we only get H61der continuity with exponent 1/2 near this point.
4
S t r u c t u r e of the R o t a t i o n Algebra .4i
In this section we will give without proofs, a description of the structure of the rotation algebra. The reader interested in the proofs will be refered to [BaBeF1]. Let us consider first the case "y = 0. The algebra 7'0 is then the convolution algebra associated to the group Z 2. Therefore by Fourier transform, one transforms it into the algebra of trigonometric polynomials with the pointwise multiplication. More precisely, if a E P0, we set :
ar
= ~
a ( m ) e ~(eml-Am2) 9
(77)
m6Z 2
This a trigonometric polynomial. The main properties of the Fourier transform are : (ab)r
A) = ac,(O, A)bcl(O, A) , a, b E Po ,
(78)
16
and (a*)r
A) : ar
A)* , a e 7)0 .
(79)
It follows t h a t for every (0, A) E T 2, the m a p a E Po ~ ar tation. Therefore : sup
la~l(0, A)l <
(0,A) E T 2
Ilallo
A) E C is a represen-
, a E Po 9
(80)
In particular, the Fourier transform a E Po ~ ar E C(T2), extends to fl~ as a *homomorphism. As a consequence of the Gelfand theorem we get the following result [BaBeF1] : Theorem
5 The Fourier transform a E Po ~ p h i s m f r o m ,40 to C(T2).
Let us now consider the case 7 = 2rcp/q each others. As we have seen in section 2, can assume t h a t 0 < p < q without loss of analysis to this case by introducing two q • uq = I = vq ,
ad E C(T2), extends as a *-isomor-
where p, q are positive integers prime to A z is isomorphic to Az+2~ , so t h a t one generality. One can extend the previous q unitary matrices u and v satisfying : (81)
uv = e2i~P/qvu .
The Fourier transform of a E P~ is then given by the following matrix valued function ar
=
~
(82)
a(m)e'(em,-Am~)w(m) ,
mEZ 2
where : w ( m ) = UmlVm2e -i~pmlm2/q .
(83)
We r e m a r k t h a t in this last expression, w ( m + q m ~) = w ( m ) , namely m is defined modulo q. An example of such pair of matrices is given by : 0 0
1 0
0 1
... ...
0 0
0 0
1
0
0
...
0
0
0
A
0
.--
0
0 ,
0 1
0 0
0 0
... ...
0 0
1 0
0 0
0 0
0 0
-.. .'.
Aq-~ 0
(84)
0 A q-1
where A = e 2~p/q. Actually, any such pair is unitarily equivalent to this l a t t e r one. Let us also set : (85)
w ' ( m l , m2) = w ( - m ~ , m 0 ,
to get the following characterization of J42~p/q [BaBeF1] : 6 The Fourier transform a E 7)2~p/q ~ acl C C(T 2) ~ Mq, extends as a * - i s o m o r p h i s m f r o m .A2,p/q to the subalgebra Ccov(T 2, q) of C(T 2) | Mq, the element of which being continuous functions aci f r o m T 2 into Mq satisfying the covarianee condition
Theorem
w ' ( m ) a d ( O , A ) w ' ( m ) - l = ad((0, A) + 2 ~ m p / q )
.
(86)
17 The main interest of this result is that it makes it possible to compute the spectrum of an element a E A2,~p/q. For indeed for every (0, A) E T 2, the map rr(0, A) : a 9 9A2,~p/q ~ acl(O, A) 9 Mq is a representation, and the family {Tr(0, A); (0, A) 9 T 2} is faithfull. Therefore, thanks to prop.3, denoting by e~(0, A), 1 < k < q, the eigenvalues of acl(O, A), the spectrum of a is : Sp(a) = Ul
(87)
Each set Im(ek) is called a "band". The computation of the eigenvalues can be done numerically by matrix diagonalization. In many important examples, such as the "Harper" model (see Fig. 1) given by : HHarper = U ~- U* + V -I- V* ,
(88)
it is possible to compute analytically the points in the 2-torus for which the band edges are reached. In these cases, the numerical computation of the spectrum requires to diagonalize only few matrices for each value of p/q [BaBeF1]. The theorems 1 and 5 can be rephrased by characterizing the set of (closed twosided) ideals of .A2,~p/q. If p = 0, any ideal J is given by the space of continuous functions on T 2 vanishing on some closed subset f~j of T 2. The map J ~ 12j is actually one-to-one. If p ~ 0, the same is true if we demand that ftj be invariant by the translations of period 27r/q in the 2-torus [BaBeF1]. However If ~//27r is irrational, we get the following result [Sla, BaBeF1] : T h e o r e m 7 If 7/27r is irrational the algebra .A~ is simple, namely there is no other ideal than {0} and ~ itself.
C o r o l l a r y 3 If 3,/27r is irrational every representation of the algebra A z is faithfuU . A nice proof of theorem 6 was provided by Slawny [Sla], and the reader will find it in [BaBeF1]. The corollary is an immediate consequence of that theorem, for if 7r is a representation of the algebra ~ its kernel is an ideal, namely it is either the algebra .A~, in which case 7r = 0, which is not possible since 7r(I) = 1, or it vanishes, namely 7r is falthfull. Thanks to this last result, we can choose any representation to produce explicit calculations. The three theorems of this section are sufficient to characterize the algebra ~4i for any compact subset I of R. For indeed thanks to (55) (prop. 2), J is an ideal of ~4x if and only if for any 3' E I, ~ ( J ) is an ideal of . ~ . The ideal structure is sufficient to characterize any C*-algebra B which is a homomorphic image of .AI.
5
Semiclassical a s y m p t o t i c s for the s p e c t r u m
In this section we will denote by H = H* a selfadjoint element of AI, and we intend to given a description of its spectrum when I is a small open interval around "), = 27rp/q. The same kind of results can be obtained for a unitary operator, and this will be left as an exercise to the reader.
]8 5.1
2D
lattice electrons in a magnetic
field
In this subsection, we will describe a physical situation where the rotation algebra enters as an essential tool. It will give us a different intuitive description of the problem which may be useful. Let us consider a two dimensional lattice, with lattice spacing ~, that we will identify with Z 2, on which charged particles like electrons or holes, are supposed to move. Their quantum states will be wave functions r = (r C g2(Z2). We suppose in addition that a uniform magnetic field B is applied on this lattice, perpendicularly to the plane of the lattice. Let A = (A1, A2) be the corresponding vector potential, namely a vector field on R 2 solution of the equation alA2-02A1 = B. In the Hilbert space g2(Z2), we consider the "magnetic translations" T1, T2 studied by Zak [Zak], associated to the two basis vectors el = (1,0) and e2 = (0, 1) of Z ~, namely the unitary operators defined by : T~r
: e 2 ' ~ f(m-~,)6dl'Ar m6
-- et~) ,
# : l, 2 ,
(89)
where e is the electric charge of the particle, h is the Planck constant and the integral in the phase factor is computed along the segment joining the sites m - e~ and m. These two operators satisfy the following commutation rule :
T1T2 = e2i'~r162176 ,
(90)
where r is the magnetic flux through the unit cell, and r = h/e is the flux quantum. Therefore these two operators generate a representation of the C*-algebra A~ where now
V = 2rrr162 = const.B ,
(91)
is proportional to the magnetic field. In practice, if the lattice is given by the positions of the ions of a metal, 6 is of the order of 1/~ so that even with the highest kind of magnetic field that can be produced in laboratories, namely B ~ 18 Teslas , we get 3,/2~r ~ 0.5 10 -4 which is fairly small, and shows that in this situation a semiclassical approximation will always be valid. However during the last ten years, networks with lattice spacings of the order of the micrometer have been built [PaChRa], leading to values of 3,/27r of the order of unity in magnetic fields not larger than 40 Gauss. This is why it has been necessary to go beyond the semiclassical regime. /,From the band theory of metals [MeAs], the conduction properties are given only by those electrons sitting in the conduction bands, namely with energies within an interval of order kO from the Fermi level, if O denotes here the temperature, and k the Boltzmann constant. If we assume for simplicity that there is only one such band, thanks to the so-called "Peierls substitution" [Pei], one can prove rigorously that the restriction H of the Hamiltonian to that band is given by a selfadjoint element of the C*-algebra generated by the two magnetic translations [BehEva]. If several bands have to be considered, the Hamiltonian will be represented by a matrix with entries in this algebra. For B = 0, the band Hamiltonian H is represented through its Fourier transform (see section 4) by a continuous flmction Hcl on T 2. In Solid State Physics, one usually uses the quasimomentum notation k = (kl, k2) instead of (0, A) to represent a point in this 2-torus. Thus we get the following correspondence : (T1)d
:
e 'kl ,
(T2)~l = e ~k~ , ifB = 0 .
(92)
19 The advantage of this latter notation is that it restores the symmetry between the two directions in the lattice, a natural fact in the present context, even though it does not look so natural in the kicked rotor problem. In this subsection, we will prefer the use of the quasimomentum notations instead of the action-angle ones. Let us now remark that the representation given by (89) is actually a very natural one from algebraic point of view. For indeed, thanks to subsection 2.2 (34&(35), the Hilbert space g2(z2) can be seen as the completion L2(.Av, "r) of the prehilbert space P~ endowed with the scalar product (alb) = T(a*b). Let r / b e the natural imbedding of P~ i n t o L2(u4q,T). Since r(a*a) < ][a*all~, r/can be extended to Av by continuity. Then let rrCNS be the representation of A~ on this Hilbert space given by the left multiplication, namely 7rcNs(a)r/(b) = ~?(ab) , a, b E ~
.
(93)
The name "GNS" refers to Gelfand-Naimark-Segal [Dix], who defined and studied this representation in a C*-algebra. Then we claim the following [BaBeF1]: T h e o r e m 8 The representation of A~ given by the magnetic translations in (89) is unitarily equivalent to the GNS representation relative to the trace of A v. This representation is faithfull for any values of 3". Actually, the representation given by the magnetic translations depends upon the choice of a vector potential. The GNS representation corresponds to the so-called "symmetric gauge", namely A = B ( - x 2 , xl). Every other gauge can be reached by a unitary transformation. 5.2
Low
field expansion
We now consider an interval I of the form I = [-e0, e0], for some ~0 > 0, and let H = H* belong to Jr1. We will describe a semiclassical expansion near a b o t t o m well. In order to do so, let us assume that Hcl admits a local minimum or a local m a x i m u m at k = k0. Moreover we will assume that this extremum is regular which is a generic property. More precisely, and without loss of generality we will assume : (H0) H = H* E differentiable with (H1) Hcl admits a (H2) The Hessian
CN(.A1) for N > 2, and all its Fourier coefficients are N-times respect to 3`. local minimum at k0 = (0,0) and with Hcl(0, 0) = 0. D2H~I(0,0) of Hcl at k = (0, 0) is a positive definite 2 • 2 matrix.
Our goal is to describe the spectrum of r/~(H) near the energies E close to Hcl(0, 0) ----0 for 3' E I. To describe the result, let us assume that H can be written as : r/~(H) = ~
h(m;3')W~(m).
(94)
meZ:
with h(m; 3")* = h ( - m ; 3'). Since H is smooth, one can check that this series converges absolutely in norm, so that this expansion is meaningfull. Let us introduce the following function : Hsel(k;3') =
Z mEZ 2
h(m;3') e~(mlk'+'~2k~) , k E T 2 , 3' E I ,
(95)
20 which coincides with Hr for 7 = 0. Then we get the following result [BehEva, BaBeF1]
9 Let H satisfy (HO), (H1)gJ(H2), and let Hscl be defined by (95). Then there are 6 > 0 and 0 < ~ <_ ~o such that if [3`1 <- ~, the set S p ( ~ ( H ) ) N (-6, +6) is contained in the union over n E N of the intervals J.(3`) = [E.(3`) - A(3`), E.(3`) + A(3`)] where E~(3`) admits a Taylor expansion in 7 up to the N th order of the form : Theorem
En(3`)
]3`1 ( n + 1)(detD2Hr
(0"~r
----
+3`\
(96)
+ ' ' " + O("/N) ,
03` ]7=o,k= 0
0 < A(3' ) _< const.13`l N' , for some N ' > N .
(97)
To understand more intuitively this result let us introduce a faithfull representation ~r of .A~ in a Hilbert space in which one can find two selfadjoint operators K1, /s such that [K2, K1] = isgn(3`), where sgn(x) denotes the sign of the real number x. That such a representation exists is a well known fact [BaBeF1], and is a consequence of the Weyl theorem on the canonical commutation relations. In this representation, one has : 7r(U) = e iI'ylI/~N1 , 7r(Y) = eiN1/'K2 , (98) Therefore we get from (94) : M r = 7r o r/~(H) =
~ h(m; 3`)eq~l(m~Kl+'~2K~) , 3, e I . mEZ 2
(99)
Let us expand this expression formally in powers of 13`]1/2 to obtain :
H7
=
1
3`oqTHscl(0;0) + ~I3`[O.O.H~,(O,O)K.K~ + 0 (13`13/2) ,
(100)
where we have used the Einstein convention on the repeated indices (here #, v E {1,2}). By a unitary transformation, the quadratic term can be transformed into w(K~ + 1s where w is the determinant of the Hessian matrix OuOvHscl(O , 0). W e recognize here the Hamiltonian of a harmonic oscillator. Actually, if we choose the representation corresponding to a 2D free electron in a uniform magnetic field, namely the Hilbert space is L2(R2), and Ku = eonst.(P, - eA,) for some physical constant, then this Hamiltonian is the Landau one, namely the Hamiltonian describing a free electron in a uniform magnetic field. For this reason, the energy levels E,~ are called the "Landau levels" and are equal to that order in 3` to w(n + 1/2), leading to the expression (96). The proof of this theorem can be found in [BaBeF1, BehEva]. The calculation of E,~ to the next order has been done in [RaBe:Alg], in the case for which O~H = O, and leads to (for 3,' > 0) :
En = 3`w(2n + 1)/2 + 3`2A2Hc,(0 ) (1 + (2n + 1) 2)/64 ....
3`2 [9(3n 2 + 3n + 1)16Hc,(0)l 2 + (3n 2 + 3n +
2)103Hc~(0)12]/288w + 0(3` 3)
,
(101)
21 where :
cO co --
cokI
.cO CO .co ? ~ 2 ' ~ = ~ + Z8-~2 ' A = C05.
(102)
These formulm have been checked numerically on several models. The calculation to the third order in powers of V havsbeen computed for the Harper model (see (88)) [RaBe:Alg] and gives for the minimum (Fig. 1) : E. = -4
+
(2n + 1) -
V2 [1 + (2n + 1) 2]
+
~/~ In 3 + (n + 1) 3] + O(0/4)
.
(103)
Another example of interest has been investigated in [BeKrSe] and concerns the nearest neighbour model on a triangular lattice with two fluxes (Fig. 2). The corresponding Hamiltonian is given by : H~ = 7'1 + T2 + T3 + T~* + T2* + T~ , with
TIT2T3 -- e ~2~r162176 ,
(104)
and (89). Here r represents the flux through the "up" triangles, while r 1 6 2 represents the flux through the "down" triangles. The corresponding classical counterpart is given by : V H+,,r = 2 cos kl + 2cos k2 + 2 cos(kl 4- ks 4- ~ - ~/) , (105) if we set 7 + = 27rr162 The minima and maxima occur at the points kl = ks = 21ra/3+V'/3 -- 0o, where ~ = - 1 , 0 , + 1. If 0, ~ 7r/2, one gets the following expression [BeKrSe] : E~,~ = 6 cos 0~ - 7vf3(2n + 1) cos 0~ + 7 sin 0~ + 72 [1 + (2n + 1) 2] cos 0~/8+
9.. + ~/2sin20~cosO~[3(2n + 1) + 5]/72 - V2/12 cos 0~ - V2x/~sin 0~(2n + 1)/6 , (106) giving rise to three bundles of Landau levels. For Vp ~ 0 two of these bundles are very close and actually intersect each other (Fig. 3) . The comparison between this formula and the exact spectrum obtained by matrix diagonalization is very good : they agree up to four digits for the coefficients of the power expansion in V [BeKrSe]. The assumptions (H0, HI, H2) concern the generic case, for which the extremum is regular. However, some non generic case has been observed. For example, in [Wil:Cri, BaKr], the case of a square lattice with second nearest neighbour has been studied. The corresponding Hamiltonian is : HWBK = TI + T; + T2 + T~ + t2 (T214- T?" + T~ + T~ *) .
(107)
For t2 < 1/4, the classical Hamiltonian has only one absolute minimum, like in the Harper case. At the value t2 = 1/4, this minimum bifurcates to give four degenerate minima for e > 1/4. At the bifurcation value, this minimum becomes fiat namely the Hessian actually vanishes identically, giving rise to a normal form like : 72 HWBK,~ = --3 + -~- (K~ + K~) + 0(74) 9
(108)
A Bohr-Sommerfeld quantization condition gives at the lowest order in 7 : 727r . En = - 3 + 4F(1/4)4 (2n + 1) 2 + O(V 4)
for n large ,
giving parabolic Landau levels, as can be observed in (Fig. 4).
(109)
22
5.3
E x p a n s i o n n e a r a r a t i o n a l field
The method outlined in the previous subsection for a low field expansion of the spectrum, can be extended to the expansion near a rational field, or also to the case of a matrix Hamiltonian, as can happen if several different bands contribute to the conduction. We will give here the method for the rational fields, leaving to the reader the case of a matrix Hamiltonian as an exercise. We consider now an interval I = [2rp/q - ~o, 27r + r and H = H* 9 .AI. Using the matrices u, v given in section 4 (84), letting Uz, V~ be the generators of the algebra .Az for 17] <- e0, we consider the elements U', V ~ in A~ | Mq defined by : U' = U~ |
u , v ' = V~ |
v .
(110)
They are unitary and satisfy the commutation rule :
U'V' = e~('~+2~v/q)V'U' ,
(111)
showing that they generate in A~ | Mq a subalgebra ,-isomorphic to .A~T27rp/q. Letting 7 vary in I(0) = I-co, +e0], we get a *-isomorphism between .AI and a closed subalgebra of .AI(0) | Mq. Assuming that H is smooth enough, one can expand it as :
l]7+2.p/q(S )
:
H7 =
~
h(m;7)W~(m) @ w(m) , 7 C I(0) ,
(112)
mEZ 2
and this series converges in norm. So we are left with the same problem as in 5.2, with now matrix Hamiltonians instead. Following the same scheme, the classical counterpart is the matrix valued function : g~cl(k;7) =
~
h(m;7)ei(mlkl+'~k2)w(m) , k 9 W2 , 7 9 I(0) .
(113)
mEZ 2
As we already indicated, we must first diagonalize this matrix at "y = 0, giving q real eigenvalues (since H is selfadjoint), el(k), e 2 ( k ) , . . . , eq(k), and therefore q bands B1, B 2 , . . . , Bq, namely the set of values of the e~(k)'s as k varies in the 2-torus. Since H is selfadjoint, it is always possible to choose the es(k)'s smooth. We will also denote by P1 (k), P 2 ( k ) , . . . , Pq(k) the corresponding eigenprojections; they are also smooth with respect to k. We will now assume the following : (Hq0) H = H* 9 CN(AI) for some N > 2, and all coefficients in the expansion (113) are N-times continuously differentiable with respect to % (Hql) The eigenvalue e~ admits a minimum at k = 0, and e~(0) = 0. Moreover, no other eigenvalue of H~r 0) coincides with e~(0) = 0. (Hq2) The minimum of ej is regular, namely the Hessian matrix 0,0ve~(0) is positive definite. Then we get the following result [Bel:Eva, BaBeF1] : T h e o r e m 10 Let H satisfy (HqO), (Hql), (Hq2). Then there are 6 > 0 and 0 < ~ < ~o such that if ]71 < e, the set Sp(~+2~v/q(H)) n (-6, +6) contains a subset Es which is itself contained in the union over n e N of the interval J~,j(7) = [E~,~(7) -
23 A(7), E~,~(7) + A(3`)] where E~,i(3`) admits a Taylor expansion in 3" up to the N th order of the form :
E,~,A3`) =
13'1(~+ 1/2) (det D2ej(O)) 1/2 + 3" \(OeJ O~/ I] 7=o,k:O + 3`Eaw + O(72),
(114)
where 0 < A(3') < const.13"[N', for some N' > N, where ERw is the "RammalWilkinson" term given by the following expression : ERw
=
-
,
(115)
The strategy used to prove this theorem is based upon the so-called "Schur complement formula". Let H = H* be a selfadjoint operator acting on a Hibert space of the form 7-I = 7) @ Q. Let P, Q be the orthogonal projections on each subspace of that decomposition and let D be a partial isometry from 7"/to P such that DD* = Iv and D*D = P. We define on P the family of operators :
He~(z) = DHD* + DHQ(zI - Q H Q ) - I Q H D * ,
(116)
whenever z is a complex number which does not belong to the spectrum of QHQ. Then it is possible to show that z E Sp(H) - Sp(QHQ) if and only if z E Sp(HefF(z)). Moreover E is an eigenvalue of H not in Sp(QHQ) if and only if E is an eigenvalue of He~(E). We then denote by P -- I - Q the projection I | P~(0) of .AI(0) @ Mq. For (k;3') ~ (0,0), it follows that there is a small neighbourhood O of ej(O) such that if z E O, z ~ Sp(QHscl(k; 3`)Q). Since the eigenvalue e~(k) is simple for k ~ 0, the projector P3(k) is one dimensional for k ~ 0, and therefore there exists a partial isometry D : C q ~ C such that /)/)* = I, and D*/) = Pj(O). If D is the partial isometry I | let us introduce the effective Hamiltonian :
hi(z) = DtI~D* + DH.~Q (zI - QH.~Q)-' QH.yD* .
(117)
By construction this is an analytic family of elements in AI(0) now. We can therefore analyze it by the method developed in 5.2, and will give rise to a bundle of Landau sublevels En,j(z) near the lower edge of the band Bj. The corresponding part of the spectrum of H~ near e3(0) = 0, will then be given by solving the implicit equation E = E,~j(E). The solution can be computed explicitely order by order in powers of 3`, thanks to the hypothesis made on e~. The Rammal-Wilkinson term comes from the first order contribution of the second term in (117). It reflects the fact that the matrices H~cl(k; 3`) do not mutually commute for various values of k in general, namely it reflects the existence of a curvature in the fiber bundle over the 2-torus defined by Pj(k). The calculation of this term can be found in [RaBe:Alg, BeKrSe].
5.4
Qualitative analysis of the spectrum
Let us now comment on the formulae (114)&(115). Due to the absolute value of 3' appearing in the first term of (114), the right and left derivatives of the band edge with respect to 3` are different, showing that the band edges eventhough continuous
24
functions of 7 by the theorem 3, have nevertheless a discontinuous first derivative at each rational point. On the other hand, even if O~H = 0 the Rammal-Wilkinson term may not vanish. This is the case for instance in the Harper model for p/q = 1/3 (see Fig. 1). We can see the effect of this term by the fact that the left and right derivative of the band edge are not symmetric around 7 = 27rp/q. The difference between them reveals the occurrence of curvature effects. On the other hand one can recognize whether the band edge is a maximum or a mini at the slope of the Landau sublevels emerging away from 7 = 27rp/q. For most values of p/q, all bands are separated by gaps. However, many non generic situation can be observed on examples. (i) Two bands may overlap without touching each other. Then, each minimum or each maximum of the corresponding band will reveal itself by the occurence of a bundle of Landau levels emerging on both sides of 7 = 27rp/q (see Fig. 5), and given by the formula (114)&(115). (ii) two bands Bj, By, with or without overlap, may touch each other. In this case, generically they will touch on a conical point (see Fig. 6). This situation leads to a different canonical form. For indeed the previous analysis can be extended by replacing the projector Pj(0) by P~(0) + Pj,(0). Then the effective Hamiltonian becomes a 2 x 2 matrix unitarily equivalent to the Dirac operator [HeSj:Har2, RaBe:Alg] : HDirac = 17[ 1/2
I g l ~iK2 -
-
0
+ 0(7) .
(118)
This case will give "Dirac levels" which are parabolic namely : E ~ = +const.lnT] 1/2 , n G N ,
(119)
which is for instance what happens in the Harper model at E = 0 and p/q = 1/2 (see Fig. 1). This formula must usually be corrected by a P~mmal-Wilkinson term, giving a slope to the sublevel n -- 0. This is what happens in the WBK-model (108), at
p/q = 1/2 (see Fig. 7). (iii) Two bands can also touch with a contact of order 2. There is another example proposed by M.Wilkinson [Wil:Cri] and studied in details by Barelli and Fleckinger [BaF1], which is the following :
Hw : T1 + T2 + t3 (T21T2e-{~ + T[2T2e {~ + TiT~e -i~ + T,T;2e {') + b.c..
(120)
At E = O,p/q = 1/2, we do get two families of Landau sublevels on either side of p/q = 1/2, corresponding to the bottom wells of the two bands. The generic parabolic touching can be seen on Fig. 8. (iv) A maximum or a minimum can also be reached on a curve. This has been observed in the W B K model at p/q = 1/2. This case has been investigated in details by Helffer and SjSstrand [HeSj:Har3], who remarked that the "subprincipal symbol" may break this degeneracy and create what they have called "miniwells", namely local extremas with deepness of order 0(7). Such an example has never been investigated numerically, but there are indications that such a phenomenon should occur on the W B K model.
25 At last we must point out the occurrence of tunneling effect. For indeed, the classical model gives a Hamiltonian on the phase space given by a 2-torus. This is equivalent to choosing R 2 instead, but requiring that the Hamiltonian be periodic in both directions. This will be called the "extended picture". In this picture, each local extremum is repeated periodically, giving rise to an exact degeneracy. Therefore a tunneling effect should occur between the corresponding wells, ending into a broadening of the Landau levels or sublevels. The width of this broadening can be computed by the WKB method, and will give rise to terms of order O(exp (-S/3`)) where S is some constant equal to the real part of the tunneling action between two neighbouring wells. This effect has been studied in great details in the Harper model by Helffer & SjSstrand [HeSj:Harl, HeSj:Har2, HeSj:Har3], and for the corresponding model on a honeycomb or triangular lattice by Kerdelhud [Ker]. By evaluating precisely the tunneling matrix representing the effective Hamiltonian restricted to each of the Landau sublevel, they could prove that it is again represented by a ttamiltonian with nearest neighbour interactions, having the symmetry of the original lattice (e.g. a Harper model for a square lattice), with a small correction. Therefore, each Landau sublevel is itself decomposed into subbands, and this explain the occurence of the fractal structure. This tunneling effect has also been exhibited in a spectacular example by Barelli & Kreft [BaKr], in the WBK model for t2 > 1/4 and 3` ~ 0. As we already said, after the bifurcation the unique minimum splits into four degenerate minima surrounding one maximum. Since these four wells are very close to each other in each unit cell of the extended phase space, compare to the distance between cells, the tunneling effect between these four wells within the unit cell is likely to dominate over the other sources of tunneling. Each well gives rise to its own bunch of Landau levels, but the splitting due to the tunneling will separate them. It turns out that the tunneling action in this case is not purely imaginary, so that the Landau levels can be represented by if n c N, t2 = 1/2 and i = 1,2,3,4 : 3 2n + 1) + 0(3 ,2) E,~,i = E,~(3`) + dE,~,i(3') , E~(3`) = - 3 + 53`(
(121)
where the splitting is given by [BaKr] :
dEn,i = 3`3e-Im(S2)/~ cos(Re(S4)/43` + 7r/4) + O(e -s'/~) , 7r
(122)
where $2 represents the action lAB kldk2 for a path A B in the complex energy surface Ha(k) = E~(3`) joining two neighbouring wells A and B, while $4 is the tunneling action for a closed path in the same energy surface going through the four wells once. Moreover, S ~ is some action larger than $2. Even though there are usually many non homotopic such paths in this complex energy surface, only the "shortest" ones (in terms of the corresponding action integral) do contribute to this order. In this formula the width of the splitting is controlled by Im(S2) which gives an exponentially small term. But the occurence of a non zero real part produces a nice braiding between these four sublevels as can be seen in (Fig. 9). In a recent work, Barelli and Fleckinger exhibited a braiding of Dirac sublevels near the half flux ( see Fig. 10)[BaF11.
26
6
Elementary
6.1
The
Properties
Furstenberg
of the
Kicked
Rotor
Algebra
As we have seen the Floquet operator for the kicked rotor cannot be seen as an element of the rotation algebra. This is because the kinetic part is not a continuous function of U and V. However, we have seen that it defines a *-automorphism of the rotation algebra. To deal with that we have two choices. The first is to ignore the Floquet operator itself and to stick with its action on the non commutative torus. This is fine as long as we are interested only in the evolution of observables. However, in many occasions do physicists need to know more on the spectrum of the Floquet operator itself, the so-called "quasi-energy" spectrum. One of its most important property is the "dynamical localization", a phenomenon similar to the Anderson localization in Solid State Physics of disordered metals [FiGrPr]. In order to deal with this latter problem, we can simply enlarge our algebra by brute force, adding the missing unitary F0 equal to the kinetic energy defined in section 1 (12) by : Fo = e -~A2 /2"~ . (123) As we have seen in section 1 (19) this operator satisfies the following commutation rules (i) F o V F o ' =: V (ii) F o U F o I = U V - l e -'~/2 . (124) As before we will denote by BI the C*-algebra generated by the polynomials in U, V, Fo with coefficients in the set of continuous functions of 7 in I. This algebra can be rigorously constructed along the line developed in section 2. However one can use the general method of C*-algebras, namely the notion of crossed-product [Ped], to construct it. One can indeed see B1 in two ways : (i)-the first one comes from the previous definition, namely F0 acts on the rotation algebra Jti by mean of the *-automorphism ~o(a) = Foa~o 1
a e A .
(125)
Therefore BI can be seen as the crossed product ,41 • Z of the rotation algebra AI by the Z-action defined by ~0. Using Weyl's operators defined in section 2 (32), we notice that /3o (W(m)) = W(Gm) , m e Z2 , (126) provided G is the element of S L ( 2 , R) given by : G=[
-
11 0
]
(127)
1j "
(ii)-the second one consists in considering first the subalgebra generated by functions in g(I), together with the operators V and F0. This is an abelian C*-algebra isomorphic to g ( I x T~). This isomorphism associates to V and Fo respectively the functions fv(7, x, y) = ei~ and f F o ( % X , y ) = e% Actually, the inner automorphism associated to U leaves this algebra invariant. This is because the commutation rules (19) can be written as (i)
U V U -1 = ei~V ,
(ii)
UFo U-1 = e~'Y/2VFo .
(128)
27 In other words, for f ~ C ( I • T2), we get : U f U -1 - f o r
(129)
where r is the "Furstenberg" map acting on I • T 2 as : r
= (%x + %y + x + 7/2) ,
( % x , y ) E I • W2 .
(130)
This m a p was used by Fhrstenberg to study the ergodic properties of diophantine approximations in number theory. Thus/3i can be seen as the crossed product C ( I • T 2) xr Z by the Fhrstenberg map. This is why we propose to call this algebra the "Furstenberg algebra". We see that r leaves each fiber {7} • T2 invariant and we will denote by r the corresponding restriction. It is well-known that whenever 7/2~ is irrational, r is a minimal diffeomorphism [CoFoSi].
6.2
Calculus
o n Bx
As for Az, a calculus can be defined on the Furstenberg algebra. Since the trace on .4i is/3o-invariant, it defines a trace on the crossed product in a natural way. It is actually defined by the formula : T (W(m)Fg)
=
(~m,0.(~/,0 ,
m E Z2 , 1 C Z.
(131)
Since we have defined originally (cf. section 1) U, V, F0 in term of action-angle variables in the classical case, one also gets an angle average (.) namely : (W(rn)F0 l)
=
~ml,oVm2F~ ,
if
m = (ml,m2) C Z ~ , 1 E Z .
(132)
Thus, if a E 131, (a) E C(I x T2), and this average satisfies the properties described in (32). In much the same way, a differential structure can be defined. The derivation COo can be extended immediately to/3i by : OoU = i U ,
COoV = 0 ,
00F0 = 0 .
(133)
We notice however that COAcannot be extended as a derivation in B/ because COAFo would be unbounded, namely outside/3i. But a new derivation COyappears defined by COuU = 0 ,
COyV = 0 ,
COuFo= iFo .
(134)
Both 0o and COyare the infinitesimal generators of the following two-parameter group of *-automorphisms : P0,u (W(m)F0 l)
ei(ml~
(135)
which leaves the trace invariant. At last, the definition of a Poisson bracket is not obvious because for 3 / = 0 the algebra B0 is no longer commutative. Even though it is in principle possible to define such an object, we will not use it, and we skip this part of the calculus.
28 6.3
Representations
and
structure
of BI
Among the representations of/3I, we will select one family of special interest in view of the original definition of the kicked rotor in the physical Hilbert space L~(T) given in section 1. It is actually simpler to work in the momentum space, namely in ~ ( Z ) where the integers of the chain Z are simply the quantum numbers for the angular momentum. This family {%,~,u; (7,x,y) E I • T 2} is indexed by points in I • T 2 and acts on g2(Z) as follows : (i)
(ii) (iii) (iv)
(Tr%x,y(f)r (r~,x,y(U)r (~r~,x,y(V)r (Tr.~,x,y(F0)r
(n) = f(7)r , (n) = r - 1) , (n) = e~(x-n*)r , = ei(y-nx+n='/2)r
f C C(I) (136) ,
if
~b E g2(Z) .
Comparing with the equation (12) & (13), 7 appears as an effective Planck constant, x as an effective magnetic field, and y as a phase factor entering in the definition of F0.
With these definitions, the following result can be easily proved by standard technics
P r o p o s i t i o n 7 1)-The family {~r~#,y; (7, x,y) E I x T 2} is faithfull. In particular, the norm of a E/3I is given by : ]lalI
--
sup
sup
~EI (z,y)~T 2
II=~,~,~(a)ll
9
(137)
2)-The map ( 7 , x , y ) E I x T 2 ~-+ %,z,y(a) is strongly continuous for all a E BI. 3)-For ~/ E I, the trace is given by : r~(a) =
fT dxdy 2 4r
(138)
Moreover if "y/27r is irrational, we get 9
1
= 1,m 27- T
L,),
uniformly in (x,y) E T a. 4)-If T is the translation operator in g2(Z), namely if (Tr = r - 1) for r E g2(Z), then :
TTr~,,.y(a)T-' = ~r~(~,~,~)(a), 5)-If N is the position operator in s have : %,x,y(Oea) = i[N, rr~,~,y(a)] ,
a ~ I3i, ('r,x,y) e I x T 2 . defined by (Nr
= nr
(139)
, r E g2(Z), we
0 7r~#,y(Oya)= ~yr%x,y(a) .
(140)
Thanks to this result the elements of BI can be described as follows. For a E 131, we set :
a('~, x, y; n) = (01rr~,~,y(a)ln) .
(141)
29 This is a continuous function on I • T 2 • Z converging to zero at infinity. In terms of such functions the product and the * in BI can be expressed as follows : ab(7, x , y ; n ) = y ~ a ( 7 , x , y ; l ) b ( 7 , x
- 17,y - lx + 127/2;n - l) ,
(142)
IEZ
a* (7, x, y; n) = a(7, x - nT, y - n x + n27/2; - n ) * ,
(143)
for a, b E BI. Moreover, the representation 7%,x,y is given by : (Try,x,y(a)r (n) = ~ a ( 7 ,
x-nT, y-nx+n27/2;l-n)r
,
r 9 g2(Z) .
(144)
IEZ
In particular, due to the faithfullness of this family, a = 0 if and only if the function a(7, x, y; n) vanishes identically. If we denote by By the algebra Bl for I = {7}, the following theorem characterizes its structure : T h e o r e m 11 1)-If 7/2~r is irrational, 13y is simple. In particular, every non zero representation is faithfull. 2)-For 7 = O, the algebra Bo is isomorphic to the universal rotation algebra A . 3 ) - I f 7 = 27rp/q where p, q are positive integers p r i m e to each others, B2,p/q is isomorphic to the sub C*-algebra of Mq(C) | Bo generated by : ~]=u|
l/=v|
F0 = w |
(145)
where Uy, Vy, Fo,y are the generators of B~, and u, v, w are three unitary q x q matrices fulfilling the following conditions : u q = v q = w 2q = I , u v u -1 = e2i'P/qv ,
(146)
u w u -1 = ei'P/qvw ,
vw = wv .
(147)
P r o o f : 1)-For 7/27r irrational, the Furstenberg m a p r : (x, y) E T 2 ~-+ (x + 7, Y + x + 3'/2) E T 2 is a minimal diffeomorphism of the torus [CoFoSi]. Therefore, the crossed p r o d u c t By = C(T 2) x r Z is simple [HiSk]. 2)-For 7 = 0 the commutation rules become : UV
= VU
,
VFo
~- F o V
UFoU -1 = VFo .
,
(148)
These rules are precisely the ones defining the universal rotation algebra .4 if we identify V with the m a p 3' c T ~-~ e ~y C C (cf. section 2). 3)-If one chooses the matrices u, v as in (84), the matrix w becomes : 1 0
0 A'
0 0
... -.-
0 0
0 0 (149)
W =
0 0 where ~'
=
e iTrp/q.
0 0
0 0
... .-.
~,(q_~)2 0 0 )r
30 It is easy to check that U, I), F0 satisfy the commutation rules for the algebra B2~p/q. Hence they define a *-homomorphism p from B2,p/q into Mq(C) | Bo. 4)-To achieve our result it is sufficient to prove that p is one-to-one. For (x, y) C T 2, let ~'~,~ be the representation of Mq(C) | Bo given by id | 7ro,~,y acting on C q | g2(Z). Any a E Bo can be seen as a function on T 2 x Z as (see (141)), and for r E C q | andAEMqweget : q-1
(150)
(n) = ~ ~ A3,j,a(x, y - nx;1 - n)r
[~'~,~(A | a)r
j=0 IEZ q-1 Let {ej}j= o be the canonical basis of C q with the convention that e~+q = ej, and let
{Sn; n E Z} be the canonical basis of g2(Z). We set : IJ, n) = ej | 5,~ .
(151)
(j, 01#,,y(A | a)IJ', l) = A j d , a(x, y, ;l) .
(152)
Then : It is not difficult to check that if now b c B2~p/q and r E C q N g2(Z) we get : [5,,y (p(b))r
(n) = ~ b(x - 2rjp/q, y - nx + j2~p/q; l)r
+ l) ,
(153)
lEZ
where j + l is defined modulo q. It is actually sufficient to check this formula on the generators Ue~p/q, V2,p/q, Fo,~p/q since #x,y and p are .-homomorphisms. In particular (0, Ol~x,y(p(b) )Il, l) = b(z, y; l ) .
(154)
Thus p(b) = 0 if and only if b(x, y; l) = 0 for any (x, y; l), namely b = 0. Hence p is one-to-one. Using the same strategy we can easily get : C o r o l l a r y 4 1)-for "7 c R the algebra B2~p/q+~ is isomorphic to the subalgebra of Mq @ B 7 generated by u | U~, v | V~, w | Fo,~.
2)-for "7, 7' C R the algebra B~+~, is isomorphic to the subalgebra of B~ | by U~ | U~,, V~ | V~,,Fo,~ | Fo,~,. 6.4
Algebraic
Properties
of the
z, generated
Rotor
Kicked
In section 1 we have expressed the Floquet operator of the kicked rotor as : -1
~ e-iA2/27e-iKc~176
if1
(155)
where 7 = l i T / I is the effective Plan& constant and x = - # B T is the effective magnetic field. Moreover in the momentum space representation, A = "/N - x if N is the position operator (see prop. 7). Using the previous algebraic framework, it follows that : FK,-r,x = 7r,y,~,o(rK) , (156) with : FK
eiK(U+U-1)/25Fo ,
(157)
31
where "~ : 3' E I ~ 3' c R. In this special case we notice the following property :
(158)
~r~,x,~(F~:) - e'~FK,~,~ ,
so that one can set y = 0 without loss of generality. It follows that FK belongs to/3i for any compact set I in the real line not containing the origin. Our first set of results concerns the spectrum as a set of this Floquet operator. Since it is unitary its spectrum is necessarily contained in the unit circle S~. Actually the following results are still valid if we replace cos(O) = (U + U-1)/2 by any real valued 27r-periodic continuous function g(O) on the real line. T h e o r e m 12 1)-For any 3" ~ O, the spectrum of FK,7 = ~ ( F K ) is the full circle.
2)-If 3"/27r is irrational, the spectrum of FK,~,~ is the full circle for any x e T. 3)-If "y/2~ is rational, but x/2~r is irrational, the spectrum of FK,~,~ is the full circle. 4)-If 3"/2~r and x/2~r are rational, FK,~,~ admits a band spectrum. P r o o f : 1)-Since the family {n~,~,~; (x, y) e T 2} of representations of/3~ is faithfull, we get : SPB,(FK,~t) = U(x,y)Ew2eiUSp(FK,.~,x) 9 (159) Taking the union over y clearly gives the full circle. 2)-If 7/27r is irrational, B~ is simple. Thus each of the ~r~,~,y's is faithfull, in particular S1
=
SpB~(Fg,~ ) = Sp(FK,~,~) ,
Vx ~ T ~ .
(160)
n C Z
(161)
Vn e Z .
(162)
3)-If 3' = 2top~q, the covarianee condition (145) gives : T"%,~,~(Fu)T-"q
= ~,~,~+.~x(FK)
,
In particular Sp (Tr~,z,y(FK)) = Sp (Tr~,~,u+~(FK)) ,
If in addition x/2~r is irrational, given any y' E T we can find a sequence (nl) of integers such that yP - y = liml~r nlqx mod 27r. By the strong continuity of ~r~,~,y with respect to y, it follows that : Sp (Tr~,,,r
C Sp (Tr~,x,y(Fg)) .
(163)
Since y, y~ are arbitrary, the same result holds after exchanging them. In particular for any y we have : Sp (Tr~,~,u(Ft()) = e~YSp (n~,~,o(Fg)) = Sp (7c~,x,o(Fg)) , showing the result. 4)-If 3' = 2rp/q and x = 2rr/s, the periodic. By the Bloch theorem we get using the corollary 4 that the algebra Mq | Ms | r generated by u | u' | u ~, v ~) are the q x q matrices (resp. s
(164)
covariance property shows that rc~,x,o(Fg) is a band spectrum. Actually one can easily see, 7r~,x,0(/3x) is isomorphic to the subalgbera of eik, v | e ~ | 1, w | v ~| 1 where u, v, w (resp. x s) defined in the theorem 12, and k is the
32 quasimomentum. Here we used the fact that r%,~,O(FK)Q = I if Q = 2(q v s). This gives the band spectrum by diagonalizing the finite dimensional matrices and varying k. The next set of results concerns the density of states. Let A be an interval in the unit circle, namely the image by w ~-+ e i~ of an interval of the real line. Let us also call gi the restriction to the finite set [-L, L] of g(O). This is a self adjoint matrix of dimension 2L + 1. Let also ~(L) 0,"/,x be the restriction of F0,~,x to the same interval. Because it is diagonal it is a unitary (2L + 1) x (2L + 1) matrix. Then we set FK(L) 1:2(L) piKgL(O)/"/ Again, this is a unitary matrix of dimension 2L + 1. Let then ,'y,x = "tO,~',xv nn(A) be the number of eigenvalues of this matrix contained in A. As L ~-~ c~, this number increases like O(L), so that we can define the Integrated Density of States (IDS) as the following limit, if it exists : nL(A)
''
2L+l
(165)
The first important property is the "Shubin formula" [Bel:Gap] P r o p o s i t i o n 8 If'y/2~r is irrational, the limit defining the IDS exists uniformly with respect to (x, y) E T 2 and is independent of (x,y). Moreover it is equal to : Aft(A) = % (x~(FK)) ,
(Shubin's Formula) .
(166)
where Xzx is the characteristic function of the interval A. The proof of this proposition can be found in [Bel:Kth, Bel:Gap, BeBoGh] for self adjoint operators. It can be easily adapt for the Floquet operator. We notice that the limit is reached uniformly with respect to (x, y). This is because the Furstenberg map is minimal and not only ergodic. Another remark is that the eigenprojection xA(FK) does not belong in general to the algebra B~. However, it belongs to the von Neumann algebra Lcc(Bz, %), namely the weak closure of B~ in the CNS representation associated to the trace. Thus the Shubin formula is meaningfull. Thanks to the Shubin formula, the IDS can be written as : Aft(A) = f a dAft(E) ,
(167)
where dA/'.r is a probability measure on the torus T (which we identify with the unit circle) called the Density of States (DOS). We can actually compute the DOS namely
P r o p o s i t i o n 9 If "y/27r is irrational, for any continuous real valued 2rr-periodic function g on the real line, the DOS of the kicked rotor is equal to the normalized Lebesgue measure on the torus, namely :
dE
KAf~(E)-- 27r
(168)
33 P r o o f : The Shubin formula implies that the DOS is the unique probability measure on the torus such that :
fw d'N"Y(E)ei'~E =
T7 (F~) ,
n E Z.
(169)
We claim that T7 (F~) = 0 unless n = 0 which will prove the result. For indeed, the trace is invariant by the,automorphism group/~0,k. On the other hand, we have :
~o,k (FK) = e~kFK.
e"~kF~
It follows that ~3o,k(F~) =
(170)
showing that
T~ (F~) (e '"k - 1) = 0 .
(171)
Our last result concerns the algebraic way of writing the kinetic energy. In order to study numerically the spectral properties of the kicked rotor, several physicists [CaChIzFo] have iatroduced the averaged kinetic energy. Giving an initial state r c t?2(Z), it is given by (see (9) & (11)) : s
=
(r162
L2
,
t e Z,
(172)
where L is the angular momentum, I is the moment of inertia and F the Floquet operator. Thanks to the definition of the position operator N (see Prop.7) and introducing the period T of the kicks, one can write it as :
gc(t) = 2~(r162
.
(173)
In order to keep only dimensionless quantities, we will redefine this kinetic energy by forgetting the prefactor I/2T 2. Moreover physicists usually choose an initial state localized on one value of the initial angular momentum. Using the covariance condition, it is always possible to choose r = 10) by changing the value of (x, y) if necessary. This why we will rather define the mean kinetic energy in the following way : s
= 72 (01F~ ~,,xN2F~ t, ,xlO)
.
(174)
We notice that varying y will not change this definition. Using now (146), it follows immediately that if IAI 2 = AA* :
C~,~(t) = 72(01~r~,~,u(10oFk,~,~1210) . The choice of the initial average over the position generic properties of the to taking the trace. This
(175)
value of the angular momentum being arbitrary, we may of the initial state in momentum space, in order to get the system. This is equivalent to averaging over (x, y), namely why we will also consider the quantity :
C~(t) = 7%~ (1OoFk,~,J 2) 9
(176)
34
7 7.1
Localization and D y n a m i c a l Localization Anderson's
Localization
The localization phenomena was predicted in 1958 by Anderson [And] for conduction electrons in a disordered metal. The main idea underlying this effect is that the electronic wave in an infinite medium is reflected by the obstacles (ions, defects,etc,...). If the medium is a perfect crystal, the total reflection coefficient may not be equal to one due to constructive interference effects and allows the wave to travel freely towards the boundary. This happens whenever a Bragg condition is fulfilled, for special values of the total energy of the traveling particle, defining a band spectrum. This is the essence of Bloch theory for perfect metals. In such a case, the conductivity is infinite, if one neglects the influence of phonons and of the electron-electron interaction. If the medium is not periodic but quasiperiodic, such as quasicrystals, one may have also free Bloch waves if the quasiperiodic potential describing the influence of the ions on the travelling particle is not too strong [DiSi, BeLiTe, ChDe, BeIoScTe, BenSir]. However, in a disordered medium, the Bragg condition is unlikely, namely destructive interferences may force the electronic wave to vanish at infinity. Thus, the electonic wave is trapped in defects : in other words it is localized in a bounded region. Anderson proposed a tight binding model of such medium and could predict that 1-dimensional disordered chains always exhibit localization [Pas, Cyc]. Later on [AbAnLiRa] it was argued that in 2D the same effect occurs. But in higher dimension, localization holds only for strong disorder or at the band edges [FrSp, FrMaScSp]. Then if the disorder is not too strong, Ohm's law holds, leading to a finite conductivity, even if we ignore the phonons and the electron-electron interaction. The Anderson model is extremely simple but contains most of the properties necessary to describe such a medium. In a tight binding representation, the electronic states can be represented as elements of the Hilbert space g2(zD), if the crystal we start from is the D-dimensional lattice Z D. If there is no disorder, in the one electron approximation, the conduction electrons are approximately described by the free Laplacean •D namely if r ~ ~2(zD) :
A~r
~
r
(177)
In-,~q-1 where t is the "hopping" parameter which measures the energy required for an electron to hop from one site to the next one. The energy spectrum is then given by the band
[-2Dt, 2Dt]. Adding one defect in the crystal can be described by adding to the previous Laplacean a local potential in the form of a sequence Vd~f~ct = (Vd~/~ct(n); n C zD), as was shown in 1949 by Slater. To get a homogeneous distribution of defects it is therefore sufficient to replace Vd~I~r by a homogeneous sequence V. To take into account the randomness of the defect distribution we will assume that the values V(n) of this potential at each site are identically distributed random variables. Even though we expect some correlation between them in realistic systems, at least at short distances, Anderson proposed to consider the simplest case for which they are independent and uniformly distributed in an interval [-W, W 1. Then W is a measure of the disorder strength. Let s be the corresponding probability space (in this example, 9t = I-W, W] zD) and let P be the corresponding probability measure (in this
35 example, P = (~,~ez D d V ( n ) / 2 W ) . The potential becomes a function of the random variable ca E f t so that the Anderson Hamiltonian can be written as :
H~ = / % + w~.
(178)
The probability space (f~, P) can be seen as the configuration space for the disorder. The translation invariance of the original lattice is not completely lost. For indeed, translating this new system is equivalent to translate the distribution of defects back. More precisely, there is a measure preserving action of the translation group on f~. For the Anderson model this action is given by Trcan = ca,~-r. If we denote by T ( r ) the translation by r c Z D in the Hilbert space, namely for r c t~(zD), T ( r ) r = r r), we get the following "covariance condition": T ( r ) H ~ T ( r ) -1 = HTr~ .
(179)
We will complete this framework by adding two conditions. The first one is the ergodicity of the probability measure P. Thanks to Birkhoff's ergodic theorem, it expresses the fact that space averages coincide with P-average. In this way, P can be constructed in practice simply by taking space averages, an unambiguous process. The second one concerns the existence of a topology on f~ which makes it a compact Hausdorff space, and such that the P-measurable sets are generated as a a-algebra by the Borel sets, namely P is a Radon measure. In the Anderson model the product topology will do it. Actually an intrinsic definition of homogeneous system has been proposed in [BehKth, Bet:Gapl leading to the definition of a canonical topology on the disorder configuration space. For this topoloKy, the mapping w E f t ~ H~ is strongly continuous (in the resolvent sense whenever H~ is unbounded self adjoint). To summarize, homogeneous media, such as crystals, quasicrystals, glasses, amorphous, aperiodic or disordered systems, may be mathematically described by the following axioms. (D1)-The disorder configuration space is a compact Hausdorff topological space ~t endowed with a probability Radon measure P (D2)-Tbe translation group is a locally compact abelian group G acting in ft by mean of a continuous group of homeomorphisms ca ~ gca. The probability P is G-invariant and ergodic. (D3)-The quantum state space is a separable Hilbert space 7-i in which G acts through a projective unitary representation {T(g); g c G}. (D4)-The Hamiltonian is a strong-resolvent continuous family H = {H~; ca ~ ft} of self adjoint operators acting on ~ with a common G-invariant domain Z). (D5)-A covariance condition is satisfied, namely: T ( g ) H , , T ( g ) - ' = Hg~ .
(180)
In general we will prefer a projective unitary representation. For indeed there are concrete examples for which the translation group does not act as a true representation. This is the case for a crystal in a uniform magnetic field [Bel:Gap]. We have restricted ourself to abelian translation groups because no concrete useful example have been studied till now with non abelian groups. However, systems living on a Cayley tree admits a non abelian translation group which is usually a free group. We
36 can also include in G other symmetries like rotations, reflections, if necessary. This has never been investigated in detail yet, even though we believe that it should be useful: classification of defects in crystal may be related to such groups. The smallest observable algebra that can be of interest for physics, is the one constructed with the energy. In more concrete systems however, other observables like spins, may be relevant. For simplicity, we will consider the simplest case for which the only relevant observable is the energy. In a homogeneous medium, the choice of the origin is arbitrary, since the systems reproduces itself under translation. So that the physics of the system is described by any of the elements of the family H = {H~o;w C l-l} representing the energy. In order to avoid choosing arbitrarily one of them, we will include all of them. We then define a non commutative C*-algebra C*(H) as the smallest one in the space of bounded operators onT-/ containing the resolvent of each of the elements of H. In general, we do not know the structure of such an algebra. However for most concrete examples construct till now, namely by using the Schr5dinger operator for one electron systems [BehKth, Bel:Gap], like the Anderson model, this algebra is nothing but the crossed product g(ft) x G defined by the topological dynamical system (fl, G) describing the disorder configurations in the original medium. This algebra must be slightly modified if a uniform magnetic field is turned on. We will ignore this latter case here. Thanks to this framework, there is a very close analogy with aperiodic media in Solid State Physics and the dynamics of a kicked rotor. Even though the physical interpretation is very different, the C*-algebra used to describe the observables is also a crossed product. However, in the kicked rotor model, the lattice G is the quantized momentum space instead, and the space ~t admits a fairly different interpretation since the variable -y plays the role of an effective Planck constant and is related to the period of the kicks, x plays the role of a magnetic field, whereas y represents a generic translation in momentum space. We also notice that the ergodicity of the measure holds only if 7/2~r is a fixed irrational number. There is also a very close analogy with 2D-dimensional lattice electrons in a uniform magnetic field. We have already seen that the observable algebra is the rotation algebra AI which can also be seen as the crossed product g(I • T) •162Z if r : (% x) E I • T ~-* (% x + ~') E I • T. Then 7 plays the role of a dimensionless magnetic flux per plaquette, whereas x is a generic position of the origin in the xdirection of the lattice. Again, the ergodicity of the measure on gt = I • T holds only if I = {-y} where V/2~r is a fixed irrational. The main question now is whether this formal analogy between so different problems will produce phenomena similar to Anderson's localization. The common belief is that if H is a selfadjoint operator belonging to this algebra, with short range interactions, namely if it is smooth enough with respect to the differential structure that will be described in the next subsection, it will exhibit such phenomena at least if the dynamical system (~, G) is "sufficiently aperiodic". The precise meaning of "sufficiently aperiodic" is not completely understood yet. Several numerical studies have investigate this question, but they are far from having given a precise criterion yet [FiHuXX]. More precisely we define a 2-point function by C(g) = {FFg) - (F} ~, where F is a continuous function on ~t and Fg(w) = F(g-lw) while (.) is the ergodic average. If any 2-point function converges to zero fast enough as g ~ oo, the localization is expected to occur. This is certainly not the case for a periodic or an almost
37 periodic dynamics, describing for instance a perfect crystal with or whithout a uniform magnetic field. And indeed we do not expect in this case localization to occur. Still, a 1D model like the Almost Mathieu Hamiltonian [AuAn, ChDe, BeLiTe], has been proved to exhibit a metal-insulator transition at large coupling. But the Furstenberg map for instance, which satisfies this criterion, should give rise to localization. This is the basis of an argument by Fishman, Grempel and Prange [FiGrPr] predicting that localization occurs in the kicked rotor problem. The next problem therefore is to describe mathematically w!lat we expect to characterize the localization. One of the first criterion used by Anderson was connected to the time evolution of quantum states : if the time-average of the probability for the initial state to come back after time t is positive, then localization do occur. We will see later on, thanks to an early result of Pastur [Pas] that this criterion is related to the existence of a point spectrum for H~, P-almost surely. This is essentially why mathematicians describe localization in term of the existence of a point spectrum. It is related to the finiteness of the so called "inverse participation ratio" (see below). Another way consists in defining the localization length: roughly speaking it gives a measure of the diameter of the region where a typical eigenstate is localized. One of the main problems in dealing with the spectral property of the Hamiltonian, is that in many situations, this requires the choice of a fixed representation of the observable algebra. While in the Anderson model, this choice is quite natural, thanks to the description of the original disorders medium, in other models for which we would like to use the localization theory, it is not necessarily so. Two inequivalent representations of the same algebra may give different type of spectral measure for the same Hamiltonian. This happens for instance in the problem of Bloch electrons in a magnetic field. Therefore if this latter point of view were correct, localization would require to distinguish physically between different representations. However, the computation of the localization length requires a space average, in order to get a quantity insensitive to the specific configuration of the disorder, and therefore as we will see, it can be interpreted in a purely algebraic way. There is therefore an apparent contradiction between the two points of view. This is actually nothing but the usual opposition between the SchrSdinger and Heisenberg point of view in Quantum Mechanics. Our main purpose in this section is to show how to reconcile them, and to show that in some sense they are equivalent. Our last comment concerns the semiclassical limit. While this limit is meaningless in the Anderson problem, since the starting point is the band theory for perfect crystals, a fairly strong quantum theory, the kicked rotor problem gives a nice example where the semiclassical limit exists indeed together with a localization effect. It is therefore natural to consider what happens to the localization phenomena in this limit. The main discovery of Chirikov, Izrailev and Shepelyansky [ChIzSh] was to relate this limit to the diffusion constant in phase space of the classical kicked rotor. Even though this relation has not been proved to hold rigorously, many numerical studies show that it is probably correct at least under some unknown "generic condition". Therefore we have reached here one point of the so-called "quantum chaos". We will give only some pieces of this puzzle here.
38 7.2
The
Observable
Algebra
To avoid useless technical difficulties, we consider now the C*-algebra C(f~) x G where G = Z D . D will be called the dimension of the lattice. However, most of what will be described here can be extended to more general groups such a s R D for instance. As for the rotation or the Furstenberg algebra, we can develop a calculus as follows. Elements of C(f~) x Z D are continuous complex functions a ( w , n ) on the space f~ x Z D vanishing at infinity. To define this algebra properly, it is more convenient to start with the dense sUbalgebra G(f~ x Z D) of continuous functions on f~ x Z D with compact support, endowed with the following operations: ab(~; n) = ~
a(~; l)b(T-Iw; n - l) ,
(181)
IEZ D
a*(w; n) = a(T-~w; - n ) .
(182)
Since the functions a and b have compact support, the sum above is finite. Remarkable elements are given by : I(w;n) = 5~,0 ,
U ( r ) ( w ; n ) = 5n,-r ,
(183)
r C ZD .
The first one I is a unit, whereas U(r) is a group of unitaries namely U ( r ) U ( r ' ) = U(r + r'), U(0) = I and U(r)* = U ( - r ) = V(r) -1. A family of representations in the Hilberts space &(Z D) indexed by w E f~ is given by: 7rw(a)r
=
~
a(T-'~w; n' - n)r
,
a E Cc(~~ X Z D) ,
/~ E g 2 ( z D ) 9
n I EZ D
(184) In particular we get a(w; n) = (0brw(a)in). Then a C*-norm is defined by: I[all = sup II~rw(a)l[ ,
a e G(f~ x z D) .
(185)
weft
Then C(f~) x T Z D is the completion of Cc(f~ x Z D) under this norm. To shorten the notations we will denote it by .4. Given an invariant probability measure P on f~, a normalized trace wp (or T for short whenever no confusion arises) is defined by: v(a) = I n dPa(w; O) ,
a c .4.
(186)
It is easy to see, by using the Birkhoff ergodic theorem, that if P is ergodic, r(a) = lira
1
ATZ D ] - ~
TrA 0rw(a))
for P - almost all w .
(187)
At last, the differential structure is related to the group action and defined as follows. If n = ( n l , . . . , n o ) E Z D, we define the *-derivation c9, by: a,a(w; n) = i n , a(~; n) ,
# = 1,..., D .
(188)
39 These derivations commute together and are the infinitesimal generators of the Dparameter group of automorphism {po; 01T D } (the so-called dual action of Takasaki [Ped]) defined by: po(a)(w; n) = e'~
n) ,
On = Olnl + . . . + ODnD 9
Moreover, denoting by N, the position operators defined by N , r g2(zD), we get: 7c,,(O~,a) - i[N~,, 7r~(a)] . 7.3
Localization
(189) = hue(n) in (190)
Criteria
In this subsection we give several criteria for the localization and discuss the relation between its finiteness and the nature of the spectrum. We will consider a self adjoint element H = H* in the algebra A = C(9t) • zD previously described. In view of the study of a Floquet operator we may consider a unitary element F = (F*) -1 of this algebra instead. This latter case reduces to the former provided we identify F with e i T H for some T > 0 and the Borel sets A are subset of the unit circle. In the physical representation 7r~ we consider the operator 7r~(H) = H~ instead. If A is some Borel subset of R we denote by PA t eigenprojection of H corresponding to energies in A namely : (191)
PA = Xzx(H) ,
where X~ is the characteristic function of the interval A. Again, we notice that in general P/, may not belong to A. However it always belongs to the so-called Borel algebra B ( A ) [Fed], formally generated by Borel functions of elements of A. The Borel functional calculus permits to extend any representation of .4 to its Borel algebra. Hence the previous definition makes sense. The price we pay for it is that the mapping w E ~t ~-~ Try(a) may not necessarily be strongly continuous any more, but it is always strongly borelian if a E B ( A ) . If H~ has a pure point spectrum in A we get the following decomposition:
7r~(Pa) = ~ HE(w),
(192)
EEA
where HE(W) is the eigenprojection of H~ corresponding to the eigenvalue E. If E is a simple eigenvalue, one gets He(w) = [r162 where eE,~ is a normalized eigenstate namely: HeE,~][2 = ~
IeE,~(n)[ ~
1 < +C~.
(193)
nEZ D
The first quantity measuring the localization is the probability of staying at the origin. It was introduced by [And] and studied by Pastur [Pas]. To define it let us first consider the time-average A,~,~,(A, w) of the probability for an initial state at n to be localized at n t after time t: A,,n,(A,w) = lira f T dt [(n[Tc~(eitr4 PA)In')]2 T~---,(~JO T-
(194)
40 If H~ has a pure point spectrum, the decomposition (192) leads to : A.,n,(A,w) = ~
(195)
[(nlIIE(W)ln')l ~ .
EEA The covariance condition implies A~,n,(A, w) = A0,,,_~(A, T-~w), so that the staying probability is entirely given by the function A0,0(A,w), provided we consider it as a function of the disorder. We remark that if the eigenvalues are simple, since the eigenstates are normalized we get:
Ao,o(A,w ) =
EEe~ICE,~(O)I 4 (E~
1r
(196) 2 '
namely A0,0(A, w) is the mean inverse participation ratio for energies in A. To get a quantity insensitive to the disorder, let us average it with respect to P defining the averaged inverse participation ratio : (197)
~A = fn dP A0,o(A, w) .
Using now the automorphism group defined in eq.(189) and the eq.(184,186), an elcmentary calculation leads to the following expression for {~: ~a =
lim[T -Tdt fT~
r~
Jo
dDO (2~) ~
~-(e"n PApo(e-"n P~)) .
(198)
So we see that the staying probability or the inverse participation ratio, admits a purely algebraic expression. The Pastur theorem [Pas] can then be established as follows: T h e o r e m 13 For almost all w E 12, the number of eigenvalues of H~ in A is either
zero or infinity. The latter is realized, namely H~ has some point spectrum in A, if and only if the averaged inverse participation ratio ~a is positive. C o m m e n t : this criterion is not sufficient to eliminate continuous spectrum. We now introduce a stronger notion of localization giving a measurement of the localization length. Whenever Try(H) has pure point spectrum, the eigenstate may decay faster at infinity on the lattice. We are led to introduce quantities like: ] 1/p
g(P)(E,u;) =
E IeE,~(n)121nlp n~Z D ]
1
(199)
for p > 1. If the eigenstates decrease exponentially fast one can also consider the quantity g(E, w) = lira sup --In[r
(200)
However such expressions are very badly behaving with w in general and they are not suited for comparison with experiments or numerical calculations. The following
41
definition will be more convenient and will give rise to a quantity independent of w. We consider the averaged fluctuation of the position in the form: AX~,,(T) 2 =
(nl(N~(t) - N)2ln) ,
(201)
where N,~(t) = @H~tNe-'u~*, and N = ( N 1 , . . . , N o ) is the position operator. The covariance property gives A X ~ , , ( T ) = AXT-,~,o(T), so that after averaging over the disorder, we get a quantity independent of n, namely AX(T) 2 = fn dP(w)AX~.o(T) 2" An elementary calculation shows that: AX(T)
--
(202)
We will generalize this expression by considering, for every Borel subset A of the real line, the corresponding quantity AXz~(T) 2 obtained in the same way if we replace e iHt by e i H t p A . The main result in this respect is the following: T h e o r e m 14 If g2(A) = lim sup AXA(T) 2 < c~ ,
(203)
T~--*~
then H~, has a pure point spectrum in A for almost all w E ~. Moreover, if]q" denotes the density of states of H, there is an N-measurable non negative function g on R such that for every Borel subset A t of A,
e~(A ,) = f,, dN'(E)e(E) ~ ,
(204)
C o m m e n t : we will see in the proof that if ~m,(w) denotes the set of eigenvalues of
H~: g2(A ') = ~ d P ( w )
~ nc=ZD
n2
I(01HE(~)In)I ~ .
~
(205)
E6app(w)oA
In particular, letting A shrink to the point E, the function g(E) 2 represents a kind of average (over the disorder and over a small spectral set around E), of the quantity ~ e Z D n21r 2. Namely it is a measure of the extension of the eigenstate corresponding to E. This is the reason for the definition below. Definition 2 The function g will be called the localization length for H. P r o o f of t h e t h e o r e m : (i)-The basic argument we will use here is due to Guarneri [Gua, Bel:Tre]. We will denote by a~(w) the set of eigenvalues of H~ (the point spectrum), whereas II~(w) will denote its spectral projection on the point spectrum and IIc(w) = I - II~(w) will be its spectral projection on the continuous part of its spectrum. Using the definition of the trace in A, we get : nZpT(~,n) ,
(206)
p~(~, ~) = f f ~-I(Ol~(e*"P~)l~)l ~ 9
(207)
AXa(T)~ = / n dP(w) ~ nEZ D
where,
42 We will set pT(n) = fn dP(w)pT(w, n). By definition, we have :
0 < pT(W, n) < 1 ,
(208)
~_, pT(W,n)= 1,
(209)
nc::ZD
whereas the Wiener criterion gives lira pT(W, n) =
T~--*oo
~
I(01IIE(w)In)]2.
(210)
EEapp(w)NA
In particular, if L is a positive integer, and Inloo = maxl_<j
(211)
If we set r = fn dP(w)(01II~(w ) ]0) we obtain after averaging over the disorder lim
E
pT(n)_
(212)
Since r _> 0, one can find TL > 0 such that if T >__TL, EI-I~
A X a ( T ) 2 >_ L 2 s dP(w)
pT(W,n) > L2(1 in[c~>_L
~
pT(n)) > L2r
in]oo
--
(213)
2
Taking the limit T ~-~ c~ leads to L2r <_ 2e2(A) < oc for any L E N. Thus r = 0 showing that for almost all w's, (0]II~(w)]0) = 0. Using the covariance condition we also get for all n's {nlH~(w)]n) = 0 almost surely, and since Z D is countable, there is IT C ft of probability one such that for any w E f t ' , II~(w)]n) = 0 for all n E Z D, namely the continuous spectrum is empty. (ii)-Given two Borel subsets A1, A2 c A, we define the following expression: s T,~ t,( A 1,t-a2} A ~ = JOT "~dt
~
n2(Olr,.(eiHtpA1)ln)(Ol~r,~(e'Htpa2)]n
) .
(214)
I,~I~
~(L)/A ~ , ~ ,, A2) =
E Inl~
n2
~ I(01n~(~)ln)l 2 EEapp (~)FIA1FIA 2
= c(L)(A1AA2) .
(215)
From this definition of $(L)(A') whenever A' C A is Borel, it follows that (a) 0 _< s _< L 2, (b) If A, A A2 = q}, then s 0 A2) = s + s (c) If (Ai),eN is a decreasing sequence of Borel susbets of A converging to the empty set, namely NieN A, = q}, then s decreases to zero, (d) g(L)(A') _< s (e) g(L)(N) is a Borel function of w as a pointwise limit of Borel functions.
43 After averaging over the disorder we obtain E (L)(A') = fa dP(w)$~(L)(N) which fulfill (a), (b), (c) using the dominated convergence theorem, and (d). From (202,206), we also get: fo dP(w)~(T~J(A" A') -
(216)
Using the dominated convergence theorem we conclude that $(L)(N) < ~ ( N ) and also thanks to the property (b), g(L)(A') _< re(A) for A' C iX. It follows that limL,oo g(L)(iX,) = g(iX,) exists and defines a non negative a-additive set function over the set of Borel subsets of iX, namely a Radon measure. Moreover it satisfies g(iXt) < tr2(iX,), and by the monotone convergence theorem, eq.(215) above implies: $(iX') = / dP(w) E nEZ D
n2
E
I(0lI]E(w)ln)l 2'
(217)
EEapp(W)CIA
(iii)-On the other hand the definition of g2(iX,) and Fatou's lemma imply: e2(iX') <
/odP(w)
~ nEZD
n21imsup
]{Obr,(e~HtpA,)ln)]2.
(218)
T~--*o~
By the Wiener criterion the right hand side is nothing but $(iX') showing that g2(iX,) = $(A') < gi(iX) for iX' C iX. Hence it is a nonnegative Radon measure on iX. (iv)-To finish the proof it is sufficient to show that this measure is absolutely continuous with respect to the DOS A/" of H. Let then iX' C iX be such that A/'(iX') = T(iX') = 0. From the definition of the trace it follows that (0]Tr~(P~,)[0) = 0 almost surely. By covariance and because Z D is countable this gives ~r~(Pa,)In) = 0 for all n's almost surely, namely r~(Pn,) = 0 almost surely. Then (214) above implies s T,~[iXt, iX')= 0 for any L, T, and almost every w. Consequently $(iX') = 0, and the representation (204) holds.
7.4
L o c a l i z a t i o n in t h e K i c k e d R o t o r
As claimed previously, one can use the same formalism for investigating the localization properties of the kicked rotor. It is then sufficient to work with the Floquet operator instead of a Hamiltonian, and with Borel subsets of the circle. However the C*-algebra we are using, B~, is parametrized by the effective Planck constant 7, an additional parameter here. Apart from this remark, we get the previous structure if we set ft = T 2, D = 1, and the action is provided by the Furstenberg map. The Lebesgue measure dxdy/47r 2 gives the probability measure, which is ergodic whenever V/27r is irrational. In view of the theorem (14) above, we cannot expect any finite localization length otherwise, because the action is no longer ergodic and from a result of Izrailev and Shepelyanski [IzSh] it follows that we get an absolutely continuous spectrum for 7%~,y(F) whenever ~//2~r and x / 2 r are rational. Now, we remark that the definition of the localization length coincides with the definition of the mean kinetic energy given by (176) up to the constant 3,2. Hence FK,.y,x will have a pure point spectrum in A whenever the mean kinetic energy $.~(3') = v21imsup T~ T1 r~- , ~-~ (I O~ t=0
) '
(219)
44 is finite. Moreover we get the elementary formula 72t~2~(A) = ~7a(7 ) whenever g~ denotes the localization length in term of 7. The case of the kicked rotor permits to go a little bit further. First of all, the definition of the Floquet operator permits to show that it is C~ with respect to 00. Moreover we get the folowing result: P r o p o s i t i o n 10 If the localization length g, exists for the Floquet operator FK,~ of the kicked rotor model, it is constant over the circle. Proofi Clearly r/y = t)0=0,y (see (135)) commutes with the derivation 00. Moreover r/~ translates the spectrum of FK,~ by y along the circle because for any Borel subset A ofT: u~(FK,~) = e'YFK,,~ , ~/y(Pa) = Pa+y. (220) Thus
(100(r
2) :
2):
,
(221)
because r/y is an automorphism. It implies ~ ( A ) = ~ ( A + B) for any Borel set A, and therefore g(E) = eonst. Thus A is not needed anymore so that : C o r o l l a r y 5 For the kicked rotor model the following formula holds
2 g~/
_--_~*f ,~2
1 T-I lim -- ~ % (IOo(F*K)]2) T~--*ooT t=0
(222)
A result by Casati & Guarneri [CaGu] shows that, the spectral measure of FK,~,~ is purely continuous generically in 7. Thus : P r o p o s i t i o n 11 For the kicked rotor model there is a dense G~-set F of zero Lebesgue measure in [0, 1] such that for any 7 c F the localization length diverges. However many numerical calculations [CaChIzFo, BeBa] have shown that the mean kinetic energy for the quantum kicked rotor model is bounded in time. So we expect the localization length to be finite on a "large set" of 7's, presumably for almost all 7's in [0, 1]. Before discussing this question let us mention without proof another result which supplement the previous one namely P r o p o s i t i o n 12 For the kicked rotor model the localization length is a lower semicontinuous function of 7. We may also expect V292(7) to converges to some finite quantity as 7 ~-~ 0. This is the content of the Chirikov-Izrailev-Shepelyansky formula [ChIzSh] found on the basis of a numerical work. The well-known observation is that despite the diffusive behavior of the classical model (namely for strong coupling) the quantized version exhibits, up to a certain breaking time T*, a diffusion-like motion in phase space and then for t > ~-* its kinetic energy saturates as a function of time. This numerical result allows us to write = g~(r*) ~ Dr* , (223) where D is the classical diffusion coefficient.
45 There is here a mathematical difficulty. First of all, never was a diffusion coefficient shown to exist rigorously for the standard map. Moreover, averaging it over all possible initial conditions will not give a finite quantity due to the "Pustilnikov acceleration modes" (or "islands of stability"). This means that we should not average over the full torus. It raises the question of which quantum average should be considered. However, recent works [BeVa, Vai, Cher] have shown that for the sawtooth map, a diffusion coefficient does exist. Moreover, a conjecture states that for the standard map there is a "large" set of values of K for which no island of stability occurs, and a diffusion constant does exist. To get the Chirikov-Izrailev-Shepelyansky formula, we argue as follows. Since the eigenstates of the Floquet operator are localized, only a finite number g -- g~ of eigenvalues contribute effectively to the evolution of the initial state 10). Therefore we can approximate this Floquet operator by a g • g matrix F (t). The existence of classical chaos will lead to a strong level repulsion. Hence one can consider that the mean distance between the quasienergies is A E ~ 2~r/g = 0(3') on the torus. For times short enough, the discrete spectral sum arising from the previous approximation can be approximated by an integral, which will be precisely the classical approximation. Hence for t small, g~(t) ~ gcl(t) ~ Dt. This is fine as long as t A E << 27r. But after a breaking time r* ~ 2~r/AE ~ g, the quantization dominates and gives an almost periodic function of time for g~(t). Thus, ~3`2 = ~ ~_ g~(T*) ~-- DT*. Since g ~- g~, we get : D g~ _~ ~ - ~ . (224) Numerical calculations are in a fairly good agreement with this prediction, but no rigorous mathematical work has been produced to justify this formula yet. We may expect that : lira0g~/2 = D at K large, (225) under certain conditions. For indeed, we have seen that g~ diverges on a generic set of 3`'s. Moreover, D does not exist for all K's. For the moment we do not know how to define mathematically the breaking time T*. We would like now to study the behavior of the kinetic energy for the quantized version of the kicked rotor model as the effective Planck constant 3' tends to zero. For that, we perform a numerical calculation giving the classical and quantum energies of the KR for K = 4 corresponding to the diffusive regime (Fig. ll). We computed the quantal energy for different values of Planck's constant 3' in both cases; it is easy to see that as 3' is decreased the quantal curves tend to the classical one. One could think that this energy converges to its classical limit as 3' ~-~ 0 but a problem arises because of the uniformity of the semiclassical limit with respect to time. That the breaking time be O(72) can be shown by the following heuristic argument [HeTo]. The semiclassical approximation [Gut:Hou] for the evolution is correct modulo error terms of O(h(3'2). Therefore, the quantum and classical evolutions for observables should agree up to time O(h-~). Whenever the semiclassical approximation is exact, however, such as in the hydrogen atom, the harmonic oscillator, the Arnold Cat map, we should not see any breaking time.
46 Appendix Our aim in this appendix is to prove the theorem 1. The following theorem 15 actually implies the theorem 1. Let 7-/be a separable Hilbert space and H be a self adjoint operator on 7-/with domain D. This domain becomes a Hilbert space when endowed with the norm [ICl[ = I[r 2 + ][Hr 2. Let also V be a bounded self adjoint operator on 7-/leaving the domain D invariant and bounded on it for the domain norm H" ]IH. Let also f be a periodic continuous function on R with period T. Then the solution of the Schrbdinger equation : ihCt = (H + f(t)V)r , (226) with r
= r
is given by r
=
u(t, s ) r
(227)
where U(t, s) is a unitary operator such that : (i) it is strongly continuous with respect to s, t , (ii) U(s, s) = I for all s e R, (iii) V(t, s) = V(t, t')Ut', s) for all t' E R, (iv) U(t + T, s + T) = U(t, s) for all s, t E R, (v) for any r E D, the vector U(t, s)r belongs to D, is strongly differentiable with respect to s and t and is a solution of the Schrbdinger equation. The operator Fs = U(s + T, s) is called the Floquet operator for the family H(t) = H + f ( t ) V . Notice that if t = s the corresponding Floquet operators are unitarily equivalent thanks to (iii) and (iv). Now for ~ a positive real number, let p~ be a non negative function on R with support in the interval [-~, ~] and of integral equal to one. We will set :
f~(t) : ~
p~(t - nT).
(228)
nEZ Let F~ be the corresponding Floquet operator with t = - e . Then the following result holds : T h e o r e m 15 As ~ tends to zero, the Floquet operator F~ converges strongly to the
unitary operator F given by : F = e-'(~).e-~(D
.
(229)
P r o o f : Denoting by U~(t, s) the evolution operator, it is a classical result that it admits the following Dyson expansion, which converges in norm :
U,(t, 8) = ~-~n)O(__~)n ~s(sn(...~_s,~_tdSl.., dsnfe(sa)Z(s2).., f~(s.) (230)
e-(t-sl)~-Ve -(s'-82)~- V . . . Ve -(8"-s)~- . Each term is a well defined strong integral. Taking t = T - ~ and s = - ~ we get an expansion for the Floquet operator.
47 As ~ tends to zero, the restriction of the measure f~(s)ds to the interval [-r T - ~] converges weakly to the Dirac measure supported by (0}. Since the integrand is strongly continuous, the term of order n in the Dyson expansion of F~ converges strongly to ( _ ~ ) n e_,(_~__)V,Vn!" Summing up all these terms gives the result.
48
References [AbAnLiRa]
E. Abrahams, P.W. Anderson, D.C. Licciardello, T.V. Ramakrishnan, Phys. Rev. Lett. 42 (1979) 673.
[And]
P.W. Anderson, Absence of Diffusion in Certain Random Lattices, Phys. Rev. 109 (1958) 1492-1505; Local Moments and Localized States, Rev. of Mod. Phys. 50 (1978) 191-201.
[AuAn]
S. Aubry, G. Andre, Ann. Israeli. Phys. Soc. 3, 133 (1980).
[AvSi]
J. Avron, P.H.M. van Mouche, B. Simon, On the measure of the spectrum for the Almost Mathieu Operator, to be published in Commun. Math. Phys (1992).
[BaBeF1]
A. Barelli, J. Bellissard, R. Fleckinger, in preparation (1993)
[BaF1]
A. Barelli, R. Fleckinger, Semiclassical analysis of Harperlike models, to be published in Phys. Rev. B (1992)
[BaKr]
A. Barelli, C. Kreft, Braid structure in a Harper model as an example of phase space tunneling, J. Phys. I France 1 (1991) 1229-1249.
[Bel:T're]
J. Bellissard, Stability and Instability in Quantum Mechanics, dans Trends and Developments in the 80's, S. Albeverio & P. Blanchard Eds., World Scientific, Singapour (1985).
[BehKth]
J. Bellissard, K-theory of C*-algebras in Solid State Physics, Lect. Notes in Phys. 257, Springer, Berlin, Heidelbe New York (1986).
[BehEva]
J. Bellissard, C*-Algebras in Solid State Physics : 2D electrons in a uniform magnetic field in Operator Algebras and Application, Vol. 2, D.E. Evans & M. Takcsaki Eds., Cambridge University Press (1988).
[Bel:Gap]
J. Bellissard, Gap labelling theorems for Schr6dinger's operators, in Number Theory and Physics, J.-M. Luck, P. Moussa and M. Waldschmidt Eds., Springer Proceedings in Physics 47, Springer, Berlin, Heidelberg, New York (1993).
[BeBa]
J. Bellissard, A. Barelli, Dynamical Localization : Mathematical Fb~mework, in Quantum Chaos, Quantum Measurement, P. Cvitanovic, I.C. Percival, A. Wirzba Eds., Kluwer Publ. (1992) 105-129.
[BeBoGh]
J. Bellissard, A. Bovier, J.-M. Ghez, Gap labelling theorems for one dimensional discrete Schrgdinger operators, Rev. Math. Phys. 4 (1992) 1-37.
49 [BeIoScTe]
J. Bellissard, B. Iochum, E. Scoppola, D. Testard, Spectral properties of one dimensional quasi-crystals, Commun. Math. Phys. 125 (1989) 527-543.
[BeKrSe]
J. Bellissard, C. Kreft, R. Seiler, Analysis of the spectrum of a particle on a triangular lattice with two magnetic fluxes by algebraic and numerical methods, J. Phys. A 24 (1991) 2329-2353.
[BeLiTe]
J. Bellissard, R. Lima, D. Testard, Almost periodic SchrSdinger Operators in Mathematics + Physics, Lectures on Recent Results, Vol. 1, L. Streit Ed., World Scientific, Singapore, Philadelphia (1985) 1-64.
[SeVa]
J. Bellissard, S. Vaienti, Rigorous Diffusion Properties for the Sawtooth Map, Commun. Math. Phys. 144 (1992) 521-536.
[BeVi]
J. Bellissard, M. Vittot, Heisenberg's picture and non commutative geometry of the semiclassical limit in quantum mechanics, Ann. Inst. H. Poincar@ 52 (1990) 175-235.
[BenSir]
V.G. Benza, C. Sire, Phys. Rev. B 44 (1991) 10343.
[Bou]
N. Bourbaki, Thdories Spectrales, Hermann, Paris (1967).
[CaChIzFo]
G. Casati, B.V. Chirikov, F.M. Izrailev, J. Ford, Stochastic behavior of a quantum pendulum under a periodic perturbation, Lect. Notes in Phys. 93 (1979) 334-351.
[CaCu]
G. Casati, I. Guarneri, Chaos and special features of quantum systems under external perturbations, Phys. Rev. 50 (1983) 640-643.
[Cher]
N.I. Chernov, Ergodic and Statistical Properties of Piecewise Linear Hyperbolic Automorphisms of the 2- Torus, to appear in J. Stat. Phys. (1992).
[ChIzSh]
B.V. Chirikov, F.M. Izrailev, D.L. Shepelyansky, Dynamical Stochasticity in Classical and Quantum Mechanics, Soy. Sci. Rev. C 2 (1981) 209-267; B.V. Chirikov, F.M. Izrailev, D.L. Shepelyansky, Quantum chaos : localization vs ergodicity, Physica D 33 (1988) 77~
[ChE1Yul
M.D. Choi, G. Elliott, N. Yui, Gauss polynomials and the rotation Algebra, Inventiones. Math. (1991).
[ChDe]
V. Chulaevsky, F. Delyon, Purely absolutely continuous spectrum for almost Mathieu operators, (1990)
[CoFoSi]
I.P. Cornfeld, S.V. Fomin, Ya.G. Sinai, Ergodic Theory, Grundlerhen, Bd. 245, Springer, Berlin, Heidelberg, New York (1982).
50 [Cyc]
in H.L. Cycon, R.G. Froese, W. Kirsch, B. Simon, SchrSdinger Operators , Texts and Monographs in Physics, Springer-Verlag, New York, 180 (1987).
[DiSi]
E.I. Dinaburg, Ya.G. Sinai, On the One Dimensional Schr6dinger Operator with a Quasi periodic Potential, Funct. Anal. Appl. 9 (1975) 279-289.
[Dix]
J. Dixmier, Les C*-alg~bres et leurs reprdsentations, Paris, Gauthiers-Villars (1969).
[FiGrPr]
S. Fishman, D.R. Grempel, R.E. Prange, Chaos, Quantum Recurrences and Anderson Localization, Phys. Rev. Lett. 49 (1982) 509-512.; D.R. Grempel, S. Fishman, R.E. Prange, Localization in an incommensurate potential : an exactly solvable model, Phys. Rev. Lett. 49 (1982) 833-836.
[FiHuXX]
S. Fishman, N. Hurwitz, J. Phys. A (1991).
[FrSp]
J. FrShlich, T. Spencer, Commun. Math. Phys. 88, 151-184 (1983).
[FrMaScSp]
J. Fr6hlich, F. Martinelli, E. Scoppola, T. Spencer, Commun. Math. Phys. 101 (1985) 21.
[Gua]
I. Guarneri, Private Communication, Trieste (1986).
[Gut:Hou]
M.C. Gutzwiller, The semiclassical quantization of chaoti Hamiltonian systems, NATO ASI, Les Houches, Session LII, Chaos and Quantum Physics, M.-J. Giannoni, A. Voros, J. Zinn-Justin Eds., North Holland, Amsterdam, London, New York, Tokyo (1991) 201-250.
[HeSj:Harl]
B. Helffer, J. Sj5strand, Analyse semi-classique pour l'dquation de Harper I avec application ~ l'dtude de l'dquation de SchrSdinger avec champ magndtique, Bulletin de la Soci~t~ MatMmatique de France, Tome 116, Fasc. 4, M~moire 34 (1990).
[HeSj:Har2]
B. Helffer, J. SjSstrand, Analyse semi-classique pour l'dquation de Harper II, Bulletin de la Socidt~ Math~matique de France, Tome 118, Fasc. 1, M~moire 40 (1990).
[HeSj:Har3]
B. Helffer, J. SjSstrand, Analyse semi-classique pour l'dquation de Harper III, Bulletin de la Soci~t~ Math~matique de France, Tome 117, Fasc. 4, Mdmoire 43 (1989).
[HeTo]
E.J. Heller, S. Tomsovic, Private Communications (1991).
[HiSkl
M. Hilsum, G. Skandalis, Invariance par homotopie de la signature ~ coefficients dans un fibrd presque plat, Preprint (1990).
51
[IzSh]
F.M. Izrailev, D.L. Shepelyansky, Quantum Resonance for a rotator in a non linear periodic field, Teor. Mat. Fiz. 43 (1980) 417-428.
[Net]
Ph. Kerdelhu~, Equation de Schr~dinger magndtique pdriodique avec symdtries triangulaires et hexagonales. Structure hidrarchique du spectre, Thesis, Universit~ de Paris-Sud, Orsay (1992).
[MeAs]
D. Mermin, N. Ashcroft, Solid State Physics, Saunders, Philadelphia, Tokyo (1976).
[PaChRa]
B. Pannetier, J. Chaussy, R. Rammal, Experimental determination of the (H, T) phase diagram of a superconducting network, J. Phys. France Lettres 44 (1983) L-853 - L-858; B. Pannetier, J. Chaussy, R. Rammal, J.-C. Villegier, Experimental Fine Tuning of the Frustration : 219 Superconducting Network in a Magnetic Field, Phys. Rev. Lett. 53 (1984) 1845-1848.
[Pa l
L.A. Pastur, Spectral Properties of Disordered Systems in One Body Approximation, Commun. Math. Phys. 75 (1980) 179.
[Bed]
G. Pedersen, C*-algebras and their automorphism groups, Academic Press, London, New York (1979).
[Pei]
R.E. Peierls, Zur Theorie des Diamagnetismus von Leitungelectronen, Z. ffir Phys. 80 (1933) 763-791.
[PiVol
M. Pimsner, D. Voiculescu, Exact sequences for K-groups and Ext groups of certain cross-product C*-algebra, J. Operator Theory 4 (1980) 93-118.
[RaBe:Alg]
R. Rammal, J. Bellissard, An algebraic semiclassical approach to Bloch electrons in a magnetic field, J. Phys. France 51 (1990) 1803-1830.
[mel
M.A. Rieffel, Irrational rotation C*-algebra dans Short Communication to the Congress of Mathematicians (1978); M.A. Rieffel, C*-algebras associated with irrational rotations, Pac. J. Math. 95 (1981) 415-419.
[Siml
B. Simon, Almost periodic SchrSdinger operators. A review, Adv. Appl. Math. 3 (1982) 463-490.
[Sla]
I thank M. Rieffel for this information.
[Tom]
J. Tomiyama, Topological representations of C*- algebras, Tohoku Math. J. 14 (1962) 187-204.
52
[v i]
S. Vaienti, Ergodic Properties of the Discontinuous Sawtooth Map, J. Stat. Phys. 67 (1992) 251.
[Wil:Cri]
M. Wilkinson, Critical properties of electron eigenstates in incommensurate systems, Proc. Roy. Soc. Lond. A 391 (1984) 305-350.
[za ]
J. Zak, Magnetic Translation Group, Phys. Rev. A 134 (1964) 1602-1607; Magnetic Translation Group II : Irreducible Representations, Phys. Rev. A 134 (1964) 1607-1611.
53
Figure captions Fig.1 Spectrum of Harper's model (Hofstadter's butterfly). Fig.2 Magnetic translations and fluxes through elementary cells ; upper figure : square lattice ; lower figure : triangular lattice, from [BeKrSe]. Fig.3 Spectrum of triangular lattice with 77 = 27r0.0175 around half flux, from [BeKrSe]. Fig.4 Spectrum of square lattice with second nearest neighbour interaction, from [BaKr]. Fig.5 Asymmetry of the central band edges for the Harper model near ~ = 1/3. Fig.6 Conical contact between bands at half flux in the Harper model. Fig.7 Spectrum of the Hamiltonian with second nearest neighbour interaction near half flux, from [BaKr]. Fig.8 Parabolic contacts between bands at half flux in a Harper-like model, with third nearest neighbour interaction, from [BaF1]. Fig.9 Braiding of Landau sublevels in a model with second nearest neighbour interaction, from [BaKr]. Fig.10 Braiding of Dirac sublevels near half flux in a model with third nearest neighbour interaction, from [BaF1]. F i g . l l Time evolution of the kinetic energy for the standard map in the chaotic regime K = 4; thc staight line corresponds to the classical energy and points represent quantum curves for different values of the effective Planck constant.
54 !..t.~\'x'\"~li
,,/~/ i*
.~ !
~i 'l
i
.." "."7
' !:'7
"~IY
: '"; Z~ "-;:.7
il]I'
i. /
--'~1
ili-: i
~
I~
.'.:7
~",,, ',..: -~
7/ ,,,., . ~/. "lj~
".,Q
~----~.~;~ xnu opou~IN
.~-----~.m~
z
55
T~
!
5G
';::i ;' ~i ,l:i, '~ .JiiJ ~- 1",
I
d
II
I I ~'
~,1~
I. t I
II
'1.
'
"! ~l
'Ji.
%
'i,
'
,. 4.
'1 i,, ,. ..
~e
.
,
0...~1
,
.. 9
II
,
..
i
.
~.
.
q,~"
I
Z
T
9 ,,o
"~I~,'o. "'OI; I f:o "tl;~
,' I ""::"
I ,.t 'l
..... O;d..6.a-/o .........
":1; I .... "J"l::l:l,
f
~..
".,
"g
~~
. f~
~~
W
~
~ .o "~
~.; % !
,.
e,~..
!
! !
qr~
U~ 0
11a:~ 4!un Jad O|uonl~ xnl. I
0
57
\
/
,.... \.,
/ !" ! i
/
i. ',,
c5
,::5
~5
,:5
c5
d
d
d
/
58
/
.//
/" -!
/
\-
,,"i-/i
\
,I~!~,,,,)~'
m,l
:ill~i"--
'.,.-".."::: 7/i t .... '..../- i /' :
,. "
.:.
'"
i-
! i ,:
:
.
,"if~{
i
--------
~1~
~i,
o
59
-2
Pi
00
-o, \
x \
:. . - i ! ".. : ~ - :
\ I
\
', ,.
-\
- ii \ ~::~
0q. -<9,
o I
J .1 J'~
i
,,.,,' ,,:" /XtiL~:
~.,"" /"///!!~I
I
I
o
1~----~ Ih , m
. . . .
d~ tr~
,A ~
d~ -~-
,g
o
o
r
o
u~ d
~ o
x n u 0!l~Ug~lAI
~ o
~ o
~ o
u~ o
61
t3=0.5
k2
)
Pi kl
Energy
62
--,~..: .
::(...;'-...,
. . . . . . . . . . . . . . .
"",~l ;:'t.~'l~ i~ 7,~ ;li ~ ...... "~Z"~ I ~,ll~l~:,~l,~'"',.."::i :::::'..' .... '.:'I '..."
!':'.'." ", " ~,".".,'. '~ %-, ~..:::,;k i,,, ~ ,,. ~ir ~ '1, ' i~,,,, ~-~
~,..-,',l~
.",<
~-.
I".<1.. ',,'1 Ii!.~,., --' 'qllt,.~l"- I l~l'7"lllW .....
I
.....
\ ~i!II';I.,,.,,
I
~till,,.'l.
~l.flli,.l'
.....
~.lll'l.lli'lllli 9: : : r ,~ "7 :: :
I
,.-I~-"~il'~,~.,rr
~,,,,,-
.,~,..
'~l,,.:l[,,<~lii,~l > "~ '-'tL"~t:t'lll -",:~l~!-Iilidi' [~1!~ '~,;'"lliil:.;.~, ,'~{.~ ,*
.~ i . , .
~'1
"-';" ", --
,~1~,
i~"
,.
"-I",,,.."
I"
,.1"~r
""r'i i"'l
9
~
.". :.
r
'1-'11]~4
.-., ',, -'.....
'
/
I"~lg:l
, I
.-,~l!~,f,] I ~.~
,t[~l '"
I
J ~': .
&
A
&
-,,-
...i e,.,.,'"
'1"
x.~
"l~ll.!,li",,"~(,!.. i~'%.'~.~,~,\I 'lL, llll
&
&
&
Ii.
~ %.
',
-i~
'~".,
-IIIlillli,,,,.
': :'~
", ".--
". " "-" ".
". '..~ ,. " -.."
"~
.... \
.
~> $
I I. ~%'~AI
llllll.~ '~i~
,
I
-..,.~
A
xnL4[o!lais.~l~l,1
&
A
.,.~..::." ff
" t.--, I
.
. ~"~' .... '"i",,.
.i,,l~,
..Ix
/ I " /':1'.."', / :', ": ",, ,. :.:: :.:.-. .~..--
.
I !l'}l ~'~1~1 ~-
I
.
&
~""
,,".': ~.~"
,,~ I ~ - ......... --. I ~.-
9. : . . . .
,'"~:"
,,,i,.,'~l."/
i
"
\-,,.
L~j:~'lll'li ~'" ,~l,d t .'..',:,7,','1~'~ ' I,[~. " ;?"t' l r ~ ; ~-':;, 9 /-:,
:.... " ,
....
.#, ~ii'~
, I,
'",,,,
,
,,,,,I,:,~1
/
..
~
i'I,4 ..,I.X{ ;.
".~ .~,*"-i
,~ ~
.,,.'k',
I I,~11'7~' ' S t
I --,1r
,wl! ,~l:l.d, ~l
~'~
<~.
.....
.
-.
,.,- ~,.
,..,
r
"<.'.::k"
:
. 'l'~
:,',ii.",'..:"~.." ,~, "..~." ~':>~
"'-.~t.."~
~"~
I..,,./
, .... ."
l t/itii7
h.
,
/"/i!
./ / :..-s
.il I i i" i ''77:
i~,71 t-
' I ~
-...
,,i t
:~i:
,~lllll i
Iliik~l. ill,_ ~li i~
.,2..
..." .. i i
.....
"-,.~li;,',.
I
,~
-.." /
..,.~,"
,.".- I -" ...... . I,'",,..".i 17. ",.,
I
.I,..--I'
" ."
.,/ ./~ , ...... .... I, ,,," /-'/
,,iili~"l~ ~iltl,,~l~, '-ii~'~%,mi].,ll"
,~.1~ ~ lllJiil~. I .--,,
/
A
&
A
<~ A
\
&
A
63
;
I
\
-
9
.
9
.
.
.
......
. . . . . .
.
;
,'~ "."
9 9
.,-"
I
-
i,_ ";
;
:
:
9 ....
,
,
"
9
~
"
.
,
9
::
~ ,,~' . . . . .
~~
"
.:.
.:
/
'....
.,~. .....(..."
...... "" v"..'~'
,
;::,2
... ~..- ...:
\
\
,,,,
..:- ...~ 9 .-....~
9.
"
",
'J.". 9 ";(i~;
9 .
~
,
9 ...,."/ 9
" ~
7..
9 ,,,:,,,,",
.."..L.S " ^-'. 9..,..... '., : " ' "-.a..=..:.
'"" '~' ..(. ~
\, '~
...:~:~~
"
II
m--~m~gm~:m ~:~
o xnH
0!10u~IAI
o
o
o
64
20
15
iiiii iiiiiiiiii!iiiiiiiiiiiiiiiii
10
.
.
.
.
|
20
.
.
.
.
9
40
9
~
,
i
i
60
,
,
, .
t
~
8
i
i
i
I
100
t
E Q U I D I S T R I B U T I O N OF P E R I O D I C ORBITS: A N O V E R V I E W OF CLASSICAL V S Q U A N T U M R E S U L T S
Mirko Degli Esposti, Sandro Graffi, Stefano Isola
Dipartimento di Matematiea, Universith degli Studi di Bologna, Piazza di Porta S.Donato 5, I-~0127 Bologna, Italy
1. Introduction. Two mathematical problems, arising in classical and quantum dynamics respectively, are intimately connected to the chaotic behaviour and seem to have intriguing, though so far somewhat hidden, connections to each other. We state first the classical problem. Let Ct : M --~ M be a time evolution on a compact manifold M where the time variable t may be discrete or continuous. Denote by P the set of all primitive (i.e. non repeated) periodic orbits 3` of it. Let p(3`) be the (minimal) period of 31 and assume that there are p.o.'s of arbitrarily large period. Denote by r(x) the number of such 3`'s with period less or equal to T, i.e. r ( T ) = #{3` E P ]p(3`) _< T} (1.1) The first part of the classical problem can therefore formulated as follows C1 Find the asymptotic behaviour of re(T) as T ~ ce. To formulate the second part, given a periodic orbit 3` let us define a the probability measure #v in the following way: for any f E C(M), let
f
1
f(r
f d#~ - p(3`) Jo
dr,
x.y E 3`
(1.2)
if t is a continuous variable, whereas p(-~)-i 1 f d # ~ = p(3`) E f(r
x. r E 3'
(1.3)
t-----0
for discrete time evolutions. Then the second part of the classical problem is: C2 Given the sequence {3`k} of all periodic orbits of Ct, ordered according to non-
decreasing periods, find (possibly in some suitably averaged form) all weak* limit points of the sequence {P'yk} as k ~ oc. In othcr words, one would like to know how the periodic orbits become distributed in a spatial sense as their period increases.
66 These two questions are intimately related and have been partially solved only for a quite narrow class of dynamical systems satisfying strong hyperbolicity conditions, like the geodesic flow on a negatively curved compact manifold or, more generally, systems satisfying Smale's Axiom A [Sm]. Concerning the first question, consider for instance an hyperbolic flow restricted to its basic set Ct : A --~ A (see e.g. [PP], chap. 9, for definitions) and assume mixing. An example is the geodesic flow on a compact Riemannian manifold with sectional curvature < 0 everywhere, not necessarily constant. Then given a function B : A --~ ~ one defines the Ruelle zeta function [Rul]: ~(s) = I I
-
1 - exp
"~EP
( s - B((ot(x~)))dt
(1.4)
JO
and also the pressure P(B) of the function B through the variational principle (see [Ru2], [PP], [WAD:
P(B) = sup { hp(~l) + / Bdp }
(1.5)
JA
P
where the sup is taken over the set of all probability measures that are invariant under the flow ~t and hp(~l) denotes the measure theoretic entropy of p with respect to the time one map 41 (see, e.g., [Wa]). One then shows that for B H61der continuous the zeta function is non-zero and analytic for Re s >_P(B), except for a simple pole at P(B). Then, considering the particular case B = 0, and using the above zeta function in the same way as the Riemann zeta function is used to prove the prime number theorem (that is through Wiener-Ikehara Tauberian theorem, see, e.g., [Ko]), Parry and Pollicott [PP1] proved the following 'prime orbit theorem': e hT
7r(T),~
as
T~oe
(1.6)
where h = P(0) is the topological entropy of the flow. In the special case of the geodesic flow on a manifold of negative curvature one recovers an earlier result of Margulis on the distribution of lengths of closed geodesics. However for a precise understanding on how formulae of the type (1.6) can be established in different contexts we refer the reader to the following literature: [Hu], [SI1], [Mar], [B2], [PP1], [Po]. In chapter 2 we will obtain an analogous result for the linear hyperbolic diffeomorphism of the 2-torus. We observe that far beyond the questions formulated above there is the problem of characterizing the detailed structure of the set P and in particular to obtain corrections (i.e. error terms) to the leading asymptotic behaviour (1.6). This is in general a very di~cult problem belonging to an area of active mathematical research where very subtle questions are studied as testing grounds for techniques of analytic number theory. Concerning the second question, one generically finds that the periodic orbits of a hyperbolic system not only do proliferate exponentially but their average distribution in phase space becomes so uniform that any nonperiodic trajectory can be approximated arbitrarily closely, uniformly over arbitrary long time intervals, by some periodic orbit [B2].
67 Consider again an hyperbolic flow restricted to its basic set A. Then for B H61der continuous there exist a unique equilibrium state for B, i.e. an invariant probability measure #B which attains the supremum in the variational principle (1.5): P ( B ) = h~B (r + fA BdpB [Ru2]. In particular, if B = 0 one finds the measure of maximal entropy Po of Ct. Then using some analytic properties of the pressure and again the Wiener-Ikehara Tauberian theorem one proves the following result ([Pal, [PP]): T h e o r e m 1.1. Let Ct : A ~ A be a hyperbolic flow and let B : A ~ IR be a HSlder continuous function with equilibrium state pl?. If f E C(A) then
Z(B,
E
f d#'r' ep@) f B d,,.~
f d#B
--,
as
T---*oo
(1.7)
p(9/)<_T
where P(g)<_T
In particular ( Z(O, T) : rr(T) ), re(T)
f dp.y ~
f d#o
as
T~oc
(1.S)
p(7)<_T
This result provides a weak* convergence of (weighted) averages of the sequences of measure {#~} as the period of the 7% increases. One may then wonder if some stronger results can be proved under suitable assumptions. For instance, if it is possible to extract subsequences {>~ } of full density converging individually to the maximal entropy measure. In chapter 2 we shall prove this kind of results for the particular example of the linear diffeomorphism of the 2-torus. On the other hand, it is known that for hyperbolic systems any invariant probability measure can be approximated by suitable subsequences of the type {#~ } [HPS], so that one cannot expect in general to obtain convergence to the equilibrium state of any sequence of periodic orbits with increasing periods. We now state the corresponding quantum problem. Consider a compact Riemannian manifold V equipped with the standard form metric ds 2 = 9ijdxidx j. Let M = T*V be the cotangent bundle of V and Ct : M ~ M the geodesic flow on it, generated by the Hamiltonian
H(x, ~) = gij~i(j
(1.9)
By standard quantization procedure ([LL], [La]) to the Hamiltonian (1.9) corresponds the Laplace-Beltrami operator A : C a ( V ) ---, C~176 given by 1 A--
0
/
v'~ Oxi t v ~ g
ij 0 \ -~xJ)
(1.10)
68 where g = det(g~j). The stationary states of a quantum particle moving on V are then described by the eigenfunctions of A given by
(l.ii)
-h~ACk = ~kCk where the ~k's are the eigenvalues of A labelled in the increasing order: 0 <: A1 <: As < ... < Ak ~
(1.12)
Wee see that in this case the classical limit tt ~ 0 corresponds to the large eigenvalue limit k --* ~ . Now, let N(~) be the number of Ak's less or equal to A, i.e. N(A) = #{Ak [ ~k _< A}
(1.13)
The Weyl theorem asserts that
g ( ) 0 = CdVol(Y))~d/2 + R()O,
where
R(,k) = o()~d/2)
(1.14)
Here C d I : '_) d ~_r d / 2 ]_~ (~d + 1) and et = dimV. Then the first part of the quantum problem can be formulated as follows: Q1 Characterize the asymptotic behaviour of R() O. One important item in the theory of quantum chaos is that generically R()0 behaves as a random function so that its detailed study constitutes a very difficult mathematical problem. One way to describe this randomness is to introduce the family of measures: 1 p~ (a,b) = N------~#{)~ < )h Ak -- ~ - i
E (a,b)}
(1.15)
for given 0 < a < b < oc, and to study its weak* limit points as )~ ~ co. Such limiting distributions describe the asymptotic statistical properties of spacings between nearest eigenvalues and are expected to be completely determined by the ergodic properties of the classical geodesic flow. However, we shall not discuss further this problem here, referring the reader to the relevant literature (see, e.g., [SI2], [Sa], [UZ], [GVZ] and references therein). Now, let dv(x) = v/'~dx be the Riemannian measure associated the the metric ds 2 and let dp(x, ~) = dv(x) 9d)~(~) be the normalized Liouville measure on M = T* V. Take moreover the Ck's forming an othonormal basis of L 2 (V). Then, suppose we are given a positivity preserving quantization procedure for the observables (see e.g. [Ta]), i.e. such that to any non-negative ] E C~176 it associates a (pseudodifferential) operator ] > 0 acting on L2(V). Then the correspondence
< Ckf,r
>=/M
f(x'r162162
(1.16)
defines a probability measure #r on M. Thus, the second part of the problem can be formulate exactly as problem C2 above Q2 Find (possibly in some suitably averaged form) the weak-, limit points of the sequence of measures {#r } as k -~ oo.
69 Using Egorov's theorem IT] one can preliminarily show that any limit of the #k's must be invariant under the geodesic flow. Hence the problem reduces to investigate which invariant measure can be recovered as a classical limit of the measures #r This problem is of course well defined regardless of the ergodic properties of the classical flow. However the results proved so far concern mainly the two opposite situations: 1) The geodesic flow is completely integrable or quasi-integrable. In this case 9one is led to study how the measures #r localize on invariant lagrangian submanifolds. Since our main interest in this paper is to understand the possible relations between the classical equidistribution problem stated above and the present one, we shall not discuss this case here, referring the reader to the monograph of Lazutkin [La]. 2) The geodesic flow is ergodic with respect to the Liouville measure (in particular is a hyperbolic flow). In this case we have the following result of Schnirelman [Sehn], Zelditch [Z] and Colin de Verdiere [CdV]: T h e o r e m 1.2. Assume that the geodesic flow on M is ergodic, then there exist a subsequence {)~k~} of density one such that lira [ ~c~
fd#r
J M
----[ fd#
(1.17)
JM
It is worth noticing that a preliminary step to prove (1.17) is a result on the average analogous to (1.8):
E
fd#r
~
fd#
as
~~ ~
(1.18)
which follows from classical symbolic calculus and the Karamata Tauberian theorem. Using the ergodicity of the classical flow and the Egorov theorem one is then able to extract a subsequence of density one converging to the r.h.s (see, e.g., [CdV]). Thus, the above theorem asserts that almost all of the measures #r become equidistributed in the classical limit with respect to the Liouville measure. One interesting problem is then to find the conditions under which the Liouville measure is the only weak* limit point ('quantum unique ergodicity' in the language of [Sa]) so that one may avoid to take the limit on subsequences. In the next chapter we shall show that this happens in the case for the linear hyperbolic diffeomorphism of the 2-torus and its quantized version. For other results and/or conjectures in this direction see [Z1], [Z2], [RuSa]. The two problems (C) and (Q) stated above will be referred to in sequel as the classical and quantum equidistribution problem, respectively. Besides their own interest as individual mathematical problems one is also interested in their possible connections. A first known fact is the following: consider the particular situation of a compact manifold V of dimension d and constant negative sectional curvature, say -1. Then, every conjugacy class of the fundamental group of V contains exactly one closed geodesic so that one can arrange the set of all closed geodesics as a countable family { ~ - k } ~ with non-decreasing lengths {gk}k~v. Moreover, there is a
70 one-to-one correspondence between the closed geodesics 7-k of V and the primitive periodic orbits 7k of the associated geodesic flow Ct : M ~ M. In particular 7j will have least period ~k, i.e. P(3'k) = fk. Now, a remarkable relation between the two sequence of real numbers {g-k} (often referred to as tile length spectrum) and {~k} is provided by the Selber9 trace formula [Hell: E
k
f(x/~-
ak) - Vol(V)
-d-~T
471"
sinh(T/2) + ~
k
~
n=l
sinh
k/2)
where f : /R ~ ~ is any C ~ function of compact support and f is its Fourier transform. A great deal of mathematical work has been made around this formula and several applications and generalisations have been proposed (see [He2], [He3], [BalVor] and references therein). On the other side, very little is known about the possible connections between the two sequences of measures {#.~ } and {PCk } defined respectively in (1.2) and (1.16). One remark is the following. It is known that for a geodesic flow on a negatively curved manifold, the Liouville measure coincides with the measure of maximal entropy when the curvature is constant (see, e.g., [K]). Thus, one may argue that in this case there should exist some direct relation between (1.8) and T h e o r e m 1.2 (in the next chapter we will examine a discrete time dynamical system where these two measures coincides as well and the connection can be established explicitly). On the other hand, for the more general case of variable curvature the problem seems much more involved.
2. A n e x a m p l e :
the hyperbolic linear automorphism
of the torus.
We now consider the discrete time dynamical system T : M ~ M where M is the 2 - t o m s T 2 = 1R2/Z 2 (points on T 2 are denoted by x = (p, q) C [0, 1] x [0, 1]) and T is the hyperbolic automorphism of T 2 generated by the matrix
such that (a, b, c, d) c Z, ad - bc = 1 and la + dl > 2. The Lebesgue measure # is invariant because det A = 1. Moreover, the condition ITrAI > 2 makes this dynamical system an Anosov one and hence, in particular, ergodic and mixing with respect to #. Now, an orthonormal basis in L2(T 2, d#) is given by the set =
e z 2}
(2.2)
and A acts on points x = (q,p) and on suitably smooth functions f(x) on M respectively as: Ax = ( (aq + bp)(mod 1), (cq + dp)(mod 1))
f(Ax) = E nEZ 2
fnT(Atn)
(2.3)
7"1
where A t is the transposed matrix of A (notice that we have used the same symbol A to denote the matrix A and the map T; this will be repeatedly done in what follows without fear of confusion). Consider then in L2(T 2, dp) the unitary Koopman operatorblA defined by
(l.r
= f(Ax)
(2.4)
and recall (see, e.g., [A.A]) that T is ergodic iff 1 is a simple eigenvalue of b/A. Otherwise stated, if there is h E L2(T 2, dp) such that blAh = h then h is constant p-almost everywhere. Moreover, it is mixing iff, for any pair f , g E L2(T 2, dp), lira
k--* oo
= <
f, 1 > < 1,g >
(2.5)
This property makes a(UA) continuous on the unit circle, but for the eigenvalue 1. Now, if A is a continuous map of a compact metric space X then htop(A), the topological entropy, satisfies the restricted variational principle:
htop(g) =
sup h,(A) t~E2~A(X)
(2.6)
where A J A ( X ) the set of the probability measures on the Borel a-algebra of X which are A-invariant and h , ( A ) is the measure theoretic entropy (see [M], [AY]). Moreover, A is intrinsically ergodic if there exist a unique p E .A4A(X) such that htop(A) = ht,(A ). In this case p is called the intrinsic measure of A, or maximal entropy measure of A. Thus, for a linear automorphism of the torus the Haar measure, i.e. the Lebesgue one, is actually an intrinsic measure. We have the T h e o r e m 2.1 (Sinai) Let A : T d --* T d be a linear automorphism with eigenvalues )~1,... ,~d. Let # be the Haar m e a s u r e o f T d. Then
htop(A) = h,(A) = E
logjam]
(2.7)
I~l_>l
Proof. See e.g. [A.A]. Notice that this result can be easily obtained also from Pesin's formula ([M], p.265). Consider again the general situation of a continuous map A of a compact metric space X. We say that x E X is a periodic point of A, of period n, if it is a fixed point of A '~, i.e. Anx = x. We denote by Fix,~ the set of such points. It is easy to see that for linear automorphism of the torus the set of periodic points of A is dense in T 2, because it coincides with the subset of T 2 formed by all points having rational coordinates (see below). More generally we have the T h e o r e m 2.2 ( B o w e n - S i n a i ) Every topologically mixing hyperbolic homeomorphism A : X ~ X is intrinsically ergodic. If p denotes its intrinsic measure then for every continuous map f : X --~ ~ :
fdp =lim
#Fix,~
E xEFixn
f(x)
(2.8)
72
and A has topological entropy htop = lira 1 log~Fix,~
(2.9)
n--4 oo n
Proof. See [M], p.254. For the linear automorphism of the torus we have the additional result: P r o p o s i t i o n 2.1. Let )~ be the eigenvalue of A whose modulus is larger than
1. Then # F i x , -= ~ + A-'~ - 2
(2.10)
Proof. We sketch the idea of the proof referring to [If for more details. Let k be the trace of A, then Ikl > 2. Consider the numbers ),'u,~ = where D = V / ~ -
_
~-(2.11)
2D
1. These numbers satisfy the recursion
u0=0, ul=l
and
un=ku,~-l-u,~-2,
forn>l
and it can be easily checked by induction that
An=(aus-un-1 \
bus
CUrt
)
(2.12)
dun - us-1
By virtue of this formula one can easily realize that for any n > 0 the set Fix~ constitutes a regular lattice on the torus and by a simple geometrical argument ~r can be computed as the inverse of the area of an elementary cell in the above lattice (see [If), thus giving (2.10). Q.E.D. R e m a r k . Notice that from (2.9) and (2.10) one immediately recovers (2.7) for this particular case: h , ( A ) = htop(A) = log A. We now deduce some further consequences of (2.9) and (2.10). Denote again by P the set of all primitive periodic orbits (prime cycles) of A and by ~rP(n) and r ( x ) the number of them whose period is n and less or equal to x respectively, i.e. ~'(n) = ~{V e PIP(V) = n)
r(x) = #{V e PIP(V) -< x}
(2.13)
P r o p o s i t i o n 2.2.
(2.1a)
n
and
,~ ~
Xx x
where f ( t ) --, g(t) means that f(t)/g(t) ~ 1 when t ~ ~ .
(2.15)
73
Proof. We make use of the zeta function Oo
~(z)=exPE
n
z-
n=l
'D,
E
1
(2.16)
xCFixn
The strategy is to gain insight on the distribution of closed orbits out of the mewmorphy domain of ((z). In our case, by Proposition 2.1, ((z) has the simple form (1 - z ) 2 = (1 - ~ z ) ( 1 - z / ~ )
r
(2.17)
so that
~'(z__~) = _ ~ ~(z)
+ g(z)
(2.1s)
1 - Az
where g(z) is analytic in {z[Iz I < e~/A} for some e > 0. The rest of the proof proceeds exactly in the same way as in the proof for subshifts of finite type ([P.P], p.100). Q.E.D. R e m a r k . Proposition 2.2 is the analogue of the prime orbit theorem proved in the context of Axiom A flows (see [P.P]). We now prove that closed orbits exhibit a regularity in a spatial sense. In particular, we show that they are equidistributed on the average with respect to the Lebesgue measure p. Let #.y be the measure defined by p('y)-i
1 #'~ = p(3') E
8Ak(x)
x C 3'
(2.19)
k=0
and set f.~ f = fr2 fd#~. Then we have the P r o p o s i t i o n 2.3. For every continuous map f : T 2 --* ]I:l :
7r'(n)
2 fd#
f----~ p
as
n--*co
=n
Proof. We first write the number of fixed points of period n in the form # F i x . = E l. 7r'(l)
(2.21)
lIn
where lIn means that l divides n. Form Proposition 2.1 we then have
El.
~'(t) = ~," + ,~-" - 2
(2.22)
l[n
On the other hand Theorem 2.2 yields ,~-,~lim
E,l~t Ep(~)=~L f ~tl,, I. ~'(1)
f
= JT: f d #
(2.23)
74 Hence,
Let m = m a x { / c Z I lln, I < n}. If n is prime then m = 1, otherwise m > 1. One finds immediately that
p
=n
n
p(../)=/
f ~
(1 - C,~,m)
tin, l < m
3'
p(.y)=/
7
i~om which we obtain
fdp,
as
n ---* cx)
(2.25)
2
p('r) =n
where C,~,m is of order at most A"~-'~. To conclude the proof it suffices to divide (2.25) by ~-'(n) and apply Proposition 2.2. Q.E.D. Proposition 2.3 yields a result of uniform distribution on the average. We now study the behaviour of measures #~ supported on single closed orbits and we prove that they converge in measure to #. P r o p o s i t i o n 2.4. For any e > 0 and for any continuous function f lim ( # { ' ~ ] p ( ~ ' ) : n '
If'Yf-fr2fdP]>e})=O
(2.26)
~'(n)
~ - ~
Proof. For the sake of simplicity we shall use the notation fr2 f d p = f. For any k C Z+ set k--1
m~f(x) = ~ ~ I(A'x)
(2.27)
l=O
Now, given ~ > 0, introduce the sets S~,,~={Tlp(7)=n,
ff-f J-y
(2.28)
P
R6,~,k = {~IP(~) = n, Jr Im~f - ]r < VT} We have
On the other hand it is obvious that f.~(f - m k f ) = O, for any k c Z+; and thus S~,,~ D R~,n,k ~
R ~,,~,k c D S~,,~ for any
~ > 0, k E Z+
(2.29)
The ergodicity of A implies that, for p-almost every x E 1,2, lim mkf(x) = f
(2.30)
75 Hence, for any continuous f , by the Lebesgue dominated convergence theorem we can find a k0 > 0 such that r Jmkof(X) - lid# <_ di
(2.31)
Set
]d#r~
:-~_
Then Proposition 2.3 entails that d#,~ converges vaguely to the Lebesgue measure. This implies the existence of no((5) > 0 such that for any n > no r2 Imk~
- fld#~ _< ~
(2.32)
Hence by the Chebychev inequality we obtain #('YIP(~') = n, f'r Imkof - fl >- v/~} ~'(n)
< v/~
(2.33)
and therefore, by the second of (2.28):
~R~'n'k~ ( V/~, ~'(n)
n > no
(2.34)
-
Hence, from (2.29), we find #S~,,~ < v~,
~'(n)
n > no
(2.35)
and the assertion follows by taking (5 ~ O and consequently no(5) ~ c~. Q.E.D. 2.1 K o o p m a n o p e r a t o r a n d p e r i o d i c orbits on invariant l a t t i c e s . Consider any point o n / , 2 having coordinates (r/N, rl/N), with r,r~,N C ~W and 0 _< r, r ~ < N. There are exactly N 2 points of this type and they belong to the N • N subgroup of T 2 given by:
i N = ((q,P) C T2]Nq, Np E Z )
(2.36)
It is immediate to realize that LN is invariant under the action of A, so that any point in LN is periodic with period _< N 2, the origin being the only fixed point of A. Of course, any point x E Fixn belongs to a periodic orbit whose period divides n. This means that LN splits into periodic orbits (which in general may have different periods) of A. Let #N be the normalized atomic measure supported on LN. We now characterize the spectrum of UA when acting on L2(T 2, d#N). Let MN be the number of distinct periodic orbits of A t which live on LN \ (0, 0). This number is the same as that corresponding to A (see [I]). Let ~ C LN be any one of such orbits with period
76
P(7) and x = (rl/N, r2/N) e 7. Then, associated to each orbit 7 there are P(7) linearly independent vectors in ~TN2 given by: fl(k)=
v(~)-i 2ri ~ A~-Sexp-~-<(At)Sr, k>
r=(rl,r2), keZ~
(2.37)
s=0
where Al = e -~iz/p('~) and l = 0 , . . . ,p(~f) - 1, and they satisfy
5tAfl(k) = )~lfl(k)
(2.38)
Thus, there are N 2 - 1 eigenvectors of/dA of the form (2.37) which, together with the constant function 1, provide a canonical basis of L2(T2,d#N). Among them, there are exactly MN non-constant functions which are invariant. This is in account of the fact the dynamical system (T 2, A, #N) is not ergodic: the invariant measure #N obviously admits a decomposition into invariant ergodic measures of the type (2.19). The case of N prime. We now specialize now to the lattices L N with N prime. In this case a very precise characterization of the structure of the periodic orbits is possible (for which we refer to [D.G.I]) and moreover strong results on their equidistribution properties can be proved. First, it has been shown by Percival and Vivaldi [P.V] that all the periodic orbits living in LN with N prime have the same period, i.e. p(~/) = p(N) for any "f C L N \ {0, 0}. The relation among p(N), M N and N is then:
p ( Y ) . MN = N 2 - 1
(2.39)
Thus, from the above argument we then have the following result on the spectrum of the Koopman operator: P r o p o s i t i o n 2.5. Let N be a prime number and let p(N) be the period o/the cycles living on LN \ {0, 0}. Then a(LtA) is given by the eigenvalues
At -~ e 27ril/p(g)
l = O, 1 , . . . , p ( N ) - 1
(2.40)
To each )~1 is associated an eigenspace Ez of non constant functions to which all the periodic orbits o / L N \ {0, 0} contribute. Accordingly, the following decomposition holds: p(N)-I
L'(T2,dpN)--1G(
E,)
(2.41)
/=0
where 1 is the one-dimensional subspace spanned by the function 1 and dim(El) = MN V l = O , . . . , p ( N ) - 1. We now turn to the equidistribution results. We first prove a sharpening of Proposition 2.4:
77 P r o p o s i t i o n 2.6. For any f E C(T 2) and any e > 0 there is No such that, if N >_ No is prime:
# { T C L N \ { O , O } , If-r f - ] i - - - V q } #{~17 C LN \ {0,0}}
_< v~
(2.42)
Proof. It is easy to realize that, up to the additive correction vanishing as N -2 for N ~ cr (arising from the fixed point at the origin), the following identity holds true: 1 ~ MN
j=l
J
f=~
fd#N
(2.43)
Then the assertion follows by the same argument as in Proposition 2.4 where now Sr
f~f--f
R .... k :={717
C LN \
{0, 0}, j~
< V~}
I m k f - f l <- V ~ }
Q.E.D. Now, under slightly more restrictive assumptions on the sequence of primes and using the number theoretic techniques collected in the Appendix, we are able to prove the equidistribution of all periodic orbit sequences (living on prime lattices), with explicit estimates of the speed of convergence. The key fact is that since ZN becomes a finite field, this restriction amounts to operate with a (rood N) arithmetics. Recall that the prime number N is splitting with respect to the characteristic polynomial of A if there exist n E ZN such that k 2 - 4 = n2(modN) (where k denotes the trace of A) and is otherwise inert (see, e.g. [HI). It can be shown (see [D.G.I]) that if N is splitting then p(N) = (N - 1)fro for some m E /N (so that there are exactly m ( N + 1) periodic orbits living in LN), whereas if N is inert
p(N) = (N + 1)fro. Finally, let A ( T 2) be the Banach space of all functions f : T 2 ~ r such that
IIflIA
= ~
If,~l < oc
(2.44)
nEZ 2
We now prove the following stronger result: T h e o r e m 2.3. Let N E F, F being any increasing sequence of primes such that N / p ( N ) < C for some C independent of N. Set:
PN = {7 C LN \ {0, O} ] 7 periodic orbit of A},
MN = ~PN.
Then, given any f E A(T2), any sequence {Tj(N)}NEF such thatTj(N) E PN, j ( N ) E {1, . . . , MN } we have N
[
oo J~(N)
f -
fd" =0
Moreover, if f E C ~ ( T 2) there is C > 0 such that for N large enough:
(2.45)
78
f -
< IISlIAcT
(2.46)
R e m a r k . The condition N E F is equivalent to require m bounded with respect to N. On the other hand, the existence of at least a sequence of primes which satisfies this condition (actually the fact that almost any sequence of primes does it) is a consequence of the Artin conjecture, whose failure would imply the falsity of the generalized Riemann hypothesis (see e.g. [RM]). Detailed heuristic and numerical investigations on the behaviour of p ( N ) / N supporting the above genericity can be found in [K2] and [B.V].
Proof. We shall give the argument for the case of N splitting. This is equivalent to the splitting of the characteristic polynomial over ZN. In particular (see the Appendix), if we let D = V/-~ - 1 c ZN, then v -- (1, D), w = (1, - D ) E Z 2 are the eigenvectors of A acting on Z 2, corresponding to the eigenvalues AN = k/2 + D and AN1 = k/2 - D, respectively. Now, the key point is that the integral of an arbitrary character e i2~<"'~> over a periodic orbit V C LN can be written as a Kloosterman sum restricted to a cyclic subgroup of Z~v of order (N - 1)/m. Indeed, for any 7 C LN and x E 7 set N x = av + ~w where a(x), f~(x) E ZN. Then,
f~ cLN e27ri -- p(N) 1 p(N)-I E e27ri= s=O 1 p(N)--I . p(N) E ee'J~< . . . . ~+#'~
1 =
s=o
1 rap(N) E
1 e~('~r
--
p(N)
v(N)-I E ee'Frt(~<'~'v>;~+#<'~'w>'x;*) ~=o
m--1
E
E
2~,a,+b,-l~ XJ(~)e"~-' r ~
(2.47) where a ---- ~ < n,v >, b -- ~3 < n,w >E ZN and the relation p(N) -= ( N - 1)/m has been used. The functions Xj : j = 0 , . . . , m - 1 are the m distinct multiplicative characters of order m of ZN (see the Appendix). We can now apply the estimate (A.18) and obtain, for any n ~ ( 0 , 0 ) ( m o d N ) :
f~
e2~ri < K CLN
(2.48)
--
where K > 0 is independent of n and of the particular orbit 7 C LN and is uniformly bounded in N because of the boundedness of m. On the other hand, a trivial computation shows that, if n -- (0, O)(modN) then jf
CLN
e 2~ = 1
(2.49)
79 Therefore if f = ~;~,~eZ2 fne 2~i<'~,x> we can write
j(N)
2
n#(0,0) (mod N)
j(N)
k#(0,0)
Now, if f 9 A(T 2) the second term of the r.h.s, vanishes as N ~ oo and the first term also vanishes by the uniform estimate (2.48) on the Kloosterman sums. If moreover f E C~176 the second term vanishes at least as N -1 as N -~ oc and therefore we can conclude that there exists C > 0 such that if N is large enough: f -
f d" < _
llfllA(r )
C
(2.51)
Finally, the case N inert can be treated in a similar way by using the techniques of [PV] and the generalized Kloosterman sums over arbitrary finite fields (see [Kal]). Q.E.D.
2.2. The quantum equidistribution problem. Wigner function and classical limit. We now consider the quantum dynamical system (7-t,..4, VA) obtained by the canonical quantization of the former one. We now limit ourselves to recall the main ingredients referring the reader to [DGI] for the detailed construction. 1) The Hilbert space is :H = ~ N = L2(SI,vN) where h -~ 1/N and VN is the normalized atomic measure given by (C2~'iq E $1): N-1
VN(q) = ~i ~
~(q-- ~l)
(2.52)
/=0
Hence T / h a s dimension N. 2) The algebra .4 is the *-algebra of the observables on 7-/generated by the q uantization of the classical functions on the torus in the following way: let T(n), n E Z 2 be the canonical quantization of the basic observables T(n). This is based on the classification of the irreducible representations of the discrete Heisenberg group/-/1 (Z), in complete analogy with the well known construction of the SchrSdinger representation out of the Heisenberg group H,~(/Rn). The quantization of any f C A(T 2) is then given by f=
~
AT(n)
(2.53)
n~Z 2
3) The unitary bijection VA is the quantum propagator, i.e. the quantization of the action of t h e symplectomorphism A on the classical observables. This means that if f ~ f as above then f(Ax) ~ VAfVA 1. Therefore the quantum discrete dynamics is defined as
f ~ V U V Z k,
k9Z
(2.54)
80 Moreover, if N is a prime number, one easily finds that V~(N) = Id so that the quantum dynamics is periodic with period given by the classical periodic orbits living on the lattice LN. 4) We denote by e 2~iA(N) and r n = 0,... N - 1, the (repeated) eigenvalues of VA and the corresponding (orthonormal) eigenvectors, respectively. The quantum equidistribution problem can now be stated as follows: given any pair of eigenvectors r wz~(g)belonging to the same eigenspace, define a distribution d~N(r r on the phase space T 2 by
r IdaN(r
(2.55)
Then, we want to know what are the weak* limit points of such distributions when N --* c~. The main result of [DGI] is the following T h e o r e m 2.4. Let F be an increasing sequence of primes as in Theorem 2.3. Then, for any sequence of eigenvectors {r and f E A(T 2) we have
lira f fdf~N(r (N)) = Iv: f d # N ~JT:
(2.56)
and, if f is smooth enough, say f E C~(T2), the limit is attained with speed given by: Cllfh (2.57)
L/r
/r fd.l <- ----
Moreover, for any pair of sequences {r
{r ing to the same eigenspace but however distinct we have:
lira f
N--*~ JT2
of eigenvectors belong-
N)
(2.5S)
and the limit is attained with the same speed as in (2.57). R e m a r k . This result is stronger than the other results of this kind mentioned in chapter 1 in that it holds for any sequence of eigenvectors ('quantum unique ergodicity' in Sarnak's language [Sa]). In [DGI] two independent proofs of tile above result have been constructed. Here we shall just recall the main ideas of the proof using the Wigner function, because it also provides some insight into the question raised in the previous chapter about the relationships beetween the distributions d~N above and the measures d#~
(see (2.19)). We then define the Wigner transform, which is a map i~om pairs of functions in the Hilbert space into the phase-space functions, in the following way: Let r 1 6 2 E L2(ZN,I/N), (q,p) C T 2. Then their discrete Wigner transform is the function defined as W(r r
p) -
~ 7~tl ,n2 E Z N
< r ~b(n)r > e -2~i(n~q+~2p)
(2.59)
81
Since the Wigner functions have to be integrated against the measures //N on S 1 or #N on T 2, we are interested only in the values they assume on the lattice LN, and therefore we use the notation:
w(r162
Ys, r E ZN.
= w(r162
s
~)r
(2.60)
Now, using the inverse of the discrete Fourier transform we get 2~ikr
w(r162
k
= y~ ~ - - T - r
k
)~)(s+ ~)
keZN
(2.61)
= g f e-2~N2pq'r
-- q')(b(q + q')dvN(q;)
In particular for r = r we obtain the formula for the
w~(s,r) = w(r162
= ~
e- ~
Wigner distribution r
-
k)r
+ k)
(2.62)
kCZN
We have the P r o p o s i t i o n 2.7.
fT Proof.
The following representation holds:
f d~N (Ok(g), Cz(Y))- =
fJT2 fw(r ), r
(2.63)
By (2.53), (2.55) and (2.59) we have
fT ~edf~gt,~,(N) X,(N)~ _(g) ~ , x.(N) . I n < ~ k , tn)~& >
= ~
nEZ~ = ~ fnfT e2riW(r n EZ~ =
f (x)W(r
r
Q.E.D. Thus, one is reduced to investigate the behaviour of the Wigner functions in the classical limit. A first remarkable property is that the Wigner functions allow us to define a set of eigenvectors for the adjoint of Koopman operator acting on L2(T 2, dpN) (cf~. Proposition (2.6)): P r o p o s i t i o n 2.8.
Let
{Wk,t}
be the family of N 2 functions on the torus given
by w~,;(x) =
w(r ), r
82
Then, Vx E LN, UA*Wk,I(X) = Wk,l(A-lx) = e2~(~k-:')Wk,t(x)
(2.64)
and, in particular,
UA*Wt(x) = Wdx) where Wt = Wl,t, and
Ua*Wk,z(x) = whenever r
and r
Wk,dx)
belong to the same eigenspace of VA.
Proof. See [DGI], p.23. Q.E.D. Notice that we have implicitly established a relation between the degeneracy of the eigenvalues of the quantum propagator VA and the dimension of the invariant eigenspace of the Koopman operator acting on L2(T 2, d/~N) (see also [BH], [Elf):
p(N) ~-~ di = N,
v(N) ~-~ d2 = MN
i=1
i=1
where: - di is the degeneracy order of the i-th eigenvalue of VA; - MN is the number of distinct periodic orbits in LN \ {0, 0}, which coincides with dimE0; - E0 is the invariant eigenspace of/gA of Proposition 2.1; - p(N) is the common period of the classical closed orbits in LN \ {0, 0}, which coincides with the quantum period, i.e. V~ (g) = Id. As we have seen, b/A as an operator on L2(T 2, d#lv), has a point spectrum and each eigenfunction can be written as a sum of functions which are constant on the periodic orbits of LN (cfr. Proposition 2.6). In the classical limit the non ergodic, zero-entropy, invariant measures /zg weakly converges to the Lebesgue measure (with positive entropy equal to log)~ where A is the largest eigenvalue of A). This limit measure is ergodic and the Koopman operator has continuous spectrum but for the eigenvalue 1, that is, as we shall see, the Wigner functions weakly converge to constant distributions on the torus. Now, using Proposition (2.7) and (2.8) one immediately obtain the following result: P r o p o s i t i o n 2.9. We have the further representation:
Mtr ('h(N)''4"(N)'~ u
JT j=0
(2.66) :
where p(N)-i
P~0 ----5(0,0);
#~
_
1
p(N)
~ k=0
~Ak(x)'
X E "~j, j = 1 , . . . , M N
(2.67)
83 and aj(k,l)=c~.fr 3k k
, ] ~- P('YJ)W ~ k,ll'y~
wl
(2.6s)
R e m a r k 1. Proposition (2.9) yields an explicit relation between the 'quantum distributions' d~N and the periodic orbits measures d/z.y (see the remark on this point in chapter 1). In the corresponding problem for hyperbolic surfaces such a relation is at present at best very unclear [Sa]. R e m a r k 2. The coefficients ~j's may be positive as well as negative, the only condition they have to satisfy being: MN ~ 1, if l = k; Z c~j(k, l) = [. 0, otherwise. j=0
(2.69)
which follows from the normalisation condition: 2 Wk,ld#N =
O,
otherwise.
This fact makes the present problem quite different from the corresponding problem for ergodic flows on compact manifolds where a positive quantization procedure yields at once a sequence probability measures d#r k (see chapter 1) instead of the distributions d~N found here. We now state the final result after which Theorem 2.4. easily follows. P r o p o s i t i o n 2.10. There is a complete orthonormal basis of eigenvectors such that:
m--1 p--1
~lv = r ~ ) ( ~ ej,T
(2.71)
3=0 r : 0
where qhj,r with j = 0 , . . . , m - 1 are the m distinct eigenvectors corresponding to the eigenvalue e -2"~ir/N of VA (of constant multiplicity m), whereas ej,~ with r -- 0,... , p - 1 correspond to thep different eigenvalues, and VAr = r Moreover, for any pair (r el,r) one has the decomposition Wk,t = 6k,lW ~ + W~j x
(2.72)
where W ~ is a probability measure which converges in the weak *-topology of A(T 2) to the Lebesgue measure, whereas W~) x vanishes, as N --+ ~ , N E F. The speed of convergence is the same as in Theorem 2.4. Proof. After having constructed a suitable orthonormal basis of eigenvectors and used the explicit action of the operators 27(n) on such basis, the proof of this result essentially reduces (as in Theorem 2.4) to estimating sums of multiple characters over finite fields (see the Appendix) and we refer the reader to [DGI], Section 4. Q.E.D.
84 Appendix: Number Theory In this section p will denote a prime number and Yq a finite field of characteristic p with q = p~ elements (we will be interested mainly in the case q = p, Yq = Zp). Let X be a smooth projective absolutely irreducible algebraic curve over t'q defined by an homogeneus polynomial equation f(x, y) = 0 with coefficients in/Fq (whereas the varibles x, y live in the algebraic closure of Yq). Denote moreover by Nk the number of solutions of the above equation in/Fq~, so that, in particular, N1 is the number of points of X in Yq. Observe that one can also characterize the number Nk as the number of fixed points in X of the map F k where F is the Frobenius map defined by F : (x,y)--* (xq,y q) (A.1) Then, the Weil zeta function is defined by oo
Z(u, X ) = exp E
U
k
-k--Nk
(A.2)
k=l
Moreover, one esily realizes that putting u = q-8, it has the Euler product representation Z ( s , X ) = E J ~ ( h ) - 8 = I I ( 1 - A / ( h ) - ~ ) -1 (A.3) h
h prime
where the sum is over the integral divisors of the algebraic function field Yq[X] and JY'(h) = qdeg(h). Futhermore, it is known that Z(u, X ) satisfies the functional equation: Z(l-) = qg-lu2-2g Z ( u ) (A.4) q U where g is the genus of the curve X (see, e.g., [Sc]). T h e o r e m A.1. ([Well]) Z ( u , X ) is (a power series expansion of) a rational function of the form (A.5) Z(u,X) = ( 1 - uP(u) ) ( 1 - qu)
where P(u) is a polynomial of degree 29 and integral coefficients. Moreover, 2g
I~,1 = v~, vi
P(u) = l-I(1 - w~u) and
(A.6)
i=1
Notice that (A.6) asserts nothing but the validity of the Riemann hypothesis for curves over finite field, since Z(q -~) = 0 only for Re(s) = 1/2. Clearly, 1 dk Ark -- (k - 1)-------~du ---~ (log Z(u, Z))~= o
(A.7)
so that, from (A.6), 2g
Nk = qk + 1 - E w ~
(A.8)
85 and, in particular, one has the Weil estimate IN1 - (q + 1)[ < 2gv~
(g.9)
A (non trivial) consequence is the possibility of estimating certain exponential sums over finite fields. E x a m p l e . Consider the case Fq = ZN with N prime. Then it may be of some interest to generalize the trivial identity e~x = 0 x6ZN
to the sum x6ZN
where f ( x ) is, for instance, a given polynomial. Clearly, the answer is intimately related to the number of solutions of y - f ( x ) = 0 in ZN. More generally, we shall consider generalized sums over finite fields of the type:
X(f(x) )r
)
(A.10)
x6Fq where X is a non trivial multiplicative character of order dlq - 1 of Fq (i.e. X d is equal to the trivial character X0), r a non trivial additive character of Fq and f(x), g(x) 6 Fq[x] are given algebraic functions, for instance polynomials, over Fq. Before we examine some examples, let us give a further definition: let p e $V and a E Z be such that a =~ 0(modp). Then, we say that a is a quadratic residue of p if there is m e Z such that a = m2(modp). Now, given N prime, let once more Fq = ZN. Then, the only non trivial multiplicative character of order two is the Legendre symbol X2(x) = ( ~ ) defined as follows (see e.g. [Ap]): (N)
= { + 1 i f x is a quadratic residue; 1 otherwise.
(A.11)
-
Moreover ( o ) = 0 for any x =O(mod N). The Legendre symbol obviously satisfies the product law
On the other side, in the present case the group of additive characters is the set {Ca(x) = e ~ a ~ ;a e ZN}
(A.13)
T h e o r e m A.2. Let X,r be a multiplieative character ~ Xo of order d with dl( q - 1), and a non trivial additive character, respectively, o] lFq. Let f ( x ) E Fq[x] admit m distinct roots, and let g(x) E Fq[x] have degree n. Suppose that either (d, deg(f)) = (n,q) = 1, or, more generally, that the polynomials yd _ f ( x ) and
86
z q - z - g(x) are absolutely irreducible (i.e. irreducible over any finite algebraic extension of ~'q). Then IE
X(f(x))r
<- (m + n - 1)q 1/2
(A.14)
xE~q
Proof. See, e.g., [Sc], page 45. Remark the above result has been obtained by A. Weil as a consequence of Theorem A.1 and important extensions to cases where f , g are rational functions has been achieved by P. Deligne [Del],[De2]. Let us briefly discuss some consequences of this result. E x a m p l e . f ( x ) = g(x) = x, X = X2 and r = ~ba. Then we have the (generalized) Gauss sum G(r X). If, moreover,/Fq -- ZN we obtain the standard quadratic one
g(r
= E
e2~:~2
(A.15)
xEZN
and, by direct application of Theorem A.2, Ig(r mation is contained in the following
X2)I -< v r~- More precise infor-
Lemma A.1.
wh ere s
:
1, i,
i f N = l(mod4) if N = 3(mod4)
and N-1
27ri
2
Eexp-'N'(ak
[_2~rib2(4a)-I ] N
+bk)=eNN1/2(N)'exp
L
k=O
J
if a • 0(mod N), N-1
21ri 2 exp w - ( a k + bk) = N .
k=O
if a = 0(mod N). E x a m p l e . Consider once more the particular case s = ZN. Setting f ( x ) = x 2 - 4ab, g(x) = x, X = X2 and r = r we then find the Kloosterman sum
Kl(N,a,b)=
E
( x2 N-4ab) e z ~ x =
xEZN
E
e~(ax+b~-l)
(A.17)
xEZ~
Again, from Theorem A.1, one has the estimate
IKl(N,a,b)l <: 2x/N
(A.18)
The next two result are useful in reducing to the previous case some sums over cyclic subgroups.
87 L e m m a A . 2 . Suppose dl(q - 1), then d,
~(x)=
if x E ( F q ) d
0, i f x ~ ( F ; ) d x # 0
x of order d
1,
if X = 0
Proof. See [Sc], page 85. As an immediate consequence, we have: L e m m a A . 3 . YJk E Z ~ , denote A~ = < A > the cyclic subgroup generated by )~. Let # A ~ = N-1 m = p. If f : ZN x ZN ~r is any complex valued function, then p--1
1
s,t:O
m--1
j,l~-O x , y E Z N *
where {Xo,'" ", Xm-1} is a set of multiplicative characters of order m. Proof. Clearly x '~ E A,x, Vx E ZN*, because A~ is exactly the set of roots of the polynomial x p - 1 = O. Moreover, the map x , 2 * E A~ has multiplicity m. T h a t is, p-1
1
f ( ~ , ~ ) - ms ~
s,~:O
f(ym,Xm)
x,yEZN *
and the result follows immediately from Lemma A.2 because
x,yEZN *
j,l:O x,yEZlv *
Q.E.D. Using the same technique it is also possible to estimate Kloosterman sums over any cyclic subgroup of ZN, namely Proposition
A . 1 VA E Z~, denote A~ - - < A > the cyclic subgroup generated N-1 by A. Let # A ~ = ~ = p. Then, Va, b E Z ~ m
I~
2~ri exp[-~(ax+bx
-1
p-1
~ 2m. )][ = [~.~exp[-~-(aA +bA-~)][ _< C ( m ) v ~
xEA~
s=O
for some constant C ( m ) bounded in m.
Proof. p--1
2hi exp -~--(aA + bA-~) =
s:0
2~i .
• Z exp ~-(~x
,,
+ bx-~) =
xEZ~ m-1
• ~ m
2ri
1
~ ~j(x) exv -~-(~x + bx- )
j = 0 zEZIv
(A.lS)
88 where Xj J --- 0,. 9 9 m - 1 are the multiplicative characters of order m. T h e assertion now follows from the direct extension of Theorem A.2 to the rational functions of type a x + b x - 1 (see [De2], pag. 190 and [Sc], pag. 85). Q.E.D.
89 References.
[AA] V.I.Arnold, A.Avez, Ergodic Problems in Classical Mechanics, W.A.Benjamin, New York, 1968.
[Ap] T.Apostol, Introduction to Analytic Number Theory, Springer-Verlag, New York 1976.
[AY] V.M.Alexeev, M.V.Yakobson, Symbolic dynamics and hyperbolic dynamical systems, Phys. Rep 75, 287-325 (1981).
[HB] J.H.Hannay, M.V.Berry, Quantization of linear maps on a torus - Fresnel diffraction by a periodic grating, Physica D 1, 267-291 (1980).
[B1] R.Bowen, Periodic orbits for hyperbolic flows, Amer. J. Math. 94, 1-30 (1972). [B2] R.Bowen, The equidistribution of closed geodesics, Amer. J. Math. 94, 413-423 (1972).
[BalVor] N.L.Balazs, A.Voros, Chaos on the pseudosphere, Phys.
Rep. 143, N.3, 109-240
(1986).
[BY] F.Bartuccelli, F.Vivaldi, Ideal Orbits of Total Automorphisms, Physica D 39, 194 (1989).
[CdV] Y.Colin de Verdiere, Ergodicit6 et fonctions propres du Laplacien, Commun. Math. Phys. 102, 497-502 (1985).
[DE] M.Degli Esposti, Quantization of the orientation preserving automorphisms of the torus, Ann.Inst.H.Poincard 58,323-341 (1993).
[DGI] M.Degli Esposti, S. Graiti, S. Isola, Classical limit of the quantized hyperbolic toral automorphism, Commun. Math. Phys., to appear (1994).
[De1] P.Deligne, La conjecture de Weil I, Publ. Math. I.H.E.S. 48, 273-308 (1974). [De2] P.Deligne, Cohomologie Etale, Lecture Notes in Mathematics 569, 1977. [Eli B.Eckhardt, Exact eigenfunctions for a quantized map, J. Phys. A 19, 18231833 (1986).
[E2] B.Eckhardt, Quantum mechanics of classically non-integrable systems, Phys. Rep. 163, 205-297 (1988).
[cvz] M.Giannoni, A. Voros and J.Zinn-Justin eds., Chaos and Quantum Physics, Les Houches 1991, Elsevier Publ. 1992.
[H] H.Hasse, Number Theory, Springer-Verlag, Berlin-G5ttingen-Heidelberg, 1980. [Hell D.Hejhal, Duke Math. J. 43,441-482 (1976). [ie2] D.Hejhal, The Selberg Trace Formula, Vol. I, S.L.N. 548, (1976). [ie3] D.Hejhal, The Selberg Trace Formula, Vol. II, S.L.N. 1001, (1980). [HMR] B.Helffer, A.Martinez and D.Robert, Ergodicit@ et limite semiclassique, Commun. Math. Phys. 131,493-520 (1985).
90
[HPS] M.Hirsch, C.Pugh, M.Schub Invariant manifolds, Lect. Notes in Math. 583, Springer Verlag, Berlin, 1977.
[Hu] H.Huber, Zur analytischen Theorie hyperbolischer Raumforme und Bewegungsgruppen, Math. Ann. 138, 1-26 (1959).
[I] S.Isola, C-function and distribution of periodic orbits of toral automorphisms, Europhysics Letters 11, 517-522 (1990).
[K] A.Katok, Entropy and closed geodesics, Ergodic Theory and Dyn. Syst. 2, 339-367 (1982).
[Kall N.Katz, Gauss Sums, Kloosterman Sums, and Monodromy Groups, Princeton University Press, Princeton, 1988.
[Ka2] N.Katz, Sommes Exponentielles,Asterisque 79, (1980). [gel] J.Keating, Ph.D. thesis University of Bristol, 1989. [Ke2] J.Keating, Asymptotic properties of the periodic orbits of the cat maps, Nonlinearity 4, 277-307 (1991).
[Ke3] J.Keating, The cat maps: quantum mechanics and classical motion, Nonlinearity 4, 309-341 (1991).
[Ko] J.Korevaar, On Newman's quick way to the prime number theorem, Mathematical InteUigencer 4, 108-115 (1982).
[La] L.D.Landau, E.M.Lifshitz, Quantum Mechanics, (3rd edition) Pergamon Press, Oxford, 1965.
[LL] V.F.Lazutkin, KAM theory and Semiclassical Approximation to Eigenfunctions, Springer, New York, 1993.
[M] R.Mafie, Ergodic Theory and Differentiable Dynamics, Springer, New York, 1987.
[Mar] E.A.Margulis, On some application of ergodic theory to the study of manifolds of negative curvature, Func. Anal. Appl. 3(4), 89-90 (1969).
[PV] I.Percival and F.Vivaldi, Arithmetical properties of strongly chaotic motions, Physica D 25, 105-130 (1987).
[Pa] W.Parry, Equilibrium states and weighted uniform distribution of closed orbits, in Lect. Notes in Math. 1342, J.C.Alexander Ed. 1988, pp.617-625.
[Po] M.Pollicott, Closed geodesics and zeta functions, in Ergodic Theory, Symbolic Dynamics and Hyperbolic Spaces, T.Bedford, M.Keane, C.Series eds., Oxford Science Publ. 1991, pp.153-173.
[PP] W.Parry, M.Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astdrisque 187-188, (1990). [PPl] W.Parry, M.Pollicott, An analogue of the prime number theorem for closed orbits of Axiom A flows, Annals of Math. 118, 573-591 (1983).
91
[RM]
M.Ram Murthy, Artin's Conjecture for Primitive Roots, Mathematical InteUigencer 10, 59-70 (1988).
[aux]
D.Ruelle, Generalized zeta-functions for expanding maps and Anosov flows, Inventiones Math. 34, 231-242 (1976).
[Ru2] D.Ruelle, Thermodynamic Formalism, Enciclopedia of Math. and its Appl., vol. 5, Addison-Wesley, Reading, Mass. 1978. [RuSa] Z.Rudnick, P.Sarnak, The behaviour of eigenstates of arithmetic hyperbolic surfaces, preprint, 1993.
[Sa]
P.Sarnak, Arithmetic Quantum Chaos, Tel Aviv Lectures 1993 (to appear)
[Schn] A.Schnirelman, Ergodic properties of the eigenfunctions, Usp. Math. Nauk 29, 181-182 (1974). [Schm] W.Schmidt, Equations over finite fields. An elementary approach. Lecture Notes in Mathematics 536, 1976. [SIll Ya.Sinai, The asymptotic behaviour of the number of closed orbits on a compact manifold of negative curvature, Trans. Amer. Math. Soc. 73, 227-250 (1968).
[sI2]
Ya.Sinai, Mathematical Problems in the Theory of Quantum Chaos, Tel Aviv Lectures 1990, appeared in CHAOS/XAOC: Soviet-America Perspectives on Nonlinear Science, D.K.Campbell ed., New York: American Institute of Physics 1990, p. 395.
[Sm]
S.Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73, 74?817 (1967).
[Wa] M.Taylor,
Pseudo-differential operators Princeton, N J, Princeton Univ. Press,
1981.
[VN] J.von Neumann, Beweis des Ergodensatzes und des H-Theorems in der Neuen Mechanik, Zschr.f.Physik 57, 30-70 (1929).
[vz] [Wa] [z] [zl] [z2]
A.Uribe, S.Zelditch, Spectral statistic on Zoll surfaces, preprint, 1993. P.Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, Springer-Verlag, New York, 1982. S.Zelditch, Uniform distribution of Eigenfunctions on Compact hyperbolic Surfaces, Duke Math. J. 55, 919-941 (1987). S.Zelditch, On the rate of quantum ergodicity, part 1 and 2, preprints, 1993. S.Zelditch, Memoirs of A.M.S. 96, No. 495, 1992.
A C T I O N M I N I M I Z I N G O R B I T S IN H A M I L T O M I A N SYSTEMS
John N. Mather and Giovanni Forni
CONTENTS
w
Introduction
w
Area Preserving Twist Mappings. Definitions.
w
Area Preserving Twist Mappings. Examples.
w
Birkhoff Normal Form.
w
The Variational Principle.
w
Existence of Action Minimizing Sets.
w
Properties of the Variational Principle.
w
An Extension Theorem.
w
Minimal Configurations.
w
Existence of Periodic Minimal Configurations.
w
The R o t a t i o n N u m b e r .
w
Irrational R o t a t i o n N u m b e r .
w
Rational R o t a t i o n N u m b e r .
w
Application to Dynamics.
93
w
Birkhoff Invariant Curve Theorem.
w
A Survey of K.A.M. Theory.
w
Birkhoff Invariant Curve Theorem. Applications. Glancing Billiards. Non-existence of invariant curves in the Chirikov standard mapping. The variational property of rotational invariant curves.
w
Destruction
o f I n v a r i a n t Curves.
The Peierls's barrier. The destruction of invariant curves.
w
Dynamics in the Stochastic Regions. Invariant measures supported within the gaps of minimizing sets. Chaotic orbits in a Birkhoff region of instability.
w
Action Minimizing Invariant Measures for Positive Definite Lagrangian Systems. References.
w
Introduction
T h e s t u d y of the dynamics of area preserving m a p p i n g s dates fi'om the pioneering work of Poincar6 [Po]. Poincar6 showed t h a t there is an i n t i m a t e connection between the d y n a m i c s of area preserving m a p p i n g s and the dynamics of Hamiltonian systems in 2 degrees of freedom. Consider a I-Iamiltonian system in 2 degrees of freedom. Such a system is defined by a C 1 function H on a 4 dimensional manifold M provided with a symplectic s t r u c t u r e co, i.e. a closed 2-form whose square vanishes nowhere. Let X = XH denote the symplectic gradient of H , i.e. the vectorfield on M defined by the condition dH = ixw. Let {(I),}teR = {d~H,t}tER denote the flow generated by X. A n energy hypersurface {H = const. } is invariant under (I)t (conservation of energy). If c is a regular value of H , then {H = c} is a 3-dimensional submanifold of M . Consider a 2-dimensional surface S C {H = c} transverse to X , and a t r a j e c t o r y {(I)t(P)}0 0. For Q in a sufficiently small n e i g h b o r h o o d N" of P in S, we m a y find a positive n u m b e r r(Q), depending continuously on Q, with r ( P ) = r , such t h a t O,(Q)(Q) E S. Let 7~(Q) = q~(Q)(Q). T h e m a p p i n g 7~ : iV" --~ S is called a Poincard return mapping associated to the H a m i l t o n i a n system (M, co, H). It is area preserving, in the following sense. Because S C { H = c} is transverse to XH, the restriction of the 2-Ibrm co to S is non-degenerate, i.e. it defines an area form on the 2-dimensional surface S.
94 The Poincar6 return mapping preserves this area form. Moreover, the regularity properties of P reflect those of H, e.g. if H is C ~ (r _> 1), then so is 7~ (in the region where it is defined). Poincard observed that in many cases the dynamical properties of P are closely related to those of {(Ih}tER. For example, consider the case when P is a fixed point of P. In this case, {~t(P)}o
95 by Percival [Pel], [Pc2]. These sets are more plentiflfl t h a n K.A.M. invariant curves, because they exist for every twist diffeomorphism and for every r o t a t i o n number. On the other hand, the existence of K.A.M. invariant closed curves requires further hypotheses. T h e most complete t r e a t m e n t of the existence t h e o r y of K.A.M. invariant closed curves is in H e r m a n ' s book [He]; we content ourselves in w with an overview of some of the known results. Invariant closed curves in area preserving m a p p i n g s of an annulus can be classified by their topological placement, i.e. by whether they b o u n d a disk or s e p a r a t e the b o t t o m from the top of an annulus. In analogy with lunar theory, the former curves are called librational and the l a t t e r are called rotational. The theory developed in these notes deals with a generalization of the r o t a t i o n a l invariant curves, in the sense t h a t every r o t a t i o n a l invariant curve of an area preserving twist diffeomorphism is an action minimizing set, as we will show in w A n i m p o r t a n t result due to G.D.Birkhoff [Bill, [Bi3] is t h a t every r o t a t i o n a l invariant curve of an a r e a preserving twist diffeomorphism f of S 1 x R into itself is the g r a p h of a Lipschitz function u : S a --+ R , i.e. it has the form {(0, u(0)) I 0 E S 1 }. We discuss this result in w and various applications in w In the case of area preserving twist diffeomorphism f minimizing sets m i m i c invaria.nt curves in the following way: be the action minimizing set for the r o t a t i o n n u m b e r w. Then A* C S 1 and a Lipschitz m a p p i n g u : A* ---, R such that E* Moreover, f : E* -+ E*~ preserves the cyclic order on E L A~ C S 1. We discuss this result in w
of S 1 x R the action Let w E R a n d let M* there is a closed subset = {(0, u(0)) l0 E A*}. induced from t h a t on
A n o t h e r a p p r o a c h which led to m a n y of the results described in these lectures is due to A u b r y and his coworkers. A m o n g the m a n y p a p e r s written by A u b r y and his coworkers, the one which is most directly relevant to what we dicuss here is the p a p e r of Aubi'y and Le Dacron [Au-LeD]. Like m a n y of the other p a p e r s of A u b r y et at. this p a p e r is concerned with the construction of one dimensional crystals (FrenkelK o n t o r o v a model). T h e m a t h e m a t i c a l problems which A u b r y and Le Dacron s t u d y are, however, m a t h e m a t i c a l l y equivalent to some of the basic questions on area preserving twist mappings, and their theory provides a useflfl insight into some of these questions. Their theory leads to the i n t r o d u c t i o n of a class of orbits of area preserving twist mappings, which we call action minimizing orbits. We will discuss the A u b r y - L e Dacron theory of action minimizing orbits in w167 In this introduction, we will describe some of the basic results of this theory. T h e union E of the collection of action minimizing orbits is a closed set of the annulus or cylinder. Associated to each action minimizing orbit, there is a r o t a t i o n number, which measures, in a certain sense, the r a t e of advance, with respect to the S 1 factor of a point in the orbit under iteration. Let E~o denote the subset of E consisting of action minimizing orbits of rotation n u m b e r w. We show in w t h a t E* C E,,, where E* is the action minimizing set for the r o t a t i o n n u m b e r w. For irrationM w a n d a generic area preserving twist diffeomorphism, we have E* = E~, although in exceptional cases, this equation does not hohl. For r a t i o n a l w and a generic a r e a preserving twist diffeomorphism, E~, consists of a single periodic orbit, b u t according to the A u b r y - L e Dacron theory, E~, contains other orbits as well.
96
These orbits are heteroclinic to E,~, in the sense of Poincar6, and play an i m p o r t a n t role in the theory of area-preserving twist diffeomorphisms. An i m p o r t a n t application of the theory developed in these lectures concerns d e s t r u c t i o n of i n w u i a n t curves. The K.A.M. theorem asserts that, under certain condition, invariant curves persist under perturbaticms of area preserving twist diffeomorphisms. Since there are several hypotheses for tile K.A.M. t h e o r e m there are several possible converses. In w we discuss a converse of the K.A.M. theorem. Our converse, which generalizes a converse clue to H e r m a n [He] shows t h a t the hypotheses concerning the rotation n u m b e r cannot be relaxed, at least for C ~ p e r t u r b a t i o n s . We also give a p a r t i a l converse to l:[iissmann's K.A.M. theorem [Rs] in the analytic case, which deals with the o p t i m a l condition on the r o t a t i o n number, when p e r t u r b a t i o n s are required to be small in the a m d y t i c topology. A n o t h e r a p p l i c a t i o n of the theory developed in these lectures concerns orbits in a Birkhoff region of instability of an area preserving twist m a p p i n g . In ~19, we sketch how to construct certain r a n d o m or chaotic orbits in such region and various invariant measures (including positive entropy invariant measures) which arise as soon as an action minimizing set of irrational r o t a t i o n n u m b e r is not a r o t a t i o n a l invariant curve. Complete proofs of the results sketched here are given in [ M a l l ]
and [F]. F i n a l l y in w we discuss a. generalization {Ma12-13], [Mfi] of the theory of action minimizing measures to the case of Lagrangian systems in an a r b i t r a r y n u m b e r of degrees of fi'eedom. M a n y people have c o n t r i b u t e d to the theory of the d y n a m i c s of area preserving diffeomorphisms, and more generally to the theory of H a m i l t o n i a n systems. In these lectures, we have m a d e no a t t e m p t to survey the whole theory. Instead, our intention is to provide aal i n t r o d u c t i o n to the theory developed by the first a u t h o r and S.Aubry, known as " A u b r y - M a t h e r theory" . References to work of others is m o s t l y limited to related results and contributions to this theory. We hope these references are sufficiently complete to give a true picture of w h a t is relevant in the theory, b u t we have not a t t e m p t e d to survey what is known on the dynamics of area preserving mappings. In w we give an account, with detailed proofs, of basic results of " A u b r y M a t h e r theory", fMlowing the nlethod of A u b r y and Le Dacron [Au-LeD], as generalized by Bangert [Ba]. We also compare this with the m e t h o d of the first a u t h o r [Mal]. T h e rest of these notes were written by the second author, based p a r t l y on the first a u t h o r ' s lectures at C.I.M.E., and p a r t l y on the second a u t h o r ' s thesis. Some results are proved, b u t in order to survey the literature, m a n y results are given without detailed proofs. T h e a u t h o r s wish to thank C.I.M.E. for its hospitality.
97
w
Area Preserving Twist Mappings. Definitions.
We let S 1 denote the circle R/Z. A large p a r t of these lectures is a discussion of the dynamics of a class of m a p p i n g s of the infinite cylinder S 1 x R onto itself. More generally, we will want to sometimes consider m a p p i n g s of an open subset U ofS 1 xRintoS lxR. For convenience, we will use the following n o t a t i o n throughout: we will let 0 ( r o o d . l ) denote tile s t a n d a r d pa.rameter of S a and x the corresponding p a r a m e t e r of its universal cover, i . e . R . We will let y denote the s t a n d a r d p a r a m e t e r of the second factor of S 1 x R so t h a t (0, y) is a global system of coordinates for S 1 x R and (x, y) is a global system of coordinates of its universal cover. Now we describe the class of mappiI)gs which we will consider. We let J denote the set of pairs (U, f ) , whih tile following properties. We require that U be an open subset of S 1 x R which intersects each vertical line 0 x R in an open interval, which m a y be infinite at one or b o t h ends. We require t h a t f be a h o m e o m o r p h i s m of U onto an open subset fU which also intersects each vertical line in an open interval. We require t h a t f be orientation preserving and a r e a preserving. We require f u r t h e r m o r e t h a t f be exact. To explain this condition, we consider a J o r d a n curve F in U which is homotopically non-trivial in U and has zero area. We let C = S 1 x y, where y < < 0, so t h a t b o t h F and fF lie above C. T h e requirement t h a t f be exact is t h a t the area above C and below F is the same as the area. above C and below f F . Since f is area preserving and orientation preserving this condition is i n d e p e n d e n t of the choice of F. Finally, we impose a monotone twist condition. Let 7rl : S 1 x R ~ S 1 denote the p r o j e c t i o n on the first factor. A monotone twist condition is that, for each 0 E S 1, the m a p p i n g 7rlf:UM(0xR)--*S 1 is a local homeomorphism. Since the d o m a i n is an open interval and the range is S ~, this condition a m o u n t s to saying t h a t rr~ f l u M (0 x R ) is m o n o t o n e increasing or m o n o t o n e decreasing. If 7rlflg n (0 x R) is m o n o t o n e incresing for all 0 E S ~, we will say t h a t f sa.tisfies a. positive monotone twist condition. In the other case, we will say t h a t f satisfies a negative monotone twist condition. E i t h e r of these cases can be reduced to the other, obviously. We impose the positive m o n o t o n e twist condition. Thus, d is the set of pairs (U, f ) , where U and fU are a.s above a n d f is exact area. preserving, orientation preserving h o m e o n m r p h i s m which satisfies a positive m o n o t o n e twist condition, and is homotopic to the inclusion mapping. We will also consider, for each positive integer r and for r = oo, the subset J " consisting of all pairs (U, f ) E Y such t h a t f is a C" diffeomorphism of U onto fU and ~rl rio x R has everywhere positive derivative for each 0 C S 1. In the next two sections we give some examples of elements of d.
98 w
Area Preserving Twist Mappings. Examples.
In an elementary expository article [Bi2], G.D. Birkhoff discussed billiards in a Convex region as a simple and appealing example of a Hamiltonian system. For our discussion we will consider a convex, bounded region R whose b o u n d a r y OR is C 2. Billiards in R means the dynamics of a particle which moves at constant velocity in R and is reflected by the boundary, according to the rule angle of incidence equals angle of reflection. There is a simple way to prove the existence of periodic orbits in R, discussed by Birkhoff in [Bi2]. Among all convex q-gons inscribed in R, consider one of m a x i m u m perimeter. This exists by a simple compactness argument. An elementary and well known variational argument shows that two sides incident to the same vertex make the same angle with respect to the boundary, for such a maximM q-gon. Thus, a particle which travels at constant speed along the boundary of a maximal q-gon describes an orbit of billim'ds. Suet1 a trajectory is obviously periodic. More generally, Birkhoff calls a polygon inscribed in R harmonic if at each vertex, the two sides incident to that vertex make the same angle with respect to the boundary. Obviously, harmonic polygons correspond to periodic orbits. As Birkhoff remarked, the dynamics of billiards in R can largely be reduced to the dynamics of an associated area preserving mapping. If P 6 OR and 0 < r < zr, consider the ray r = r(P, r beginning at P and making an angle r with OR, counted in the counterclockwise direction starting fiom OR. Extend r until it meets OR again. Let P ' be the point other than P where r meets OR and let r be the angle which r makes at P ' with OR, counted in the clockwise direction starting fi'om OR. Let f be the mapping which associates ( P ' , r to (P, r Thus, f is a mapping of OR x (0, 7r) into itself. The dynamics of f is closely related to the dynamics of billiards in R. For example, periodic orbits of f correspond to periodic trajectories of the billiards. Let s denote the arc length parameter of OR, with the counterclockwise direction taken as positive. Let L denote the total arc length of OR. Let h(.%s') = -[[P.~ - P~,[[, for s < .s' < s + L, where P., denotes the point of OR with parameter value s and N' I] denotes the Euclidean norm on the plane. (The sign is chosen so as to agree with sign convention which we use throughout these lectures.) It is easy to see that the relation (3.1)
f(P~, r = (P.,,, r
is equivalent to the two equations (3.2)
I y = -01h(~, ~')
~' = o.,h(.~,.,')
if y = - - c o s r and y' = - c o s r In other words, given four numbers s , s ' , r 1 6 2 satisfying s _< ~' _< .s + L and 0 _< r _< r, 0 _< r _< rr, these numbers satisfy (3.1) if and only if they satisfy (3.2). Here, 0a and 0.2 denote the first partial derivatives with respect to the first and second variable.
99
From this it is easy to see that f preserves the 2-form ds A dy = sin r ds A de. Indeed, by (3.2), d.s A dy = -012 h(s, .s') d.s A d s ' = ds' A dy'. Thus, f is orientation preserving and area. preserving (for the measure ds dy = sin r ds dr Exactness is obvious in this case. The positive monotone twist condition is also obvious. A second important example arises from the Frenkel-Kontorova model of solid state physics. Let h ( x , x ' ) = (x - x ' ) 2 / 2 + u(x), where u is a sufficiently smooth real valued function of one real variable. The Frenkel-Kontorova model concerns the case u ( x ) = k sin 2rrx, where k is a real parameter. In the model, a crystM is represented as bi-infinite sequence x = (..., xi, ...) of real numbers (which are thought of as the positions of the atoms of the crystal), which minimize the energy ~-]~i~-o~ h ( x i , x i + ~ ) . Since the sum ~i~=_o~ h ( x i , x i + l ) is rarely convergent, the energy is not defined. However, the notion of minimizing the energy can be defined as follows: x minimizes the energy if for every bi-infinite sequence ~ = (...,~i, .-.) such that ~-~o~ I~i - xil < oo, we have or
> 0 --or
and the sum on the left is absolutely convergent. To the Frenkel-Kontorova model, it is possible to associate an exact area preserving, orientation preserving twist mapping f defined by f(x, y) = (x', y') if and only if / Y = -01h(.T, z ' )
(3.3)
v'
=
.
To be explicit, we m a y solve these equations and find that f is given by (3.4)
x' = x + y - d u ( x ) / d x
y' = y - du(x)/d.T~ .
In this model, a bi-infinite sequence x = (..., xi, ...) is ca.lled a configuration and such a configuration is said to be ,~tationary if
0
O~ (h(z,_~,:~) + h(z,,'~+~)) = 0 ,
i E Z.
This condition is obtained by formally differentiating the infinite sum oo
h(x. --OO
and setting the result equal to zero. Of course, this procedure has no mathematical meaning, but it is easily verified that a crystal in the model (also called a m i n i m a l configuration) is stationary in this sense. If x = (..., xi, ...) is a stationary configuration and yi = - a l h ( x , , z i + l ) = O~h(z~_~, z , ) ,
100
then f(xi, yi) = (xi+l, Yi+l), i.e. (..., (xi, yi), ...) is an orbit of the mapping f. This correspondence between orbits of f and minimal configurations of h is one-one. Since f(x + 1,y) = f(x,y) + (1,0), we obtain a mapping of S ~ x R into itself which is in the class J ~ , introduced in the previous section. Thus, the theory developed in these lectures applies to the Frenkel-Kontorova model. In fact, Aubry et al. developed their theory in the context of the Frenkel-Kontorova model, and we will use many ideas h'om their theory in these lectures. In dynamics, one wishes to understand all orbits. The interest in the minireal orbits (associated to the minimal configurations) arises from the fact that the minimal orbits form a class of orbits which can be fairly thoroughly understood. This is in contrast to the situation fl)r general orbits of even simple area preserving mappings, such as the mapping associated to the Frenkel-Kontorova model. The orbit structure of general area preserving mappings defies complete understanding. On the other hand, in the Frenkel-Kontorova model, the crystals (minimal configurations) are the main object of study, and Aubry and Le Dacron [Au-LeD] gave a fairly complete picture of their structure. The mapping associated to the Frenkel-Kontorova model was extensively studied numerically (and non-rigorously) by physicists such as Chirikov, Greene, Percival; starting in the seventies. We will not discuss these mlmerical studies in these lectures however. The reader may c(msult the book of Lichtenberg and Lieberman [L-L]. We will refer to a mapping of the form (3.4) with u(x) = ksin2Trx as a Chirikov mapping.
101
w
Birkhoff Normal Form.
Let f be a C ~ m a p p i n g of an open set in R 2 in R 2. Let P be a fixed p o i n t of f . Let A a n d # be the eigenvalues of elf(P). We will suppose t h a t f is a r e a preserving a n d orientation preserving. T h e n A# = 1. Let r = A + # denote the trace of df(P). It is t r a d i t i o n a l to classify fixed points of area. preserving m a p p i n g s according to three cases: if [rJ > 2, the fixed point is said to be hTlperbolic. If [r I = 2, the fixed point is said to be p~m, bolic. If [r] < 2, the fixed point is said to be elliptic. If one of the eigenvalues, say A, is conjugate of it. T h e n IAI ~ = A X = A ~ = so in this case the fixed point is elliptic. real, then It] = [;~ + ~l - 2, so the fixed
imaginary, then the other is the complex 1, and I#1 e = z. Clearly, I,-I = I),+ #1 < 1, If, on the other hand, b o t h eigenvalues are point is p a r a b o l i c or hyperbolic.
In the case t h a t the fixed point is hyperbolic, the d y n a m i c s of f is fairly completely described by the H a r t m a n - G r o s s m a n theorem which asserts t h a t the germ of f at P is topologically conjugate to its linear p a r t , i.e. there exists a homeomorp h i s m h of an open n e i g h b o r h o o d U of P in R 2 onto an open n e i g h b o r h o o d hU of P such t h a t h o d f ( P ) = f o h, where t)oth sides are defined. See e.g. [P-deM]. In contrast, the case when the fixed point is p a r a b o l i c is complicated, b u t it does not occur generically, and we will not discuss it. A p a r t i a l u n d e r s t a n d i n g of the case when the fixed point is elliptic m a y be o b t a i n e d t h r o u g h Birkhoff n o r m a l form, which we will discuss in this section. B i r k h o f f N o r m a l F o r m T h e o r e m [ B i l ] . Suppose that P is an elliptic fixed point of a C ~ area preserving mapping f of an open subset of the pla~e into the plane. There exists a C ~~ 1ocM coordinate system (, 71, centered at P with respect to which f has the tbnn f ( r = r where ( ' = r exp 2rri(/t0 + ~ l p 2 + ... + [iNp 2N) + O(p k)
and k = 2 N + 2 or 2 N + 3 . Here, r = ( + i q denote the complex coordinate associated to the coordinate system (, q, a n d p = ((2 + q2)1/2. If an eigenvedue A d d f ( P ) is not a root of unity, then we m a y take N = ~ (with r e m i n d e r O(roo)). If A is a primitive q-root of unity, then we m a y take k = q a n d N = [q - 2/2]. Furthermore the coordinate system (, r~ m a y be chosen to preserve the a r e a / b r m , i.e. d ( A d r / = dx A dy, where z, y axe the st~]dard coordinates ol] R 2. We will refer to the m a p p i n g N ( ( ) = ~ exp 27ri(/:r + / h p 2 + ... + [~Np2N) as the normal form of f . Thus, f is the sum of the n o r m a l form and a r e m i n d e r t e r m which is no bigger t h a n O(pk), where k = ocif an eigenvalue A of dr(P) is not a root of u n i t y and k = q if an eigenvalue is a qth root of unity. Since #0, # l , ... are real m~mbers, ]exp27ri(/t0 + [tip 2 + ...)[ = 1. Thus, the circles p = const, are invariant under the n o r m a l form N. Clearly, exp 2~riflo is one of the eigenvalues of d f ( P ) . T h e remaining [t~'s are called the Birkhoff invaria~t~ of f.
102
If at least one of the Birkhoff invariants is not zero, then f is a twist m a p p i n g in a sufficiently small p u n c t u r e d neighborhood of P . More precisely, let 0 = t a n -1 q/~. T h e n r = (O,p 2) m a p s U \ {P} into S 1 x R , where U is a. sufficiently small open n e i g h b o r h o o d of P in R 2. Let W = O(U \ { P } ) C $1 x R. Let g = r o f o ~ - 1 : W S 1 x R . If one of the Birkhoff invariants is non-vanishing and the first non-vanishing Birkhoff invaxiant is positive, then (W, g) E joo, provided tha.t U is chosen to be a sufficiently small n e i g h b o r h o o d of P . 011 the other hand, if the first non-vanishing Birkhoff invariant is negative, then (gW, g-l) E joo. This is our t h i r d example of a.n area preserving twist m a p p i n g .
103
w
The Variational Principle.
Let ( U , f ) E J (see w Let 0 = ~ - l U , where 7r: R 2 ~ S 1 • R = ( R / Z ) • R denotes the projection. Let ] : U ~ R 2 be a lift of f to the universal cover. Let V C R 2 be the set of ( x , x ' ) such that there exists y , y ' E R with (x,y) E U and f ( x , y ) = (x',y'). Clearly, V is open in R 2. Note that if ( x , x ' ) E V, there is only one pair (y, y') which satisfies this condition, by the twist condition. Moreover, (y,y') depends continuously on (x, x'). From the twist condition, it fi:,llows that V intersects ea.ch vertical line in R 2 and each horizontal line in R 2 in an open interval. In particular V is simply connected. From the area. preserving property of f , it follows that :q' dx' - y dx is a closed 1form (in the sense of distributions) and therefore there is a C 1 real valued function h = h(x, z') on V such that (5.1)
dh = y' dx' - gdx .
Moreover, if (U, f ) E J~, then h is of class C r+l on V. We will call h the variational principle associated to (U, f). We will extend h to all of R 2 by the convention that h(x, x') = + c r when (x, x') r V. Note that the condition (5.1) is equivalent to the equations
(5.2)
{ y = -0~h(~, ~') y'= o~h(x,~')
Thus the variational principle could also be defined by the condition that (5.2) holds if and only if f ( x , y) = (x', y'). The variational principle associated to (U, f ) is unique up to addition of a constant. We have already seen two examples of this variational principle. For billim'ds this is expressed by the equivalence of (3.1) and (3.2). We defined the Chirikov mapping by (3.3). The variational principle satisfies the following periodicity condition: V is invariant under the traslation (x, x') ~ (x + 1, :c' + 1) of the plane, and
(5.3)
h(:,: + 1,:,:' + 1) = h(:,:,z')
This is an easy consequence of exactness, which means that
j ( y' d x ' - Y d:c = 0 where the integral is taken over a curve which goes once around the cylinder. It satisfies the following monotonicity conditions: in V, Olh(X, x') is a strictly decreasing function of x' and 02h(x, x') is a strictly decreasing function of x. The first follows from y = -O~h(x, x') and the positive monotone twist condition (x' is a strictly increasing fimetion of y). The second follows fi'om y' = 02h(x, x') and the fact that f - 1 satisfies a negative monotone twist condition (x is a strictly decreasing function of y').
104
If (U, f ) E j 1 then the stronger condition
(5.4)
O,2h(.~,z') < 0,
(~,:,:')ev
is satisfied. We will also need the integrated form of (5.4), which is also true for (U, f ) C J:
(5.5)
h(r162 + h(:~, z') - h(~,:~') - h(:~, r if ~ < x and ~' <:c ~ .
< 0,
It follows easily from the twist condition that the domain V where h is finite has the following form: there exist functions fi_, fl+ such that fl+(t + 1) = f l i ( t ) + 1, both are increasing (i.e.s < t implies fl• < fl• fl_ (resp.fl+) is everywhere finite or identically - o o (resp.+c~), and V is the region above graph fl_ and below graph fl+. Here graph fl+ is empty if fl• = + ~ and graph fl• = {(x, x') I fl+ ( x - 0) < x' <_ fl• + 0)} if fl• is finite.
105
w
Existence of Action Minimizing Sets.
Let (U, f ) C J, let h be tile variational principle associated to it and let V be the domain oil which h is finite. Let liminf h ( x , x ' ) . (~,~')~(~,~:')
h*(x,x')=
Then h* = h except on the flontier of V, since h is continuous on V and +oo elsewhere. Moreover, h* is lower selni-contimlous. Let ]I1 denote the set of mappings r : R ---* R which are increasing (i.e..s _< t implies r _< r and satisfying the periodicity condition r + 1) = r + 1. Let Y denote Y1 modulo the h~llowing identifications: r ,-~ r if there exists a E R such that r = r + a) at all but at most countably m a n y t. ForwCRandr
let
(6.1)
F,~(r =
i
a+l
h*(O(t), r
+ w))
dt
.
To show that this integral exists (possibly with the value +co), we will show that the integrand is bounded below. First, h*(:c, x') > - o o everywhere. This is obvious except when ( x , x ' ) is in the fiontier of V. Suppose, for example, that (~,x') C graphfl+. It is easy to see that there exists x > ~ and {' < z.' such that (~,~'), (x, {') and (x, x') are in V. Then (5.5) guarantees that
h*(~, ~') ___h(~, ~') + 1,,(.~,.~')
h(.~, ~').
-
The ease of a point in graphfl_ may be treated similarly. Thus, h*(x, x') > - o c everywhere.
Since r
+~
-
1 _< r
+ ~ ) _< r
+ ~ + 1, and h*(:~"+ 1,~' + 1) = h*(~,x'),
the integrand of (6.1) is bounded below a.s asserted. Thus, the integral (6.1) exists, although it m a y be +oc. The integral (6.1) is independent of the choice of a E R, since h*(x + 1, x' + 1) = + 1) = r + 1. Furthermore, if r and r define the same element in Y, then F~(r = F,~(r Thus, we have defined a flmction
h*(x, x') and r
F~ : Y ~ R U {+or
,
which we call Percival's Lagrangian (compare [Mall). For r E ]1"1, let g r a p h r = {(x,~') I r - 0 ) ___ x, < r + 0)}. For r r C Y~, let d1(r r denote the Hausdorff distance between graph r and graph r i.e. dl(r r
= max {supiIff liP - Qll , supinf HP - QII} P
'4
Q
P
'
where P ranges over graph r Q ranges over graph ~b and ]]-I] denotes the Euclidean distance. For [r [r e Y, let d([r162
= i n f { d l ( r 1 6 2 1 6 2 e [r
r E [r
9
106
It is easy to verify that d is a metric on Y and Y is compact with respect to this metric. The topology associated with this metric is the topology of almost everywhere convergence. Let A denote the Lebesgue measure on R. If r e 111, then r is a measure on R which is invariant under the translation t --* t + 1. Let ~r : R --~ R / Z denote the projection. Then 7r,(r , a + 1)) is a p r o b a b i l i t y m e a s u r e on R / Z which we will denote (by abuse of terminology) by r This measure is independent of the choice of a 6 R and depends only on the equivalence class of r in Y. In fact, [r -~ r is a homeolnorphism of Y onto the space of probability measures on R / Z , provided with the vague topology. It is easy to check that F~ is lower senli-continuous with respect to the topology of almost everywhele convergence. (Compare [Ma.1] where a detailed proof that F~ is continuous was given under a slightly stronger hypothesis). Hence: Theorem
6.1. There exists a m i n i m i z e r r
of F,~.
It may happen, however, that F~ -- +oo. The connection with the dynamics comes fl'om the Euler-Lagrange equation: T h e o r e m 6.2. I f r = r
is a m i n i m i z e r of F~, and r
+ o~ 4- O) E V for all t E R ,
then the Euler-Lagrange equation - ~, + 0), r
O~h(r
+ 0)) + O~h(r
+ 0), r
+ w + 0)) = 0
is satis~ed for all t 6 R .
Since r may be discontinuous, it is essential to use the limits r
lim r
and
r
lim r
8~t-
,~--*t +
P r o o f . Consider a l-parameter family r
d
d---~F,~(r ----
/
/"+1 d
I~=0 = ~,,
~ h(r
with r r
r
+
w))
+
a~h(r
We have
+ w)l.~=0 dt =
,~+1
{0, h(r
= r
- w), r
d ~ r
dt ,
d a
at least formally, where we have used r + 1) = r + 1 and h(x + 1, x' + 1) = h(x, x ' ) to make the change of variables. The problem is to choose the family so that r 6 ]I1 and so that the formal operations above are justified. First, we consider the case when t - w, t and t + w are points of continuity of r and r is not constant in any interval containing t. In this case, we let u~ be a 1-pa.rameter family of diffeomorphisms of R with 'u0 = identity. We suppose that u~ dependes infinitely differentiably on all variables and u~(x + 1) = u~(x) + 1. We set r ----us o r We may suppose that u , ( x ) = x for x r [r - 6, r + 6] + Z. Then the formal argument above applies and dr I~=o vanishes outside r ([r -6, r 6]) + Z. By choosing u~ appropriately we may assume that dr always has the same sign in the non-vanishing region. By the continuity assumptions and the
107
assumption that r is not constant in any interval containing t, it will be the case that c92h(r - w), r + 01h(r r + w)) always has the same sign in the region where dr o does not vanish. Hence, the Euler-Lagra.nge equation is satisfied in this case. Next, t, but we containing to (5.4), it
(6.2)
we drop the assumption that r is not constant in any interval containing retain the other assumptions. We let [t0,tl] be the maximal interval t on which r is constant. From the monotonicity properties of r related follows that
o~h(r
-
~), r
+ Olh(r
r
+ ~)) = E(0
is a decreasing function in [to, t~] and is constant in this interva.1 only if both r and r + w) are constant in it.
-~)
If we try to apply the same argument as before, we may run into the problem that E(t) changes sign in the interval [t0, t~]. If it doesn't, we may apply the same argument as before. If it does, we choose t* such that E(r) ~ 0 for T _< t* and E(r) <_0 for r ~ t* ( and r E [t0,tl] in both cases). We choose u~ as above and for definiteness we suppose that du,(x)/d.s]~=o > 0 where it does not vanish. In this case, however, we define r = u, o r in [t*, t + 1/2] and r = u _ , o r in (t - 1/2, t*), and extend it by periodicity. We still have r E Y1, h}r .s _> 0 (but not. for ,~ < 0; the order preserving property fails). Moreover, dF,~(b.~)/d.~]~=o < 0, so we obtain a contradiction to the assumption that r mininfizes F~. Thus, we again obtain the Euler-Lagrange equation in this case. (Note that tiffs argument fails if r maximizes F~). Thus we have proved the Euler-Lagrange equation when t - w, t, t + w are points of continuity of r The set of t satisfying these conditions is dense in R. By passing to the limit, we obtain the conclusion of Theorem 6.2. []
Suppose that the hypotheses of Theorem 6.2 holds. Set ~•
= -O,h(r
+ 0),r
+ w + 0)) = & h ( r
- w + 0),r
+ 0)) .
Let M* = M * ( f ) = {(r
4 - 0 ) , r l + ( t ) ) I t E R}. Then M* C 0 (the universal covering space of U; cf. w and is invaria.nt under f. These assertions follow from the definition of V and h (w Obviously M* is invariant under the translation (x,y) ~ ( x + l , y ) . We let E* = E * ( f ) C U denote the projection of M*. Obviously, E* is invariant under f. We call E* (ol- iV/*) an action nfinimizing set of rotation number w for f (oi f). To finish our discussion of the existence of action minimizing sets, we need to describe conditions under which the hypothesis of Theorem 6.2 that (r 40), r + w 4- 0)) E V for all t E R is satisfied. In the case of the Chirikov mapping V = R 2, so this condition is satisfied trivially. In this case, we may therefore conclude that an action minimizing set E* exists for every real number w.
108
More generally, V = R 2 and consequently E*~ exists fi:n every real w when 1 xRand
U=fU=S
~l](Z,V)~+oo,
as y ~ + c e ,
where 7rl : R 2 --* R denotes the projection on the first factor. W h e n this condition is satisfied, we will say t h a t f twists each end of the cylinder infinitely. To fit billiards into our scheme, we introduce the coordinate x = s/L, where s is the p a r a m e t e r i z a t i o n of the b o u n d a r y by arc-length and L is the total arc-length of the boundary. T h e n the m a p p i n g associated to billiards is a r e a preserving for the area form dx h dy, where y = - cos r as before. In this case U = S 1 x ( - 1 , 1) and V = { ( x , x ' ) 9 R 2 Ix < x' < z + 1}. I f w < 0 or w > 1, then F~ - +c~. Also when w = 0 or co = 1, no r satisfies the hypothesis of T h e o r e m 6.2. W h e n 0 < w < 1, the minimizing r satisfies the hypothesis of T h e o r e m 6.2. Obviously, there exist r such t h a t F ~ ( r < +oo, so F~(r < +o% where r is a minimizing r Consequently, ( r + 0), r + w + 0)) 9 V, for all t 9 R . Thus, it suffices to show t h a t ( r + 0), r + co + 0)) is never in the b o u n d a r y of V, i.e. it suffices to exclude the possibility r + co 4- 0) = r 4- 0) and r + co 4- 0) = r 4- 0) + 1. For example, suppose t h a t r + w + 0) = r + 0). F r o m t h e m o n o t o n i c i t y and the periodicity of r it follows t h a t there exists an integer n > 2 such t h a t r > r Let n be the smallest such integer. Let g = r = :r, = r a n d x' = r It is easy to check t h a t 02h(~.,x) + O~h(x,x') < 0. Using an a r g u m e n t similar to t h a t used in the proof of T h e o r e m 6.2, we m a y then p r o d u c e a 1 - p a r a m e t e r family r 0 < s < e, r 9 ]I1, r = r such t h a t dF~(r < 0. This contradicts the a s s u m p t i o n t h a t r minimizes F~o. T h e other cases m a y be t r e a t e d similarly and we o b t a i n t h a t r satisfies the hypothesis of T h e o r e m 6.2. We m a y therefore conclude t h a t in the case of billiards, there exists an action minimizing set M*o~ for every 0 < co < 1. We did not feel t h a t it was necessary to give the complete details in the argument above, since t h e y were given in a more general setting in [Mal]. T h e r e we showed t h a t if (U, f ) 9 J and fU = U, then M~* exists for every co in a certain range co_ < co < co+, where co_ and co+ are the r o t a t i o n lnunbers of the m a p p i n g s fl_ a n d /3+ defining the b o u n d a r y of V. (See the end of w In the case of billiards, co_ = 0 and co+ = I. Of course, the range of admissible r o t a t i o n numbers d e p e n d s on the choice of the lift f of f . If f a n d fx are two lifts of f , then they differs by t r a n s l a t i o n by an integer f l = f + (n, 0), for some n 9 Z. T h e r o t a t i o n n u m b e r s of the action . . .l .m l z m g sets are changed by the same number, i.e. M*~o+,(Yl) = = M*~ ( J"-" mm ) . For example, in the case of billiards, the range of admissible r o t a t i o n n u m b e r s m a y be n < w < n + 1, for any integer n, according to the lift. As a final e x a m p l e of the application of T h e o r e m s 6.1 a n d 6.2, we consider an elliptic fixed point P of a Coo area preserving diffeomorphism f of a surface. According to Birkhoff's n o r m a l form theorem, we choose a C ~ complex local coord i n a t e ~ = f + it/, centered at P , with respect to which f has the form f ( r = r where
~' = N(~) + O(p k)
109
and N(~) = ~ exp27ri(fi0 + [31p 2 + ... +/3Np2N). We suppose, fln'thermore, that d~ A dT/is the given area fl)rm on the surfa.ce. We suppose that at least one of the Birkhoff invariants ~1, ft2, ... is non-vanishing. Then, as described in w f is loca.lly a, twist ma.pping in a punctured neighborhood of P. We. ma.y suppose, without loss of genera.lity, tha.t the first non-vanishing Birkhoff invaria.nt is positive. (Otherwise, replace f by f - 1 . ) Then, in the notation used at the end of w (W, g) E joo. It ma.y be shown that for an appropria.te choice of lift ~ of g, the hypothesis of Theorem 6.2 is sa,tisfied for fl0 < w < [~0 + e, for a. sufficiently small positive mtmber e. (cf. w Thus, the action minimizing sets M~* exist for w in that range.
110
w
Properties
of the Variational Principle.
In [Au-LeD], Aubry and Le Daeron proved the existence of action minimizing sets for a Chirikov m a p p i n g and noted that their method works more generally. Their method differs from the first author's [Mal] (described in the previous section) and leads to further results. In [Ba], Bangert generalized the Aubry-Le Dacron method, showing that their results still hold under weak conditions on the variation principle. In this section, we discuss Bangert's conditions and some further conditions introduced by the first author [Ma8]. T h e conditions introduced in this section are satisfied by the variational principle associated to a Chirikov m a p p i n g (i.e. the Frenkel-Kontorova model). More generally, they are satisfied for the variational principle associated to any diffeomorphism f of the cylinder such that (S 1 x R, f ) E J and f twists S 1 x R infinitely at each end, provided that an additional uniformity condition explained later is satisfied. In addition, it is possible to associate a variational principle to a finite composition of such mappings in such a way that these conditions are still satisfied. This permits the extension of the theory of action minimizing orbits to such compositions. Bangert pointed out a further application of his generalization of the Aubry-Le Daeron theory in [Ba]. In 1932, Hedhmd [I-Id] developed a theory of geodesics on the 2-torus, with an arbitrary C ~ metric, whose lifts to the universM cover globally minimize arc length. Hedlund called these "class A" geodesics. Bangert showed [Ba] that a large part of the theory of class A geodesics follows from his generalization of the Aubry-Le Dacron theory. Here are Bangert's conditions on the variational principle h: (H0) h is a continuous real valued function whose domain is R 2.
(H1) h(x + 1,x' + 1) = h(x,x'). (H2) h(x,x') ~ +oo as Ix - x' I ---* +co. (Ha) T h e inequality (5.5) holds. (H4) Consider real numbers g, x, x', 7, ~'.
Suppose both of the functions
h(g,y) + h(y,x') and h(~,y) + h(y,(') of y take a m i n i m u m value at y = x. Then either ( g - ~ ) ( x ' - ( ' ) < 0or5=~andx' = ('. For example, i f f maps S a x R onto itself, (S 1 x R, f ) E J, and f twists each end of S 1 x R infinitely, then the variational principle associated to f satisfies (H0) - (H~). We have already shown (H0), ( g ~ ) and (Ha) in w Cleaaly y,y' ~ 4-oo as x ' - x --* =t=ec; thus, (/-/2) follows from (5.2). Condition (//4) follows from the facts (w that O~h(x,x') is a strictly decreasing function of x t and 02h(x, x I) is a strictly decreasing function of x. For, if ~ > .g and ~ _> x ~ and one of these inequalities is strict, then d
<
d
+
which contradicts the assumption that both of the functions h(~, y) + h(y, ~') and h(~, y) + h(y, x') take a m i n i m u m value at y -- x. It is often convenient to consider two further conditions, introduced by the first author in [Ma8]. The first of these is a strengthening of (Ha):
111
(Hs) There exists a positive continuous flmction p on R 2 such that i
(7.1)
h(~,~')+h(x,z')-h(~,x')-h(x,~')
, p(u,u')dudu'
and ~ ' < x ' .
The second is expressed in terms of a positive number 0.
(H6o) The function 0(x
- x')~/2
- t,(x, x')
is convex in each variable x and x'. If h satisfies this condition for some positive number 0, we will say that h satisfies (/-/6). If h satisfies (H1) - (H~) and (Heo), we will say that it satisfies (Ho). If h satisfies (Ho) for some positive number 0, we will say that it satisfies (H). The conditions just considered on f are not enough to guarantee that h satisfies (//5) and (//6). However, if we impose, in addition to the other conditions, the condition that (81 x R, f ) E j1 (i.e. a dif[~rentiability hypothesis), then h satisfies (Hs), by (5.4). Indeed, we may set p = -012 h and then we have equality in (7.1). For (He) to hold, we need a fllrther strengthening of the differentiability hypothesis. If we set ]'(x,y) = (x',y'), the condition that (S 1 • R , f ) E dl may be expressed as two conditions: f is a C 1 diffeomorphism and c3x~/c3y > 0. Also, this condition m a y be expressed in terms of f - l , namely f - 1 is a C 1 diffeomorphism and 0x/cgy ~ < 0, where x' and y' are taken as independent variables. If, in addition to the conditions we have already imposed on f, we impose (7.2)
Oy'/Oy < 8 and
Oy/cgy' > - 8 ,
Oz'/Oy
Ox/0y'
-
-
then the associated variational principle satisfies (//6o). (In the first of these inequalities, we take x and y as independent variables and x' and y' as dependent variables; in the second, we take x' and y~ as independent variables and x and y as dependent variables).
Indeed, in this case h is C 2, and
(7.3)
a~ h(x, x') - Ox,/Oy ay'/ay
and
c911h -
Oy/ay' Ox/Oy' "
These equations follow fl'om (5.2) by a calculus exercise. In doing the exercise, it is important to keep in mind that x and x' are the independent variables on the left side of each of these equations, whereas x and y are the independent variables on the right side of the first of these equations and x' and y' are the independent variables on the right side of the second of these equations. From (7.3), it is clear that (7.4)
all h(x,x') <_8 and
022h(x,x') <_8.
These two conditions are equivalent to (//6o) in the case that h is C 2.
112
The conditions (7.2) may be interpreted geometrically. The positive monotone twist condition implies that f turns a vertical vector to the right and f - 1 turns a vertical vector to the left. Let 0 = cot fl, where 0 < fl < ~r/2. The first inequality in (7.2) is equivalent to the condition that f turns every vertical vector to the right by an angle at least #. The second inequality in (7.2) is equivalent to the condition that f - 1 turns every vertical vector to the left by an angle of at least ft. The second inequality in (7.2) is equivalent to the conditions that f - 1 turns every vertical vector to the left by an angle of at least ft. This leads us to introduce the following notation: for r >__ 1, we denote by Y~ the set of diffeomorphisms of S 1 x R onto itself such that (S 1 x R, f ) E j r , and f turns every vertical vector to the right by an angle of at least fl, and f - 1 turns every vertical vector to the left by at least/~, where fl = cot -1 0, 0 < fl < rr/2. We let Y2 = U0>0 Y~. We may summarize the discussion above as: 7.1. It' f E J~, then its associated variationad principle h satis//es (Ho). It" f E J2, then h satisfies (H). Theorem
It is clear that a Chirikov mapping is a member of J ~ . On the other hand, the condition (H) is not satisfied by the variational principle associated to a billiards mapping, nor by the variational principle associated to a neighborhood of an elliptic fixed point of an area-preserving mapping with non-vanishing Birkhoff invariant. Nonetheless, the extension theorem of the next section allows us to apply results about variational principles satisfying (H) to these mappings.
113
w
An Extension Theorem.
T h e o r e m 8.1. Let fl_, fl+ : R --* R be C ~-1 diIfeomorphisms satisfying fl:~(x + 1) = fl+(x) + 1, where ,' _> 1. Suppose tizrthermore that fl_(x) < fl+(x). Let W = { ( x , x ' ) [ f l _ ( x ) <_ x'<_ fl+(x)}. Let h : W --* R be a C ~+1 1hnction such that h(x + 1, x' + 1) = h(x, x') and 012 h < O. Then h extends to a C ~+~ function defined on ali o f R 2 satisfying ( H ). P r o o f . W is invariant under the translation (x, x') ~ ( x + l , x ' + l ) . Its quotient by this translation is compact. Consequently, there exists a number 6 > 0 such that 012 h < -?i. It is well known that a C ' - 1 fimction on a C ~-1 submanifold extends to the ambient space. Applied here, this implies that there exists a C ~-1 function p on R 2 such that p(x + 1, x' + 1) = p(x, x'), p <_ - ~ and p [ W = -0~2 h. In addition, we may suppose p = constant outside of the set {fl(x) - 1 _< x' _< fl+(x) + 1}. Then there is a unique extension of h to all of R 2 satisfying p = 012 h. This may be seen as follows: The va.lue of (.91h on graphfl_ and the relation 02(01h) = p determine 01 h on all of R ~. Then the value of h on graph fi_ and 01h determine h on all of R 2" From this procedure for determining h, the periodicity condition ( H i ) is obvious. Since 02(O~h) = p _< - ~ , it follows that O~h(x,z') --* 4-00 as x - x' --* 4-00. (H2) follows. (Ha) and the stronger condition (H5) follow immediately from 0~2 h _< - 6 , as does (H4) by an argument already discussed. Finally (H6) holds in {(x,x') [ f l _ ( x ) - I < x' < f l + ( x ) + l } by compactness. Since 02(011 h) = 01(c912 h) 0 outside { ( x , x ' ) ] f l _ ( x ) - 1 < x' <_ fl+(x) + 1}, a bound 01~ h _< 0 which holds in { ( x , x ' ) [ f l _ ( x ) - 1 <_ x' < fr + 1} holds in all o f R 2. []
This extension theorem may be applied, for example, to the billiard mapping f : U --* S 1 x R, as follows. Let h : V ~ R be the associated variational principle, where V = { ( x , x ' ) [ x < x' < x + 1}. Let ~ > 0 be a small positive number. Let W6 = {(z, x ' ) [ z + ~ ; < z' _< z + 1 - a}. Then h]W6 extends to all of R 2. Any orbit of the billiards which is bounded away fl'om the boundary corresponds to a stationary configuration of hIWe for some ~ > 0. In a similar way, if P is an elliptic fixed point of an area preserving mapping, and one of the Birkhoff inva.ria,nts is non-zero, then we can reduce the study of any orbit which lies in a sufficiently small neighborhood of P , and at the same time is bounded away from P, to the study of the corresponding stationary configuration of a suitably chosen variational principle (constructed by means of the extension theorem) which satisfies condition (H).
114
w
Minimal Configurations.
We have a l r e a d y briefly mentioned the notion of m i n i m a l configuration in w in our discussion of the Frenkel-Kontorova model. In this section, we will define the notion of a minimM configuration for a general variational principle. T h e definition we give in this section is different fl'om that given in w However, we will show t h a t in the case t h a t h sa.tisfies ( H ) , the two definitions are equivalent. This applies in p a r t i c u l a r to the Frenkel-Kontorova model. By a configuration, we will m e a n a bi-infinite sequence x -- ( . . . , x i , . . . ) C R z of real numbers. By a segment of a configuration, we will m e a n a finite sequence x = (xk,..., x~). If h : R 2 --* R , we set I--1
h(x) = h(x , ...,x,)
=
i=k
A segment will be said to be h-minimal if, for every segment (~k, ..., ~t) such t h a t xk = ~k and xt ----(t (but ~i can be a.rbitrary for k < i < /), we have <
9
A configuration will be said to be h-minimal if each segment of it is h-minimal. In [Au-LeD], A u b r y and Le Daeron developed a theory of h-minimM configurations for the Frenkel-Kontorova model. Bangert [Ba] generalized their theory to a variational principle satisfying (H0) - (//4). In the following sections we briefly sketch the A u b r y - L e D a e r o n - B a n g e r t theory.
Definition. Let x = (xk, ..., xt) be a segment of a configuration. By the Aubry graph G(x) of such a segment, we will m e a n the union of the line segments in R 2 joining ( i , x i ) and (i + 1,Xi+l), for k < i < l. A u b r y ' s C r o s s i n g L e m m a . Let x = (xk ..... xl) aJ~d y = (Yk,...,Yl) denote two h-minimM cont~gurations. Then G(x) • G(y) contains at most two points. If it contains two points, then these w e the endpoints of the two graphs, i.e. xk = Yk aJld xl = Yl. We introduce the following notation: (:,: v
= max(:
,yd
and
(x ^
=
T h e proof of A u b r y ' s crossing L e m m a is based on the following inequality (9.1)
h(z V y ) + h(x A y ) - h(x) - h(y) <_ O.
This inequality follows i m m e d i a t e l y fl'om (H3). F u r t h e r m o r e , it is clear from (//3) t h a t if G(x) crosses G(y) in the interior of the interval, then the inequality (9.1) is strict. P r o o f o f A u b r y ' s C r o s s i n g L e m m a . Assume t h a t the conclusion is false. By replacing x a n d y with subsegments and interchanging the roles of x and y if necessary, we m a y assulne without loss of generMity t h a t xk _< yk and xl _< yt, and at least one of these inequalities is strict. F r o m the a s s u m p t i o n t h a t x (resp. y)
115
is h-minimal, it follows t h a t h(x) < h(x A y) (resp. h(y) < h(x V y)). Combining this with (9.1), we see t h a t these are equalities a n d t h a t (9.1) is an equality. Since (9.1) is an equality, G(x) cannot cross G(y) in the interior of an interval. Since h(x) = h(x A y), we have t h a t x A y is h-minimal. Since h(y) = h(x V y), we have t h a t x V y is h-minimal. Since G(x) and G(y) do not cross in the interior of an interval, b u t they do intersect at other t h a n an endpoint, we have t h a t G(x V y) meets G(x A y) at a node which is not an endpoint. This contradicts (//4). []
116
w
Existence of Periodic Minimal Configurations.
We say t h a t a configuration x = (..., xi, ...) is periodic of type (p, q) if xi+q = x i + p . We set a+q--1
hpe,. = q
~
h(Xi,Xi+l)
This is obviously i n d e p e n d e n t of the choice of a. We assume t h a t h satisfies (H0) (/-/4).
Theorem 10.1. h--qper takes a m i n i m u m value on the set of configurations which are periodic o f type (p, q). P r o o f . We m a y view of (H~), h,~~ is Thus -h- qp ~ is defined a c o m p a c t subset of m i n i m u m value.
identify the set of configurations of t y p e (p, q) with invariant under the t r a n s l a t i o n T(...,xi, ...) = (...,xi on the quotient space R'~/T. In vie of (H2), lt ' h~ qpr R a / T , for any a E R . Since h~ ~" is continuous, it
R q. In + 1, ...). < a} is takes a []
10.2. If p, q, n E Z, q r O, n 7L O, and x = (...,xi ...) minimizes hW~ over the set of periodic con~gurations of type (pn, qn), then x is periodic of type Theorem
(p,q). P r o o f . Let Yi --- xi+,~ - p. Clearly x is periodic of t y p e (p, q) if a n d only if x -- y. Suppose x r y. It is easily seen t h a t the A u b r y graphs of x a n d y meet infinitely often. An a r g u m e n t like the proof of A u b r y ' s Crossing L e m m a shows t h a t this c o n t r a d i c t s the fact t h a t b o t h x and y minimize b.w~ --qn over the set of periodic configurations of t y p e (pn, qn ). []
Theorem 10.3. If x is a periodic configuration of type (p, q) which minimizes . . . . uqP~' over all suda configurations, then x minimizes .-qn h p~ over all periodic cont]gurations of type (pn, qn). Moreover, x is minimal in the sel2se of w Proof. By T h e o r e m 10.1, there exists a configuration y which minimizes -h2 ~r -qn over all periodic configurations of t y p e (pn, qn). By T h e o r e m 10.2, y is periodic of t y p e (p, q). Clearly, y minimizes h~;~'~ over all periodic configurations of t y p e (p, r per ,. per Hence h ~ r ( x ) = h ~ " ( y ) and hq,, (z) = n hqp e r (x). = h,l,, (y). Since y minimizes a n d x takes the same value, x minimizes. This proves the first statement. To prove the secoml s t a t e m e n t , i.e.
a segment ( x k , . . . , x l ) of x minimizes
h(xk, ..., xz) subject to the fixed b o u n d a r y condition, it is enough to a p p l y the first s t a t e m e n t with qn > l - k + 1. []
Corollary 10.4. There exists an h-minimal configuration which is periodic of type (p, ,~).
117
w
The R o t a t i o n N u m b e r .
Let x and y be configurations. We define the relations >~, >~, <~ and <~ a.s follows: x >~ y (resp. x >~ y, z <~ y, x <,~ y) if there exists i0 such that xi > yi for all i < i0 (resp. :ci > Yi for all i > i0, etc.). As before, we suppose that h satisfies (H0) - (//4). We have I. if x ,and y are h-minimal, then x >~ y, x = y or x <~ y and x > ~ y , x = y orx <~y. Trichotomy
This follows immediately fronl Aubry's Crossing Lemma, which implies that the Aubry graphs of x and y cross at most once. If x is a configuration and p, q E Z, we define the tranMate (Tv, q x)i = xi-q + p. Obviously, if Tv,,lx >~ x, then Tvt,~l~x >~ x for all positive integers I. By Trichotomy I, we m a y say "if and only if', in the case that x is h-minimal. The same comments apply to the relations >~, >,,, <~ and <~. Thus, we m a y define
A~(x) = {p/qlT,,,,l:,: <~ z , B ~ ( x ) = {p/qIT,,,vx > ~ x ,
p,q E Z,q > 0} p,,1EZ,,I >O}
and similarly A,~(x) and B,~(x) (replace >~ and <~ by >,~ and <~). It is easy to see that if x is h-minimal and p/q, p'/q' are rational numbers with p'/q' < p/q, then p/q C A~(z) implies p'/q' E d~(x) and p'/q' E B~(x) implies p/q E B~(x). The same result holds for c~ replaced by w. These observations follow from the fact that (Tv,,~,,~,vx)i < (Tvq,,~u x)i, since p'q < pq'. Moreover, when x is h-minimal, A~(x) U B~(x) (resp. A~(x) U B~(x)) is all of Q except possibly one element. Consequently, there exists a unique p~(x) (resp. p,o(x)) E { - o o } U R U { + ~ } such that p/q < p~(x) (resp. p~(x)) implies that p/q E d~(x) (resp. A,~(x)) and p/q > p~(x) (resp. p~(x)) implies p/q E B~(x) (resp. B,,(z)). Theorem
11.1. pc~(x) = p~o(x), if x is h-minimal.
P r o o f . Suppose the contrary, e.g. p~(x) < p,o(x). Let p/q be a rational number, expressed in lowest terms, with q > 0, such that p~(x) < p/q < p,o(x). By Corollary 10.4, there exists an h-minimal configuration y which is periodic of type (p, q). From the inequalities p~(x) < p/q < p~(x), it follows that x >~ y and x >~ y, and this holds for any translate of y. However, if r is sufficiently large, then xi > (T~,I y)i does not hold fi:~r all i. Hence, the Aubry graphs of x and y cross at least twice. Since x and y are h-minimal, this contradicts Aubry's Crossing Lemma. This contradiction shows that we do not have p~(x) < p~(x). Similarly, we do not have p~(x) < p~(x). []
118
We set p(x) = p~(x) = p,~(x). Theorem
11.2. p(x) E R , i f x is h-minimM.
P r o o f . Otherwise, we would have p(x) = q-c~. Suppose, for example, t h a t p(x) = +o0. If y is a periodic configuration, y >~ x and y < ~ x. By Corollary 10.4, there exists a periodic m i n i m a l configuration y of t y p e (p, 1), for any integer p. Thus, Y i + l = Yi -[- P. By choosing p very large, and replacing y by a t r a n s l a t e , if necessary, we m a y suppose t h a t Y0 < x0 and Yl > xa. The relations y >~ x, y < ~ x, y0 < x0 a n d yl > xi imply t h a t the A u b r y graphs of x and y cross at least three times. This contradicts the A u b r y ' s Crossing Lemma. This contradiction shows t h a t p(x) 7s +oo. Similarly, it m a y be shown t h a t p(x) # - o ~ . []
We will call p(x) the rotation number of x. We let AH --- M h C R ~ denore the set of all h - m i n i m a l configurations. We provide R ~ with the p r o d u c t topology and AH with the induced topology. Theorem
11.3. Ad is dosed in R ~ and p : Ad --+ R is continuous.
P r o o f . T h e first assertion folk)ws i m m e d i a t e l y from the definition of .h/[ and the a s s u m p t i o n (H0) t h a t h is continuous. To prove the second, we will use: Lemma.
Let x = ( . . . , x i , . . . ) E A/[. Let p,q E Z, q > 1. I f xi+,l <_ xi + p, tbr some
i, ti, en p(x) < (p + i ) / , r If
>
+ p, for some i, ti en p(x) >_ (p - a)/q.
P r o o f . To prove the f r s t assertion, we use the existence of a periodic minimM configuration y of t y p e (p + 1, q). By replacing y with a suitable t r a n s l a t e , if necessary, we m a y suppose t h a t yi < xi and yi+,~ > xi + p >_ xi+q. By the A u b r y ' s Crossing L e m m a V < - x and y >~ x, so p(x) < p(y) = ( p + l ) / ( t . T h e second assertion m a y be proved similarly. []
Now the continuity of p follows easily, since x,+,1 is obviously a continuous function of x. []
We let pri : R ~176 ---+R denote the projection on the i th factor. T h e o r e m 11.4. Let I, ~ be compact subsets of R . F o r m~y i E Z, p - l ( ~ ) A p r T l ( I ) is a compact subset c l a d . P r o o f . By the Tychonoff p r o d u c t theorem and the fact t h a t M is closed in R c~, it is enough to show t h a t for ea.ch j E Z, p r j ( p - l ( ~ ) (3 pi'~-l([)) is a bounded subset of l t . In the case t h a t j > i an u p p e r b o u n d for this set m a y be found as follows. Let p E Z be an u p p e r b o u n d for ~ and let y be a periodic m i n i m a l configuration of t y p e (p, 1) such t h a t yi is an u p p e r b o u n d for I. T h e n yj is an u p p e r b o u n d for p r j ( p - l ( f t ) A pr~-a(I)). For if x E p - l ( f ~ ) N pr~-i(I), then y >,o x since p(y) = p > p(x) E f~. Moreover, yi > :r,i, so yj > x j by the A u b r y ' s Crossing
Lemma. T h e other cases m a y be t r e a t e d siinila.rly.
[]
119
T h e o r e m 11.5. Let I be a closed interva.1 of unit length in R. p : pr~-I (I) N .h4 ~ R is suriective.
For any i E Z,
P r o o f . For any rational number p/q, there exists a. minimal configuration x = (...,xi,...) of type (p,q) by Corollary 10.4. Since I has unit length, we m a y suppose that xi E I. Since p(x) -- p/q, we obtain that p/q E p(pr~-~(I) n ~ ) . Thus p(pr~-X(I) n M/t) contains q . On the other hand, by Theorem 11.4, if ~ C R is compact then
# ( p r T ' ( I ) n M ) n ~ = p(pr;-'(I) n p - ' ( ~ ) ) is compact. It follows that p(pr~-~(I) rh M ) is closed. Since this set is closed and conta.ins Q, it contains ~,.11of R. []
120
w
Irrational Rotation
Number.
Let w be an irra.tionM r o t a t i o n number. Let It be a variational principle which satisfies the B a n g e r t conditions ( H 0 ) - ( / / 4 ) . By T h e o r e m 11.5, there exist h-minimM configurations x of r o t a t i o n n u m b e r w. We let AA~ = A41~,~ denote the set of hm i n i m a l configurations of r o t a t i o n n u m b e r ~v. By T h e o r e m 11.3, A4~ is closed in R ~176 In this section, we will descrihe several results concerning the s t r u c t u r e of A.4~, due to A u b r y and Le I)a.eron [Au-LeD] and later in a more general context to Bangert [Ba]. T h e first result concerns the order relation. Given two configurations x a n d y, we will say t h a t x < y (or y > x) if :ri < yi, for every i. If x and y are h-minimM configurations, then x < y if and only if x <,~ y and x < ~ y. This follows from A u b r y ' s Crossing Lemma. Trichotomy
I I . I f x a n d y are h-minima,1 contlgurat, i o n s a,nd h n v e the s a m e hTat,i o n M r o t a t i o n mm~ber, t h e n .~: < y, x -- y oz" x > y.
We will follow the t r a d i t i o n a l a p p r o a c h ([Au-LeD], [Ba.]) and prove this in several steps. For the first step, we prove this when y is a t r a n s l a t e of x, say y = Tp,q x. Since p ~ ( x ) = p~(x) and this n u m b e r is i,'rational, A~(x) = A~(:r,) and B ~ ( x ) = B~(x). This is because when p , ( x ) is i r r a t i o n a l A ~ ( x ) = { p / q < p~(x)} and B ~ ( x ) = { p / q > p~(.z)} (and simila.rly for w in place of a). F r o m the definition of A~(:r,) etc., it follows t h a t y <~, :~: (resp. y > ~ x ) if and only if y < ~ x (resp. y > ~ x). T h e n the result follows f i o m Trichotomy I. T h e second step is the definition of what A u b r y called the h u l l f u n c t i o n of a m i n i m a l configuration x of irrational r o t a t i o n n u m b e r w. Let r
= sup{(Tp,q x)0 [p - qw <_ t } .
T h u s r is a m a p p i n g of R into R. Note that Tp,,~ x < Tv,,q, x if and only if p - qw < p~ - q~w. (This is a consequence of the t r i c h o t o m y p r o p e r t y which we j u s t proved, and the definition of the r o t a t i o n munber.) Consequently, using the a s s u m p t i o n t h a t w is irrational, we see tha.t r is strictly increasing, i.e. ,~ < t ~
Furthermore r
+ 1) = r
A u b r y called r
r
< r
+ 1, as m a y be seen from the definition of ~6~.
the h u l l f u n c t i o n o f x.
Using A u b r y ' s hull fimction, we m a y define new m i n i m a l configurations as follows: if t E R , we set x'~ ~=~ = ,/,~:(t + ~ i • 0) .
Then x t+~ = b" " , x i t+o ,...) is a m i n i m a l configuration of r o t a t i o n n u m b e r w. For, it is possible to choose a. sequence of pairs ( p i , q , ) such t h a t Pi - qiw , 7 t. T h e n x t - ~ = limTv;,q; x a n d the fact t h a t x t - ~ is a m i n i m a l configuration follows from the fact t h a t A4,~ is closed. The case of x t+~ m a y be t r e a t e d similarly. T h e t h i r d step is:
121
T h e o r e m 1 2 . 1 . I f x a,nd y are m i n i m a l configurations h a v i n g t,he semae rot,ation n u m b e r , t h e n there exists a const,ant, a such that, r
+ 0) = r
+ a + 0) ,
tbr all t E R . P r o o f . Let I(ff~) d e n o t e the u n i q u e n o n - d e c r e a s i n g f i m c t i o n of R into itself which is t h e inverse of r T h e o r e m 12.1 is e q u i v a l e n t to t h e s t a t e m e n t t h a t I ( r a n d / ( f l y ) differ by a c o n s t a n t . S u p p o s e otherwise. Choose a n u m b e r a such t h a t min(I(r
) - I(r
< a < max(/(r
-/(G))
9
It is easy to see t h a t for a n y t E R , the A u b r y g r a p h s of yt+0 a n d x t+'+~ cross infinitely often. B u t , since these a.re l n i n i m a l configurations, this provides a contrad i c t i o n to the A u b r y Crossing L e m m a . Clearly, if a < 0 ( r e s p . > 0), t h e n x < y (resp. x > y). T h u s , in order to prove T r i c h o t o m y II, it is e n o u g h to consider the case a = 0. I n this case x t•176= yt=l:0 for all t. We use the a b b r e v i a t i o n x •176for x ~177176 Thus, (12.1)
x - ~ < x < x +~
and
x -~
+~ .
F r o m t h e fact t h a t r is strictly i n c r e a s i n g a n d the periodicity c o n d i t i o n r r + 1, t o g e t h e r w i t h the a s s u m p t i o n t h a t w is i r r a t i o n a l , it follows t h a t (12.2)
Ex
+ 1) =
+ -:,:~- < 1 .
--OO
F r o m (12.1 / a n d (12.2), it follows t h a t x a n d y are a.~ymptotic, i.e. [xi -y~[ ~ O, as i --~ 4-oo. More generally, we will say t h a t x a n d y are a - a s y m p t o t i c if [x, - Y~I -~ 0, as i ~ - o o a n d w - a s y m p t o t i c if [:r~ - y~] ~ 0, as i ~ +oo. T h e f o m t h step in the proof of T r i c h o t o m y II is the following: Addendum t o A u b r y ~ s C r o s s i n g L e m m a . I f x and y are h-minimaJ conf/gura, tions which are a - or ~ - a s y m p t o t , ic, t,hen :r = y or their A u b r y g r a p h s do not, lneet,. P r o o f . Suppose, for e x a m p l e , t h a t x a n d y are w - a s y m p t o t i c a n d their g r a p h s meet. Choose integers k < I such t h a t their g r a p h s meet in t h e i n t e r v a l b e t w e e n k a n d I. Let x i* = xi, for i r I a n d xt'* = Yl. T h e a r g u m e n t used to prove A u b r y ' s Crossing L e m m a shows t h a t there exists s e g m e n t s ((k, ..., (t) a n d ('qk, ..., Tit) of conf i g u r a t i o n s such t h a t (k = 9z .k, r/k. = y~:, . ~ .= 'r/l = .ct = Yl, a n d h(~k, ..., ~ ) + h ( , k , ..., ',,) _< h(:,4, ..., : q ) + h(',jk, ..., ~ ) - , . Moreover, e is i n d e p e n d e n t of I, since the c o n s t r u c t i o n used in the proof of A u b r y ' s Crossing L e m m a is m a d e in the n e i g h b o r h o o d of the p o i n t where the g r a p h s intersect a n d is unaffected b y a n y t h i n g n o t n e a r this point. T h u s , it is e n o u g h to show t h a t (12.3)
* ..., :,:~) * _< h(:~k, ..., :~:t) + h(,~:k,
,
122
if l is large enough. But (12.4)
p(.~:) - 2 _< ,:~+~ - :~ < p(.~:)
+ 2 ,
by the L e m m a used in the proof of T h e o r e m 11.3, and h(xi,xi+l) is uniformly continuous in the region defined by (12.4), since it is continuous and satisfies the periodicity condition h(x + 1, x' + 1) = h(x, x'). Therefore (12.3) holds for large l, since x a n d y are w - a s y m p t o t i c . [] Now Trichotomy II follows fl'om the a d d e n d u m , because we have a l r e a d y reduced the proof to the case when x and y are asymptotic. Since the graphs of x and y do not cross, i f x C y w e h a v e z < y o r x > y . [] We set A~ = A h = P'ui - i ~3d ~). h~ consequence:
Trichotomy II has the following i m p o r t a n t
T h e o r e m 12.2. A h is a closed subset of R aJ~d p% :./td h ~ A~ is a h o m e o m o r phism for a n y irrationM number w. P r o o f . By Trichotomy II, this m a p p i n g is injective. By T h e o r e m 11.4, it is proper. T h e two assertion then follow from elementary results in general topology.
[] We define two h o m e o m o r p h i s m s T (translation) and S (shift) of fld h onto itself, as follows:
T ( . . . , x i , . . . ) = ( . . . , x i + l,...)
and
S(...,xi,...)=(...,xi+l,...).
Obviously these commute, so S induces a h o m e o m o r p h i s n l S of A,I~/T onto itself. Obviously, 2Mh/T is compact. Recall t h a t if f is a h o m e o m o r p h i s m of a compact topological space, a closed invariant subset K in X is said to be minimal if it contains no closed invariant sets other t h a t itself and the e m p t y set. Birkhoff showed t h a t every h o m e o m o r p h i s m of a compact space has a m i n i m a l set. (This follows i m m e d i a t e l y fiom Zorn's Lemma). Denjoy [De] showed t h a t every orientation preserving homeomort)hisnl f of tile circle of irrationa.1 r o t a t i o n n u m b e r has exactly one m i n i m a l set, and every orbit tends to it under forward a n d b a c k w a r d iteration. Moreover, he showed tha.t such an f is semi-conjugate to the r o t a t i o n R~ of the same r o t a t i o n number, i.e. there exists 9 : S 1 ---* S 1 such t h a t ~ o f = R~ o ~5. Moreover, for each 0 E S 1, ~ - 1 ( 0 ) is one point or an interval. 12.3. S : Jtdh / T --* .A,lh / T has a unique minimal set, to which every orbit tends under forward and backward iteration. S is semi-conjugate to the rotation Rw.
Theorem
P r o o f . This is r e i n t e r p r e t a t i o n of what we already proved. If x is a m i n i m a l configuration, the m i n i m a l set is {x t+~ It E R}. This set is m i n i m a l because every orbit of S is dense in it. It. is the only m i n i m a l set, because every orbit of S tends to it. T h e generalized inverse of r
provides a semiconjugacy to a rotation.
[]
123
w
Rational Rotation Number.
Let w = p/q, p, q C Z, q >_ 1, (p, q ) = l . Let h be a variational principle which satisfies the Bangert conditions ( H 0 ) - (//4). In this section, we continue to describe results of A u b r y and Le Daeron [Au-LeD], generalized by Bangert [Ba], this time concerning configurations of rational r o t a t i o n number. Let x be a m i n i m a l configuration of r o t a t i o n n u m b e r p/q. We f r s t consider the case when T_p,_q x >~ x. Then xi+,fl - p l , 1 = 1,2, ... is an eventually increasing sequence. Moreover, xi+qt - pl ___ x~ + 2, since we would have p(x) >_ (pl + 1)/ql by the L e m m a used in the p r o o f of T h e o r e m 11.3. Therefore limt~+oo Xi+qt - p l exists. We denote the limit by (l~o x)i. Then l,ox is a m i n i m a l configuration and (13.1)
l~ox=
lim T _ t , z _ q t x . /~-t-oo
In the case t h a t T_ ~,_p x < ~ x a similar argument shows tha.t (13.1) exists. Similarly
(13.2)
l~x=
lira T_,l,_,flx
I~--oG
exists. Obviously, b o t h l~x and l~x are periodic of t y p e (p, q). If x is periodic of t y p e (p, q) then T_l,,_ q x = x, so l~x = l~x = x. Otherwise, the A u b r y graphs of x, I~x and l~x do not meet by the A d d e n d u m to A u b r y ' s Crossing L e m m a (w Moreover, the A u b r y graphs of T_p,_~ x and x do not meet, by the same A d d e n d u m , since obviously l~T_p,_q x = l~x a n d l~T_v,_ q x = l~x. Froln these observations, we m a y draw two consequences. T r i c h o t o m y I I I . D'x a~d y are minimal configurations which are translates of one another, then their A u b r y graphs do not cross, unless x = y. P r o o f . We have a l r e a d y shown this in the case t h a t the r o t a t i o n n u m b e r of x (and y) is i r r a t i o n a l (see the beginning of the proof of Trichotomy II in w The a r g u m e n t given in the case w was irrational shows in the present case t h a t the graphs of T~,~ x and x do not meet if r / s r p/q. Thus we are left to consider w h e t h e r the graphs of x and Tp,q x meet. But, we have just shown t h a t they do not meet unless x = Tp,,~ x .
[]
Thus, we have three alternatives: T_p,_q x < x, T_p,_q x = x, or T_p,_q x > x. We say t h a t x has rotation symbol p / q - in the first case, p/q in the second case and p / q + in the third. Clearly, l~x < x < l~x, when x has r o t a t i o n symbol p / q - , l,,x = x = l~x when x has r o t a t i o n symbol p/q, and l~x < x < l~x when x has r o t a t i o n symbol p/q+. We will use the following notation: we let ~dv/q denote the set of periodic m i n i m a l configurations of t y p e (p, q) (i.e. of r o t a t i o n symbol p/q), a n d we let A.4v/q+ (resp. 34p/q_) denote the union of the set of m i n i m a l configurations of r o t a t i o n s y m b o l p / q + (resp. p / q - ) and M p / , r It is easy to see t h a t each of these sets is a closed subset of M .
124
For the s t u d y of m i n i m a l configurations of t y p e p/q+, the following n o t a t i o n will be useful: if x a.nd y are two configurations, we set oo
Ah(1], :c.) : ~-~(]t(Yi, ?.]i*l) -- h ( x i , :(:i*l)) , -oo
(13.3)
provided t h a t the sum is convergent, i.e. N
lilTl ~(h(,i,~i+l M--*--oo, N ~ + e ~ M
) -- h ( x i , : r , i + l ) )
exists. In this ease, we will say t h a t Ah(y, x) exists. We will a.lso say t h a t Ah(y, x) exists when the sum converges to - c o or to + o c . T h e o r e m 13.1. Is x is a. minimal configuration and y is a configuration such that Ix, -Y~I -~ 0 as i -~ +oo, then Zxh(y, :,:) e~i.~ts and/,1,,(:,j, :,:) > 0, a/though A h ( y , x) =
+oc is not excluded. P r o o f . Since x is m i n i m a l (12.4) holds. Moreover, h is uniformly continuous in the region defined by (12.4), by its continuity and periodicity. Let yM,g denote the segment of a configuration, defined for M < i < N by yM,N = x~, for i ---=M, N and yM,N = Y~, for M < i < N. Since h, is uniforznly continuous in the region defined by (12.4), it follows t h a t
[h(y M'N) - h(yM, ...,yN)[-* 0 ,
as [M[, iN[ ~ -t-oo ,
since I x / - y/[ ---* 0, as i --* 4-00. Since x is minimal, h ( y M ' N ) >_ ] l ( X M , . . . , X N )
9
Consequently, for every e > 0, there exists a n a t u r a l n u m b e r N , such t h a t if IMI, IN[ > No, then h(~]M,'",YN)
>-- h ( : I : M , ' . . , X N )
-- e .
It is easily seen t h a t this is equivalent to tile conclusion of tlle Theorem.
[]
We will also use the n o t a t i o u a..FNq-1
(13.4)
Ahv,,,(y,x ) =
lira
M ~-oo, N ~c~
~
(h(y,,yi+l) - h(x,,x,+,))
,.+M V
when this limit exists (we a d m i t the possibility of +o~ oi' - o o as the limit). If this limit exists for every a and is independent of a, we will say t h a t Ahq(y, x) exists, a n d write/khq(y,.T,) for the common value. For example, if b o t h x and y are m i n i m a l configurations of t y p e (p,q), then
Ahq(y, x) =- 0, whereas Ah(y, x) does not exist.
125
T h e o r e m 13.2. Let x m2d ~ be periodic minimM configurations of type (p, q). Let y be a configuration such that lYi - xil --+ 0 as i ~ - ~ , and lYi - ~il -* 0 as i --* +oc. Then Ah~(y, x) = Ahq(y, ~) exists. Moreover, y is m i n i m a / i f and only if y minimizes this quantity over a11 such comqgurations. P r o o f . For every e > 0, there exists No such that
h(gM,...,yN) >__h(XM, ...,XN) -- e ifM, N
and
if M, N >_ No, just as in the proof of Theorem 13.1. This, together with Ahq(x, ~) = 0 implies that Ah,~(y, x) and Ah,~(y, () exist and are equal. Clearly, if y minimizes Ahq(y, x) then it is minimal. The opposite implication follows easily fi'om (12.4) (which is wdid for minimal x) and the uniform continuity of h over the region defined by (12.4). [] T h e o r e m 13.3. Let x be a minima/con[iguration of rotation symbol p / q + (resp. p / q - ) . Then there is no minima/periodic configuration y which sntist~es I~x < y < l~x (resp. l~x < y < l~x ). P r o o f . For definiteness, we suppose that x has rotation symbol p / q + , the other case being similar. If y is a periodic minimal configuration which satisfies l~x < y < l,,x, then it is periodic of type (p, q) and Ahq(y, l~x) = Ah,~(y, l~x) = O. The following inequality is analogous to (9.1) and a.lso follows immediately from (H:~): (13.5)
Ahq(x V y,y) 9- Al,.,,(x A y , y ) < A h , l ( x , y ) 9
Furthermore, just as with (9.1), if G(x) crosses G(y) in the interior of an interval, then this inequality is strict. Let i0 be such that xi < yi for i < i0 a.nd xi > Yi otherwise. Let
zi =
{ xi , i <_ io Yo , io < i <_ io + q zi_~ , i > io + q .
It is easily verified that
Ah,~(z, y) = Ah,,(:,: V y,y) + Ah,~(:,: A ~,,y). Thus, Ahq(z, y) <_ Ah,,(x, y) and this inequality is strict if G(x) crosses G(y) in the interior of an interval. But, the assumption that x is minimal implies that x = minimizes Ah,,(~,y) subject to the conditions ] ~ i - (l,x)i] --+ 0 as i --+ - o o and [~i - (l~x)i] --+ 0 as i --+ +co. This gives a contradiction in the case that G(x) crosses G(y) in the interior of an interva.1. In the remaining ca.se, we may obtain a contradiction by using (/-/4). For, there exists z* such that z[ = zi, if i r i0, and Ah,,(z*, y) < Ahq(z, y) = A b e ( x , y), sinee (Y~o-I, Y~0,Y~,,+I) is a nfinimal segment, a.nd therefore (Ha) implies that the segment (zi0-1, Zio, Zio+l) is not minimal, a,s Yio-1 < Zio-1, Yio = Zio, 9i0+1 = Zio+l. []
126
By the symbol ,space S we will mean the disjoint union of the real n u m b e r s R together with the symbols p / q - and p/q+, where p/q ranges over the r a t i o n a l numbers. By the projection rr : S ---+ R , we m e a n rr(w) = ~o, if w E R , and ~r(p/q+) = p/q. We will also call 7r(~o) the underlying number of the symbol w. We provide S with an order as follows: if 7r(w0) < ~r(col), then w0 < w~, and p / q - < p/q < p/q+. We provide S with the order topology. By the rotation ,symbol ~(x) of a m i n i m a l configuration x, we m e a n the r o t a t i o n n m n b e r p(x) when this is irrational and what we have a l r e a d y defined it to be when p(x) is rational. Thus, to every h-minimal configuration x, we have associated a r o t a t i o n symbol tb(x) E S, whose underlying n u m b e r is p(x). For every r o t a t i o n symbol w, we have defined a set .A4,o = Ad~ C -t,4h: when w is an i r r a t i o n a l n u m b e r , this was defined in w when w = p / q - , p/q or p/q+, in this section. By definition, Adp/,1 = Adp/~_ A .A4p/q+. Trichotomy x-=y.
I V . It'w is a rotation symbol and x,y E .Ad h then x < y, y < x or
P r o o f . For the case t h a t co is an i r r a t i o n a l number, this is T r i c h o t o m y II. For the case tha.t wis a rational Immber, this is an easy consequences of A u b r y ' s Crossing Lemma. T h e only remaining cases axe w = p / q - or p/q+. We suppose t h a t co = p/q+, the other case being similar. Suppose x, y E Ad h~. If ,5(x) = ~(y) = p/q, we have a l r e a d y excluded crossing. If/5(x) = p/q+, we have l~x < x < l,ox by the A d d e n d u m to A u b r y ' s Crossing Lemma. If f u r t h e r m o r e fi(y) = p/q, crossing is excluded by T h e o r e m 13.3. On the other hand, if/5(x) = p/q+, it follows from T h e o r e m 13.3 t h a t crossing implies t h a t l~x = I~y and l~x = 1,oy. T h e n crossing is exchlded by the A d d e n d u m to A u b r y ' s Crossing Lemlna. []
h We set A~ = A~ = p r 0(.Ad,o), for any r o t a t i o n symbol co.
T h e o r e m 13.4. A h is a dosed subset of R , emd pro : 2M h ~ A h is a homeomorphism, for any w E S. P r o o L This follows from Trichotomy IV in the same way as T h e o r e m 12.2 follows from Trichotomy II. [] Note t h a t Trichotomy II and III are special cases of Trichotomy IV. It is, however~ necessary to consider these special cases for the proofs. Finally, we consider the existence theory for configurations of r o t a t i o n symbol p/q=t=. We have a l r e a d y o b t a i n e d the existence theory for other r o t a t i o n symbols: by Corollary 10.4, Ap/,l 7~ 0, and by T h e o r e m 11.5, A~ ~ 0, if w is an irrationM number.
127
Theorem
13.5. I f J is a complementary interval of Ap/~ (i.e.
a component of
R \ Ap/~), then J n A~,/~+ # 0 and J n Ap/,~_ r O. P r o o f . We will show t h a t J N Ap/q+ ~ (~, the other case being similar. We will show more: We let x (1), x (2), ... be m i n i m a l configurations such t h a t p(x (0) > p/q and p(x (0) --~ p/q as i --* +oo. We let E (0 C A4 denote the set of all t r a n s l a t e s of x (0. We let L -- lira sup E (0, i.e.
L : {:c E -/~/] for every i0 and every n e i g h b o r h o o d N of x in .Ad, there exists i > i 0
such t h a t
N N E (0 r
.
We will show t h a t J [3 pr0(L ) :fl 0. This will be clearly enough, since clearly L C Mp&+. Suppose, to the contrary, t h a t J N p r 0 ( L ) = {~. Let [a, b] be a compact subinterval of J. T h e n E ( 0 n [a,b] = 0 for all sufficiently large i. Since p(x(0) > p/q, we m a y suppose, by replacing x(0 by a suitable translate, if necessary, t h a t x~i) < a and
:c~0 > b + p. By passing to a subsequence, if necesssary, we m a y suppose t h a t x (0 converges to ~ E .Mp/q+. T h e n ~u _< a and ~ >_ b+p. In fact, setting J = ( J - , J + ) , where J - < a < b < J+, we have (0 < J - and ~,1 > J + + P, since J A pr0(L ) = O. Now J - = 7o, J + = 5o, where 7, 5 E .hdp/,~ and (o _< J - , ~q _> J + + p i m p l y that l ~ < 7 < (5"< l~o(, c o n t r a r y to T h e o r e m 13.3.
[]
1 3 . 6 . I f J = ( J - , J + ) is a complementary interval of Av/q in R, with J - = 70, J + = 6o, 7, 5 E .hdp/q, then there exists x C AAp/q+ (resp. y C .h4p/~_)
Corollary
with xo (resp. yo) C J and l~:~: = 7, l~:r = 5 (resp. l , y = ~, l~:r = 7). Finally, we note t h a t if x and g are as in this Corollary, then the A u b r y graphs of :c and y cross. Thus, we cannot extend t r i c h o t o m y to the case of configurations of tile same r o t a t i o n number.
128
w
Application to Dynamics.
In w167 we have discussed basic results in the A u b r y - L e Dacron, Bangert theory. In this section, we discuss how these results a p p l y to dynamics. We suppose t h a t (U, f ) C J. We let h : V ~ R denote the variational principle associated to f (w If V = R 2 the application to dynamics is very simple. For example, V = R 2 in the case of a Chirikov m a p p i n g and, more generally, if f m a p s S 1 x R onto itself, a n d f twists each end of S 1 x R infinitely. As we p o i n t e d out in w in this case the Bangert conditions (/4o) - ( g 4 ) are satisfied, and these are the only conditions t h a t we used in w167 we (lid not use (Hs) or (H6) there. We have a l r e a d y p o i n t e d out the connection to dynamics in w in our discussion of the Frenkel-Kontorova model. A n~inimal configuration (..., xi, ...) is .~tatioT~ar:~l i.e. O/Oxi(h(xi-1, xi) + h(xi, :Ci+l) ) = O. Setting Yi = 01 h(:ci, :ci+l ) = 02h(xi-1, :~i), we have f ( x i , yi) = (xi+l, yi+l), where f is the lift of f to the universal cover, since (5.2) is equivalent to . / ( z , y ) = (:,:', :~/). This provides a one-one correspondence between minima.1 configurations a.nd orbits of f , in the case tha.t f m a p s S 1 x R onto itself, (S 1 x R , f ) E J , and f twists each end of S 1 x R infinitely. We will use the expression rr~ir~irT~ttlorbits to describe orbits of f which corresponds to m i n i m a l configurations and also for orbits of f which lift to m i n i m a l orbits of f . We let M = M] C R 2 denote the union of minimM orbits of f and E = E l C S 1 x R the union of m i n i m a l orbits of f . If O = (...,(x~,yi),...) is a, m i n i m a l orbit of f , we will define the rota.tior~ .~]rnboI/?((.9) and rotation number p(O) to be fi(:c) and p(:c), resp., where :c = (..., xi, ...) is the corresponding m i n i m a l configuration. We will also use these terms for orbits of f . T h e r e is a slight abuse of terminology here, because changing the lift f of f (by adding ('n, 0), where '~ is an integer) changes fi(O) and p(O) (by a.dding n), b u t this should cause no confusion, since (to the a u t h o r s ' s knowledge) all results d e p e n d only on the r o t a t i o n symbol (or r o t a t i o n m u n b e r ) rood.1. For w 9 $, we let M~ = M~o,] (resp. E~ = E,~,]) be the subset of M (resp. E) consisting of orbits of r o t a t i o n symbol a2. F r o m T h e o r e m 13.4, it follows t h a t M~ (resp. E~ is a closed subset of R 2 (resp. S 1 x R ) , and the p r o j e c t i o n rr~ on the first factor induces a h o m e o m o r p h i s m 7r1 : M,~ ~ A,o (resp. ~rl : E~ ~ A,o/Z), which satisfies the following property: 14.1. It" the variational principle h is C 2 (i.e. f is C 1) and satisfies the Bangert conditions (H~) - (H4), then the homeomorphism ~r~ : E~ ~ A,o/Z has Lipschitz inverse, whose Lipschitz constant only depends on h and its second Theorem
derivatives. P r o o f . Let i)r 0 : .Adh ~ A h be the h o m e o m o r p h i s m given in T h e o r e m 13.4 and let S be the shift m a p of Ad~ which has been i n t r o d u c e d in w T h e n the m a p :A,o --* A,o, 9 = Pro 1 o S o p r 0 is bi-Lipschitz. In fact, since ]:r.1- x0[ a n d [ x u - x - l ] are uniformly b o t m d e d for all x E M ~ and by (H1) it is sufficient to consider the case when 0 < :~: < .'? < 2, we m a y assume t h a t O - l ( : c ) , :r, ~ ( x ) a n d 0 - ~ ( 2 ) , 2, 9 (:?) are contained in a fixed compact interval I. By the B a n g e r t p r o p e r t i e s and the differentiability of h, there exist 5 > 0 and L > 0 such t h a t c~12]t ~ -(5 < 0 on I x I
129
and 01h, 0fl,, are Lipschitz on I x I with constant L. Then the fi)llowing estimate holds:
0 _< ~ . ( ~ - i ( ~ )
_ @-l(:,;)) + (~(:~) _ ~(:,.)) <_ O~h(@-l(:,.),:~)
,:I,(:/..)) = 0 2 h ( , ~ - l ( x ) , . ~ ) + o,h(.~, e(.~)) - o,h(:~, e ( ~ ) ) _< 2z(~ - :~), + c_31h(x,,~(x))
- a,h(x,
-
_
O~h,(@-x(~), ~ ) +
02h(C'-l(.'r), :Q+
where the first inequality is a consequence of the Aubry's Crossing Lemma, the second and the last one follow from the mean value theorem and the equality in between holds since (@-1 (x), .~', @(:r)) and (~-1 (:?), :2.,~(:?)) are stationary segments with respect to h. This proves our claim. Finally, 7r~-1 : A~ ~ M~ can be written as
d-~(.~,:) = (:,:,-o,h(:~, e(:,:)), which, in view of tile previous argunient, implies the statement of the theorem.
[]
We also get commutative diagrams of honieomorphisms:
For example, the comniutativity of the second diagram, together with the fact that the vertical arrows are homeoniorphisnls, says that f is topologically conjugate to S. Thus Theorem 12.3 may be restated as: T h e o r e m 14.2. If w is an irra.tional number, then .f : E~,I ~ E~,I has a miique minima] set (in the sense of topological dynamics). Every orbit of f l E ~ . i tends to the minimal set mlcler tbrwaz'd and backward iteration. Moreover, f : E~,f --~ E~,I is semi-conjugate to a rotation. Note that in w we assumed the standing hypothesis that w is an irrational number. The hyI)othesis tha,t w is an irrationM number is essential: if w = p / q - , p/q, p/q+, the minimal sets in E~,I- are the periodic orbits (whose union is E~,/q,I), and every orbit tends to Ep/q, I under forward and backward iteration. The embedding A~,] C R induces a total order on A j , which may be used to define a total order on M~, i via tile homeomolphisnl ~rl : M.~,] --+ A..,]. Trichotomy IV has the following flu'ther consequence: fin any rotation synibol w the mapping f : M , ] ~ Mr,/1)reserves this order. Likewise, we n i a y define a. cyclic order o n Ew,l via, tlie honieomorphism TEl : E~,I ~ A ~ , I / Z C s i = R / Z . We ha.ve the further consequence that f : E~,j E~,f preserves the cyclic order. Now we niay explain tile relation between the Aubry-Le Dacron, Bangert method, which we have been describing in the last several sections and the first author's method, which we described in w We recall that to a. minimal configuration x (if irra.tional rota.tion synibol we a.ssociated (following Aubry) in w a hull
fw,,ctio'n c/b:.
130
T h e o r e m 14.3. I f w is an irrat.ional n u m b e r and x is a n~inimal contlgurat, ion o f rotat.ion nun~ber a;, then the hull thnction r m i n i m i z e s Y~(r over the set ot" m e a s u r a b l e t h n c t i o n s r which satistly the i)eriodicit, y condition r + 1) = r + 1 and t,he condition that, r - t should be bounded. Moreover, the m i n i m i z e r is unique up t`o t`ra~slat`ion, i.e. if" r m i n i m i z e s F~, then r = r T,, almost` everywhere, tbr s o m e a E R , where T~ is t,het, ranslation T,,(t) = t, + eL. In fact, this holds for any variational principle h which satisfies (H0) - (/-/4); it is not necessary to assume t h a t h is the variational principle of a mapping. We will not give a proof. For twist mappings, this result was proven [Ma6, Prop. 11.1]; the same proof works for the forinula.tion we have given here. It follows t h a t the set ~* defined in w is the unique m i n i m a l set (in the sense of topological dyna,mics) of flew, in the case t h a t w is irr;~tional.
131
w
B i r k h o f f Invariant C u r v e T h e o r e m .
In the next sections, we will be concerned with the properties of a special b u t very i m p o r t a n t class of invariant sets for exact area-preserving m o n o t o n e twist maps of the a n n u l u s S 1 x R: rotational invariant curves, i.e. inwu'iant curves which separate the top from the b o t t o m of the annulus. We will see that the action minimizing sets E,~ introduced and studied in the previous sections, are a generalization of torational invariant curves since, as it will be shown in w every rotational invariant curve is an action m i n i m i z i n g set. Furthermore, if F is a rotational invariant curve which is invariant for a diffeomorphism f E j1, then clearly f l F is topologically conjugate to an orientation preserving homeomorphisnl of the circle S 1 and thus it has a well defined rotation number p(F) = w E R. T h e n F = E,~ in case w is irrational aml F = EWq+ , if a = p/q. The basic result on rotational invariant curves is the following theorem due to G.D. Birkhoff [Bill, [Bi3], which establishes in the case of curves the analogous of Theorem 14.1:
T h e o r e m 15.1. Let f E j1 be an area-preserving m o n o t o n e twist diffeomorphism o f the aanulus S i x R. Then every rotational invariant curve F is the graph o f a Lipschitz function u : S 1 ~ R , i.e. it ]]as tile following forln C ~-~ {(0, 'tt(0)) ] 0 ~ S l }. Furthermore, the Lipschitz constant o f u, Lip(u), can be a priori estimated from the properties o f the diffeomorphism f . This theorem is a particular case of a nlore general result:
T h e o r e m 15.1L Let f : S 1 x R ---* S 1 x R be a C 1 diffeonmrphism. Suppose that
f preserves the area, m a p s each end
of S 1 x R to itself~ preserves orientation, and deviates verticad lines {(0, y) E S 1 x R [ 0 = constant } in S 1 x R either to left or to the right. Let U be an open subset o r S 1 x R such that f U = U, U is homeomorphic toS lxRaa2dS lx(-oz,a] cUcS ~ x ( - o o , b), for some a < b,a, b E R . Then the frontier o f U in S 1 x R is the graph o f a Lipschitz function u : S 1 ~ R , i.e. U \ U = {(0, u(0)) ]0 E S1}. Furthermore, the Lipschitz constant of'u, Lip(u), ca~l be a priori estimated from the properties o f the diftbomorphism f .
Ezample 15.2. Suppose tha.t U satisfies all hypotheses of Theorem 15.1' b u t the one of being homeoniorphic to S 1 x R. T h e n Theorem 15.1' can be applied to the open set U obtained flom U by "filling the holes", i.e. U is the complement in S 1 x R of the u n b o u n d e d connected coniponent of the complement of U. Example 15.3. Let P C S 1 x R be a rotational inva.riant curve for a diffeomorphism f E j i . T h e n f satisfies the hypotheses of the theorem and one of the components of S 1 x R \ F satisfies the condition imposed on U (as a consequence of Schoenflies theorem). Thus Theorem 15.1 follows easily from Theorem 15.1'. P r o o f . The reader can consult [Bil, w fin fiu'ther details.
[Bi3, w
the2, Chap. I], [Ma4, w
A point x E U will be said to be positively (resp. negatively) accessible if there is an embedded curve 3' : ( - o % a) ~ U satisfying 3'(t)2 --~ - o % as t ~ - o o , and having positive (resp. negative) deviation from the vertical, such that 7(a) = x. We let W+ (resp. W_) denote the set of positively (resp. negatively) accessible points in U.
132
A s s u m e f deviates vertica.1 lines to the right (the case when f deviates vertical lines to the left ca.n be reduced to the this case by replacing f with .f-1. In the ca.se f deviates vertica.1 line to the right, clearly (15.1)
f(W_AU) cW_
and f - I ( W + n U ) c W + .
Since f is area-I)reserving and there exists a < b such tha.t S 1 x (-~,a]
c W_ clW+ n U c W_ uW+ uu
c S 1 x (c~, b],
then (15.1) implies W_ = W+ = U. In fact, U \ W_ and U \ W+ a,re closed sets in U, have finite area and, by (15.1),
u \ w _ c u \ f ( w _ n u) = f ( u \ w _ ) , which contradicts tile area-preserving property~ unless W_ = U. Similarly, W+ = U. As a consequence, for each 0 E S l, there exists a0 such t h a t U n ({0} x R ) = {0} x ( - o ~ , a 0 ) 9
(15.2)
T h e a r g u m e n t goes as follows: if (15.2) is false, there exist two points A, B E OU such t h a t the segment [A, B] is vertical and does not intersect OU. Therefore [A, B] divides U into two connected COml)onents: U1, honleonlorl)hic to S 1 x R , and U2, h o m e o m o r p h i c to R 2. This fl)llows fioin tile ,]orda.n-Schoenflies theorem. If U2 lies on the left of the vertica.1 interwd [A, B], then U2 is not contained ill W _ , similarly, it is not contained in W+, if it lies on the right of [A, B]. In either case we o b t a i n a c o n t r a d i c t i o n with the fact that W_ = W+ = U. T h e conchlsion (15.2) alrea.dy shows t h a t the frontier of U, OU, is the g r a p h of a flmetion. F u r t h e r m o r e , if u : R --~ R is a C i function, we let r : S 1 x R --+ S 1 x R be the C 1 diffeomorphism r
:,j) = (0 + ',,,(y), :V) -
Let ]~ = r o f o r - - 1 . T h e n r is inva.riant under .]".,, and, if 'a is sufficiently small in the C i W h i t n e y topology, f , deviates vertica.1 lines the same side as f . Therefore r satisfies tile conditions required to follow the above a r g u m e n t s to the conclusion t h a t , for each 0 E S l, there exists ao(u) E R such t h a t r
n ({0} x R =
{0} x ( - ~ , a 0 ( " ) )
9
Since this is true fi~i' any u sufficienlty small in tile C 1 topology, it follows t h a t the fiontier of U is the g r a p h of a Lipschitz fimction, which is the content of Birkhoff theorem. T h e la.st p a r t of tile argunient also shows t h a t the Lipschitz consta.nt can be e s t i m a t e d a I,'rio'ri fronl the I)roperties of the diffeoniorphisni .f and it does not d e p e n d on the inw,i'ia.nt set U. In fact it will d e p e n d on the size of the n e i g h b o r h o o d / / / o f f in the C i topology such t h a t any g E H still deviates vertical lines the same side as f does. []
133
w
A Survey of K.A.M. Theory
The Birkhoff invariant curve theorem can be thought as a regularity result for invariant curves. It establishes that every rotational inw~.riant curve for a monotonetwist exact area-preserving diffeonmrphisnl of tile annulus n).ust be the graph over S 1 of a Lipschitz function. Thus, it is natural to ask what is the minimal regularity of rotational invariant curves for a smooth or analytic dii~omorphism. In [S-Z], D.Salamon and E.Zehnder proved that, if the rotation number of the invariant curve is strongly irrational a.nd the rotational invariant curve is sufficiently differentiable, then it is as smooth as the diffeomorphism f, i.e. it is C ~~ if f is C ~, it is analytic if f is analytic. This result, whose flavour is certainly close to hypoellipticity for P.D.E., has been obtained by a K.A.M. iteration method in configuration space, i.e. for the Euler-Lagrange equation associated to a Lagrangian system. We recall that, by a result due to J. Moser [Mo2], any exact area-preserving monotone twist map f of the a.nnulus can be interpolated by a time dependent (nonautonomous) periodic Hamiltonian flow, induced by a Hamiltonian H : T*S 1 x R x R --~ R, satisfying the Legendre condition
H,~,(O,y,t) > 0, i.e. f coincides with the time-l-map of the Hamiltonian flow associated to H. Since H satisfies the Legendre condition, f can also be interpolated by the time1-map associated to the Lagrangian flow which can be obtained from the previous Hamiltonian flow by the usua.1 Legendre transfornmtion. The K.A.M. theory has been originally found by Kohnogorov and Arnold in the analytic case, by Moser in the smooth case as a perturbative method to establish the existence of invariant tori for Hamiltonian systems (close to be integrable). The differentiability requirement have been weakened from the famous 333 derivatives of the original paper by Moser to the optimal condition requiring 3 derivatives by the efforts various mathematicians as PSschel, R{issmann, Herman. As the same time, the estimates on the size of the perturbation allowed have been improved. We refer to the second volume of Herman book [He] (for the special case of invariant curves), to the paper by Salamon and Zehnder IS-Z] and references therein. On ttxe other hand, methods to prove converse results in the case of rotational invariant curves for monotone twist nla.ppings have been developed by Hernm.n {He], who in particular proved the optimality of the condition requiring the smallness of the perturbation in the C a topology, and later extended by the first author [Ma4], [Ma7], [Ma8] and [Mag] and by the second author [F]. We will give a.n exposition of these results in w and w We will now state the K.A.M. theorem for invariant curves in the version of IS-Z]. A real irrational number w is said to satisfy a Diophan~,ine condition if there exist constants C > 0 and r > 1 such that (16.1)
1'~~ ~
for any p C Z and , ~ > 0 .
This condition express the property of aJ being badly approximated by rationals. If w does not satisfy this condition, it is said to be a Liouville number. We recall that
134
the sets of Diophantine and Liouville numbers are dense in R, but the latter is of Hausdorff dimension and Lebesgue nlea.sure zero, as it is not difficult to show. 1 6 . 1 . IS-Z] Let fo E yl be an exact area-preserving monotone twist diffeomorphism of the annulus S t x R. Let 7o he a rotational invarim~t curve for f0 of rotation n u m b e r aJ E R \ Q, satisfying the Diopha.ntine condition (16.1). Suppose that fo is of class C ~ and Jo13`o is Ct+l-conjugate to the rotation R~o : S 1 ~ S 1, l > 2r + 2 and l - 2 7 - 1 , l - r - 1 are not integers. Then Ju 13`ois C~176 to the rotation R~. In addition, if fo is analytic, the co1~jugacy is anadytic. Furthermore, there exists a neighborhood ld = ld(C, r, fu, 3`0) of fo in the C I topology such that every exact area-preserving monotone twist C ~ diffeomorphism of the annulus f E ld admits a r o t a t i o n d invariant curve 7 of rotation number w. Theorem
In the analytic case, H.Rfissmann has shown in a series of papers, concluded in [Rs], that the Diophantine condition (16.1) on the rotation number co E R \ Q can be replaced in Theorem 16.1 by a weaker condition. Let (Pn/%),,eN the sequence of convergents of the contimled fraction expansion of c0. The condition (16.1) can be expressed in terms of the continued fraction expansion as (16.1 ~)
Logq,,+l _< CLogq,, , n E N .
Riissmann proves that the condition (16.2)
Z hEN
Logq,,+l
< oo,
qn
previously introduced by A.D. Btjuno in connection with problems related to classical perturbation theory in Hamiltonian mechanics and known as the Brjuno conditiou, it is sufficient for the stability result of invariant curves contained in Theorem 16.1: if 3'0 is analytically conjugate to a. rotation and its rotation number a; E R \ Q satisfies the Brjuno condition (16.2), then there exists a n e i g h b o r h o o d / . / o f the map f0 in the analytic topology, such that any exact area-preserving monotone twist diffeomorphism .f E/d adnfits a. rotational invariant curve "7 of rotation number w. Clearly the Diophantine condition (16.1) implies the Brjuno condition (16.2), thus Rfissma.ml theorem is in fact stronger than Theorem 16.1 in the analytic case. We recall the basic facts known a.bout the relations between the smoothness of the invariant cmve 3' as a curve in S 1 x R and the smoothness of the conjugacy of the diffeomorphism fl3' to the corresponding rigid rotation of the circle. It is a classical eounterexample by Arnold that there exist analytic diffeomorphisms of the circle, with irrational rotation mm~ber w, whose eonjugacy to a rigid rotation (which exists and is a homeomorphism by a classic theorem due to Denjoy [De]) is not absolutely continuous. Analogous examples have been constructed in the smooth case. Since every diffeomorphism of the circle can be embedded as rotational invariant curve of an exact area-preserving monotone twist diffeomorphism of the annulus with the same degree of smoothness, the mentioned examples give examples of analytic (resp. smooth) monotone twist maps f having an analytic (resp. smooth) rotational invariant curve 3' such that f i t is not absolutely continuously conjugate to a rigid rotation (although topologically it is, by Denjoy theorem). However, in these
135
examples the rotation number w, although irrational, is very well approximated by rationals. On the other hand, in the case of rotation numbers satisfying a Diophantine condition, Herman's theorem holds. We state the improved version due to J.C. Yoccoz [Yo]: T h e o r e m 16.2. Let r : S 1 ~ S 1 he a difI'eonmrphism ot" the circle of class k E N and k >_ 3. Suppose that the rotation nmnber co E R \ Q satisfies Diophantine condition (16.1). Then, it'k > 2r - 1, q5 is Ck-~-*-conjugate to rigid rotation R~, tbr aa~y e > 0. In addition, ii" ~) is C ~ the COl~jugacy is C ~ is r e d analytic the co11.iugacy is also analytic.
C k, the the if q~
Theorem 16.2 implies that, if 3'0 is a rotational inva.riant curve of rotation number' a3 E R \ Q, satisfying the Diophantine condition (16.1), then, if 3'0 is sufficiently smooth r ~ c~rve in S 1 x R then Theorem 16.1 applies. In the analytic case, the Diophantille condition in Theorem 16.2 can be replaced by weaker conditions. The picture of the situation, due to J.-C. Yoccoz, is described in the survey paper by R.Perez-Marco [P-M]. We just mention that there exists a condition 7"[ on the rotation number a~, weaker than the Diophantine condition (16.1) but stronger than the Brjuno condition (16.2), such that, if a; satisfies ~ , then the analytic diffeomorphism of the circle ~ is a nalyticMly conjugate to its linear part R~. On the other hand, if ~o does not satisfies the Brjuno condition, then there are examples of analytic diffeomorphisms ~ which are not analytically conjugate to R~. However, there is a significant gap between the condition 7-[ and the Brjuno condition. Thus, the situation is, at the level of optimal conditions, more delicate than in the smooth case. One of the most interesting applications of K.A.M. theorem for invariant curves of exact area-preserving monotone twist diffe,omorphisms of the a,nmllus is the proof of the .~tahilit!! of elliptic periodic points of a.rea.-preserving diff~omorphislns of surfaces. The problem can be reduced to the case of a fixed point by considering the appropriate iteration of the map. C o r o l l a r y 16.3. Suppose that P is an elliptic fixed point of a C ~ area-preservi12g mapping f of an open subset of the plane into the l)lane. Suppose filrthermore that the Birkhoff invariants (s'ee w of the raN) f at P are not ali equaJ to zero. Then P is Lyapunoff stable. P r o o f . According to the Birkhoff normal form theorem, stated in w there exists an area-preserving change of coordinates in a neighborhood of P such that the mapping f takes the form f ( ( ) = ( exp 27ri(flo + flip '~ + ... + flNp 2N) + O(p k) and k = 2 N + 2 or 2 N + 3 . Here ( = (+it! denotes the complex coordinate associated to the real coordinates (, q, and p = ((2 + q2)1/2. The rearl numbers ill, ...,fiN, ... are called the Birkhoff invariants of f at P. If an eigenvalue A of d f ( P ) is not a root
136
of unity, we m a y take N = oe. If A is a primitive qth root of unity, t h e n we m a y take k = q and N = [q 2/2]. Thus f is the sum of the n o r m a l form -
N ( ( ) = ( exp 2~ri(fl,~ + / : ] l p 2 -~ ... 71- lIND2N) a.nd a r e m i n d e r t e r m which is no bigger t h a n O(pk), where k = ec if an eigenva/ue A of elf(P) is not a. root of unity and k = q if A is a primitive qth root of unity. If at least one of the Birkhoff invariants is not zero, then the n o r m a l form N(~) and f are exact area-preserving monotone twist m a p p i n g s in a sufficiently small p u n c t u r e d n e i g h b o r h o o d of P , as explained in w F u r t h e r m o r e , this n e i g h b o r h o o d is foliated by invariant curves of the n o r m a l form N(~), since rio, ill,.., are real numbers, and the set of r o t a t i o n numbers of invariant curves of the n o r m a l form is an open interval I C R , since at least one of tile Birkhoff invariants is non-zero. Therefore, by K.A.M. theorem ( T h e o r e m 16.1), if an eigenva.lue A of dr(P) is not a root of unity or a.t least it is not a. qth root of unity for small q, there exists a sufficiently small p u n c t u r e d neighl)orhood of P , which contains a positive m e a s u r e set of inva.riant curves of the m a p f . In fa.ct, f can be considered, in a small n e i g h b o r h o o d of P , as a. sma.ll p e r t u r b a t i o n of its n o r m a l form a.nd by K.A.M. t h e o r e m invariant curves having D i o p h a n t i n e r o t a t i o n numbers (i.e. satisfying condition (16.1)) persist for small p e r t u r b a t i o n s . Since in dimension 2 invaxiant curves are separating, the existence of invariant curves surrounding P implies the stability of P . []
We conclude the section by stating the m a i n application, due to L a z u t k i n [La], of K.A.M. theory to plane convex billiards. Let R be an open convex b o u n d e d region in the plane whose b o u n d a r y is of class C 2. The billiard ball p r o b l e m in R has been briefly described in w where it is explained the basic fact, a l r e a d y known to Birkhoff, t h a t its dynamics largely reduces to the d y n a m i c s of an associated exact area-preserving m o n o t o n e twist mapping. A caustic for the billiards ball p r o b l e m in R is a closed curve C in R such t h a t any t r a j e c t o r y which s t a r t s being t a n g e n t to C stays tangent to C after bouncing onto OR, forever in tile p a s t and in the future. A caustic corresponds to a r o t a t i o n a l invariant curve for tile exact area-preserving m o n o t o n e twist m a p p i n g associated to the billiard ball problem. 16.4. [La] If R is strictb~ convex (i.e. the curvature of OR never vanishes) a.nd OR is suI~ciently difl'erentia.ble, then there exist caustics tbr the billiard bM1 problem in R (arl)itra.rily close to OR). Theorem
In the next section we will give a converse, due to the first a u t h o r [Ma2], to T h e o r e m 16.4. T h e a r g u m e n t will be based on the Birkhoff invariant curve t h e o r e m described in w
137 w
Birkhoff Invariant C u r v e T h e o r e m . A p p l i c a t i o n s
In this section we discuss converse K.A.M. results, due to the first a u t h o r [Ma2][Ma4], which a.re based on the Birkhoff invariant curve theorem described ill w Then we prove the ba.sic varia.tional p r o p e r t y of rotationa.1 invariant curves, i.e. any orbit on a rotationa.1 invariant curve fi)r a C 1 exa.ct area-preserving m o n o t o n e twist diffeomorphisnl of the annulus is a minimal orbit in the sense established at the beginning of w Tiffs p r o p e r t y ha.s m a n y of i m p o r t a n t consequences, which will be discussed in sections w and w Glancing BiIliard,~.
We present a converse [Ma.2] to the Lazutkin theorem of the last section (Theorem 16.4). Let R be a convex b o u n d e d plane open region, whose b o u n d a r y OR is C ~. In R we consider tile billiard ba.ll problem, already described in w We will say t h a t a t r a j e c t o r y is e-gla.ncing if for at least one bounce the angle of reflection (with either the positive or negative tangent of 0_R at the point of reflection) is < e. If e < ~r/2, we can distinguish between a positively e-gla.ncing t r a j e c t o r y and a negatively e-glancing t r a j e c t o r y according to whether the direction of reflection is close to the positive resp. the negative tangent to 0R.
T h e o r e m 17.1. [Ma2] I f the curvature of OR is zero at some point, then the b i l l i ~ d bMl problem in R has no caustics. As a consequence, tbr any e > O, there exist tra.iectories which are both positively and negatively e-glancing.
P r o o f . The proof depends on the formulation of the billiard ball p r o b l e m in terms of a.rea-preserving diffeonmrphisnl of the anmdus. A t r a j e c t o r y is positively and negatively e-glancing if and only if the corresponding orbit for the associated exact area-preserving m o n o t o n e twist m a p p i n g of the annulus f : OR x (O,7r) --~ OR x (0, 7r) visits e-neighourhoods of both b o u n d a r i e s of the annulus, OR x (0, e) and OR x (Tr- e, 0). It is a classica.1 consequence of Birkhoff invariant curve theorem, a l r e a d y o b t a i n e d by Birkhoff himself, t h a t the following holds:
L e m m a 17.2. L e t .f : S ~ x (0, 7r) --~ S ~ x (0, 7r) be an exact area-preserving monotone t w i s t map. A s s m n e f has no rota.tionM invariant curves. Then, for a n y e > 0, i1' V_ = S 1 x [0, e) a n d I7+ = S 1 x (Tr - e, Tr], there exists an orbit of the m a p f connecting V_ and V+, i.e. there exist P E S 1 x (0, 7r) a n d integer n _ , n+ such that I " - ( P ) e V_ and f " + ( P ) e V+. P r o o f . Suppose there exists e > 0 for which tile above s t a t e m e n t does not hold. Chose such e a n d consider the correspondin sets V_ and V+. Let V=
U J " ( i n t V _ ) U (S 1 x { 0 } ) . nEZ
Clearly we would have
VNV+ = ~ . Let B be the connected component of S ~ x [0, ~r] \ V which contains S ~ x {Tr}. Let u = S'x[O,~]\B, T h e n f ( U ) = U, V_ C U, V n V + OandS'x{O}is a deformation r e t r a c t of U. Then, by Birkhoff inva.riant curve theorem ( T h o r e m 15.1'), 0U is the g r a p h over S 1 of a Lipschitz flmction. Therefore there exists a r o t a t i o n a l invariant curve for f , contradicting the hypothesis. []
138
We continue the proof of T h e o r e m 17.1 as follows. We suppose t h a t the curvature of OR is zero at some point and t h a t there exists a r o t a t i o n a l invariant curve for f (a caustic for the billiard ball problem). This will lead to a contradiction, thereby showing t h a t no caustic can exists if the curvature vanishes at some point. L e m m a 17.2 will complete the proof. Let P 6 OR be a point where the curvature vanishes. Let Pt a o n e - p a r a m e t e r family of points Pt 6 OR in a n e i g h b o r h o o d of P and let P ~ = g"(Pt), n 6 Z, where g : OR --~ OR describes the restriction of f to the invariant curve corresponding to the caustic. We notice t h a t , since f is orient a t i o n preserving a n d also preserves the two ends of OR x [0, 7r], 9 is a orientation preserving h o m e m o r p h i s m of the circle. Let h : OR x OR + R be the variational principle associa.ted to f . We have seen in w t h a t h exists and has the following form: h(p,Q) -- -lip QII , for any P, Q e OR. -
It is a consequence of the generating equations (3.2) a a d of the existence of a caustic that (17.1)
h l ( P t , P 1) + h2(P[-1,Pt) - 0 ,
and, by differentiating with respect to t, (17.2)
h 1 2 ( P t , P : ) dPr - j ~ + hn(P~-',Pt) (IP[-~ d~-
( h n ( P , P 1) + h22(P~-',Pt)) ~dPt -
Since P ) a n d Pt -1 represent points on the invariant curve for f associated to the caustic, by Birkhoff t h e o r e m they are Lipschitz function of t, hence their derivatives with respect to t exist ahnost everywhere and can be a priori b o u n d e d from above and below in terms of the m a p p i n g f . F u r t h e r m o r e , since g is o r i e n t a t i o n preserving and ~ d t have the sa.me sign as ~d t " Since h12 < 0, by the definition of a dt m o n o t o n e twist m a p p i n g , we conclude that
(17.3)
h~(Pt,Pr + h22(W~,Pt) > 0,
for all t 6 S 1, as a consequence of the existence of a caustic. On the other hand, in the case of a billiard ball p r o b l e m it is not difficult to show t h a t (17.4)
hl~(P,P") + h22(P',P) < 0
for any P ' , P " , if P is a. point of OR where the curvature vanishes. This can be u n d e r s t o o d by considering the fa.ct that the p a t h of a billiard ball bouncing on a rectilinear sca.tterer .~tvictly minimizes the euclidean length among all possible paths. Since h was defined a.s the negative of the euclidean length, (17.4) follows, thus c o n t r a d i c t i n g (17.3) and concluding the proof of the theorem. []
139
Non-existence of invariant curves in the Chirikov standard mapping. The Chirikov mapping was introduced in w as the exact area-preserving monotone twist map fk of the infinite cylinder S 1 x R, (17.5)
k
k
h(6),y)=(8+y+~-~sin27rS, y+~-#sin2rO),
kER,
whose variational principle is hk : R 2 ~ R ,
(17.5')
hk(:~, ~-') = ~1
(x -
x') 2 -
4 -k~ cos 2~:~ .
It is in fact a one-parameter family of mappings, which is sometimes called the standard family, where k plays the role of a perturbative (stochasticity) parameter. When k = 0 we obtain the completely integrable mapping, which is the exact area.preserving monotone twist map ]0 of the infinite cylinder S 1 x R into itself, given by (17.6)
2"~(0, :,j) = (6) + y, y ) ,
whose variational principle is h0 = 8 9 x') 2. This ma.p is characterized by the property of having a rota,tional invariant curve for any rotation number, i.e. the cylinder S 1 x R is completely foliated by rotational invariant curves of the map f0. It is a consequence of the K.A.M theorem, exposed in w that, when Ikl is sufficiently small, a large measure set of invariant curves, namely those whose rotation number satisfies the Diophantine condition (16.1), persist. On the other hand, numerical results due to Greene [Gr] show that there are rotational invariant curves for [k[ _< /co and there are none for [k] > k0, where the critical threshold is estimated as k0 ~ 0.97. Here we will expose a simple rigorous result, due to [Ma4], which establishes that there are no rotational invariant curves for ]k I > 4/3. T h e o r e m 17.3. If I/c[ > 4/3, then there are no rotationaJ invariant curves in 81 x R
which are invaria~t under f k. P r o o f . The argument is similar to the one given in the proof of Theorem 17.1. Let f be an exact area-preserving monotone twist map of S1 • R. Suppose f has a rotational invariant curve. As a consequence of Birkhoff inavariant curve theorem, the invariant curve will be the graph of a Lipschitz fimction r : S 1 --* R. Clearly, there exists an orientation preserving homemorphism g : S 1 --~ S 1 such that (17.7)
f(0, r
= (g(6)), r
.
This follows fl'om the fact that the curve given by the graph of the function r is invariant under f. Suppose the variational principle h : R 2 ---* R of the lift of f to the universal cover R 2 is of the following form:
h(.T,.~") = ~1( . T - x') 2 +
u(x)
.
Consequently, in view of the generating equations and (17.7), (17.9)
dg-1 dg d---7-~ + ~
-
~"(~)
-
2=
-
d ,z:,-~(;~(~' ~(~:)) + ;"~('J-~(:~)' ~)) - 0 .
140
Here the derivatives dg/dx and dg - 1 / d x exist almost everywhere and are bounded, as a consequence of Birkhoff inw~.riant curve theorem. In fact, Birkhoff theorem assures that r is a Lipschitz flmction and, since f and f - 1 are smooth maps, the definition (17.7) of the homeomorphism g gives that g is bi-Lipschitz. Let L > 0 be the m a x i m u m between the least Lipschitz constant of g and the least Lipschitz constant of 9-1, i.e. (17.9')
L = max {sup ]g(x) - g(x')l Iz - :,:'l , sup
[~-~(x) - ~-~(~')1 }. 1.~ - x'l
In view of the definition of L, we ha.ve (17.9')
dg(x) < L L - 1 <- dx
and L -1 < d~ -1 (:~:) < L . dx
Let m = min u" and M = max u". They exist since u is periodic on R. Furthermore, for the same reason, m < 0. From (17.8) and (17.9) we obtain: (17.10)
0 < 2L -1 < 2 + m ,
and (17.10')
L + L -1 < 2 + M .
Since the function L --4 L -1 + L is monotone increasing for L >_ 1 and 1 _< 2/(2 + rn) < L, as a consequence of (17.10) a.nd rn < O, the inequality (17.10') implies (17.11)
2+m - -
2
2 + --<2+M. 2+m
In the case f = fk, it is not difficult to compute that M = ]k I and rn = -]k[. Inequalities (17.10) and (17.11) imply
Ik] _< 4/3. This has been obtained under the assumption of the existence of a rotational invariant curve, therefore the proof is complete. [] This technique was improved later by a computer aided procedure in a paper by R.S.MacKay-I.C.Percival [MK-P], who obtained that there are no rotational invariant curves in the Chirikov mapping fi:~r Ikl > 63/64. Another result related to Theorem 17.3 has been obtained by S.Bullet [B1]. He replaces the sine function in the definition of tile Chirikov map by a piece-wise lineaa' function and proves that for Ik[ < 4/3 rotational invaria.nt curves may exist, for ]k I = 4/3 all invariant curves a.ppear and they eventually disappear for Ikl > 4/3. The variational property of rotational invariant curves. We will prove below, following [Ma3], the basic variational property of rotational invariant curves. This property is the keystone of most of the results which will be exposed in the remaining sections, regarding in particular the destruction of invariant curves with given rotation number in w and the w~.riational construction of chaotic orbits in w
141
17.4. Let f E j1 be a1~ exact a.rea-preserving monotone twist C 1 diffeomorl)hism of the annulus and F a rotationM invaria,nt curve. Then every orbit of f ill F is a 1ninhllM orbit, in the sense of w Theorem
P r o o f . The rotational invariant curve F is the graph over 81 of a real valued Lipschitz function r by Birkhoff invariant curve theorem (w Let
-/~ /"
(17.12)
H ( x , x ' ) = h(x,.~:')
XI
r
where h is a variational principle for the lift of f to the universal cover R 2 of the a,lmulus. Let g : S 1 --* S 1 be the homeomorphism of the circle induced by the restriction of f on F, i.e.
f(x, r
(17.13)
= (g(x), r
.
As previously remarked, g is a bi-Lipschitz homeomorphism, since r is a Lipschitz function and f, f - 1 axe C 1 maps. Furthermore g is orientation preserving, since f is orientation preserving and also preserves both ends of the annulus (infinite cylinder). At a. point (x, x'), where x' = g(x), the following holds: (17.14)
g~(x,x') = hl(x,x') + r
= 0 and g.2(x,x') = h2(x,x') - r
= 0.
This follows from (17.13) ~md flom the generating equations (5.2) for f , i.e. f(x, y) = (x', y') if and only if y = - h i ( x , x') and y' = h2(x, x'). Consequently, (17.15)
d ~xg(x,g(x))
dg = H l ( x , g ( x ) ) + H 2 ( x , g ( x ) ) ~ = O,
and this equation holds a.lmost everywhere, since the derivative dg/dx is defined ahnost everywhere (and positive), being g a Lipschitz increasing function. Therefi)re, H(x, g(x)), being a. Lipschitz function whose derivative vanishes ahnost everywhere, is constant equa.1 to C E R. Since H12(x, x') = hr2(x, x') < O, a.nd Hi(x, g(x)) = H2(x, g(x)) = 0, it follows that (17.16) Hl(x,x') <0, when x' > g(x) and H l ( x , x ' ) > 0 when x' < g ( x ) ,
g 2 ( x , x ' ) < 0 , when x' < g(x)
and
H2(,T,,X') > 0
when z' > g(x) .
Consequently, H(:Gx' ) > C, when x' # g(x) (a.nd H ( x , x ' ) = C, when x' = g(x)). Suppose O = {(xi, Yi)}iez represents an orbit of the diff~omorphism f on the rotational invariant curve F. By (17.13), the stationary configuration x = (x~)~ez associated to the orbit O (see w and w satisfies g(xi) = x~+1. Let m < n be integers and let y = (Ym,...,Yn) be a segment of a configuration subject to the constraint Ym = x,,, and y,~ = xn. Suppose that yi # xi for some i, m < i < n.
142
Then jfxx n
n--]
h ..... ( x) = E
g ( x i , :l;,+l ) Ji-
r
d~ = (1~ - ~)C-]-
(17.17) -}-
r rn
< Z H(yi,yi+l) + ~TIt
r
= hmn(y) ,
rn
where hmn : R ...... --* R is the flmction n -- 1
(17.1s)
hm.(v) = Z h(y,,y,+,).
Thus, we ha.ve shown that x is a. minimal configuration in the sense of w and, consequently, O is a minima.1 orbit, a.ccording to the definition given in w []
143
w
D e s t r u c t i o n o f Invariant C u r v e s .
Tile variational p r o p e r t y of rotati(mal invariant curves, described at the end of w explains the relevance in the context of area-preserving m o n o t o n e twist m a p p i n g s of a notion of a barrier flmction, originally i n t r o d u c e d in solid s t a t e physics in connection with the Prenkel-Kontorova model mentioned in w [Au-LeD]. This barrier function provides a tool which is sensitive to the existence of a r o t a t i o n a l invariant curve of given r o t a t i o n number.
The Peierls ~ barrier. T h e Peierl's barrier is a real valued fimction depending on a r o t a t i o n s y m b o l w (defined in w a n d on ~ E R , where R is seen as the universal cover of the circle S t. It measures to which extent the s t a t i o n a r y configuration (~i)iEZ, subject to the condition Y0 = ~, is not minimal. In w we i n t r o d u c e d the q u a n t i t y (13.3)
+~ /,h(v,
=
h(v,, v , + , ) - h(x,,
X,+l),
for the s t u d y of m i n i m a l configurations of r o t a t i o n symbol p/q:t:. To inchtde the case of r a t i o n a l n u m b e r s as r o t a t i o n symbols, we introduce the q u a n t i t y A~,t~.(y, x), as h~llows: (18.1)
Awh(y, x) = ~
]t(~i, ~]i+l ) - h(:ci, Ti+l ) , I
where I is equal to Z when co is an irrational r o t a t i o n symbol or is equal to p/q+ and it is equal to {0, ...,q - 1}, when co = p/q. The above q u a n t i t y can be shown to be convergent (possibly to + c o ) whenever x is a m i n i m a l configuration of r o t a t i o n symbol w and the configuration y is a s y m p t o t i c to x (i.e. lYi - xil -~ 0 as i --~ +oc). The proof of this fact is contained in T h e o r e m 13.1, in the case co = p / q + , b u t it works as well in the general case. We recall t h a t , for any r o t a t i o n s y m b o l co, we i n t r o d u c e d in w A~ as the subset of the reM line which is the union of all m i n i m a l configurations of r o t a t i o n symbol co. In T h e o r e m 12.2 a n d 13.4 we proved theft A~ is a closed subset of R , for any r o t a t i o n symbol co. T h e Peierl~'~ barrier is defined as follows. Let co be a r o t a t i o n symbol and ~ E R , then P~o(~) = 0, if ~ E A~. In the case ~ r A,,, hence it belongs to a c o m p l e m e n t a r y interval J = ( J - , J + ) to A~ (i.e. a connected c o m p o n e n t of R \ A~), P ~ ( ( ) is defined as (18.2)
P~(~) = rain {A~,h(y, z - ) l y o
= ~} ,
where the m i n i m u m is taken over the set of all configurations satisfying x ~ <_ yi <_ x +, for all i E Z, and x + are the nfinimal configurations of r o t a t i o n symbol co satisfying :c~ = J • (which certainly exist being A,~ closed). In the case co = p/q, the m i n i m u m is taken under the additiona.1 periodicity constraint yi+,~ = yi + P. We recall t h a t , under the above constraints, the q u a n t i t y A ~ o h ( y , x - ) is finite, as a consequence of (12.2) (which holds true if w = p / q • in the case co r p/q). F u r t h e r m o r e , the m i n i m u m in (18.2) exists since the topological space given by the infinite p r o d u c t of the intervals [x~-, x+], i E Z, is compact, with respect to the p r o d u c t topology, and Ao~h(y, x - ) is a continuous flmction of y. In the case w = p / q
144
there is no difficulty related to the finiteness of A~oh(y, x - ) or to the existence of the minimmn in (18.2). In the definition of the Peierls's barrier, we may take the m i n i n m m over the larger set of configurations y of rotation symbol co, subject to the constraint Y0 = (, for which the quantity A j ~ ( y , x - ) is defned. In fact, any such configuration can be replaced by one satisfying in addition the condition x~- _< Yi _< x + decreasing A j t ( y , x - ) . This is a consequence of the minimality of x • and of the inequality (9.1) (Aubry crossing lemma). It is sufficient to consider instead of y the configuration y' defined as y' = x_ V y A x +. L e m m a 18.1. P~(() _> 0, for any ( 6 R, and P~(() = 0 i f and only i f ( 6 A,,. h~ particular, ifw 6 R \ Q, P~(() _-- 0 if and only i f there exists a rotational invariant curve o f rotation n u m b e r ca. P r o o f . It is part of the definition that P~,(~) = 0 if ( 6 A~. If ~ ff A~o, suppose ~ 6 J = ( J - , J + ) , where J is a complementary interval to A~. Let x :k be minimal configurations of rotation symbol w such that x~ = J • Suppose that y is a configuration satisfying yo = ~, x ~ <_ Yi <_ x +, for all i 6 Z, and Yi+q = Yi + P, in case co = p/,~. Then, A~,(y,,~ - ) > 0. F u r t h e r m o r e , P~(~) is given by (18.2) for all 6 [ J - , J+]. In the case that co = p/q, we already proved in Theorem 10.3 that if y is a periodic configuration of type (p, q) and minimizes a-bq-1
hPerl ~ q (,:*:)
~
h('T'i,Xi+l)
over all configurations of type (p,q), then it is mininml. This clearly implies the above assertions in ca.se w = p / q , since the configuration y is not minimal and the configurations x + are minimal. When w is not a rational number, x - and x + (and therefore y) are asymptotic, by (12.2), and w - 2 < [xi+ • 1 - : c~[ < od ~- 2, by (12.4). Therefore A,o(y, x - ) is convergent and finite. Since y is not minimal, there exists a configuration v, asymptotic to x +, such that &~(y,:,:-) > A~('o,:,,-) . Let m < n be integers and consider the segment of a configuration v m,'~ defined as ....... = f v i
if i ~ m , n , :c~- if i = r n , n .
Since h satisfies property (H1) and it is continuous, it is uniformly continuous in the region defined by co - 2 _< Ixi~+l - :c~l _< co + 2. >u-thermore the series defining A~o(v, x - ) is convergent. Therefore, for any e > 0 it is possible to choose N 6 N so that, for [nl, [m[ >_ N, n--I
(18.3)
IA'(v'x-)-
E
h('u i.............. , vi• 1 ) -- h('J: 7 ' x~-+l)l
<e
"
i=nt
Since :,:- is a minimal configuration, (18.3) clearly implies A~o(v,x-) >__ 0. Thus, A~o(y, x - ) > 0. Furthermore, since x • are minimM and asymptotic, it follows easily that A~(x +, z - ) = A ( x - , x - ) = 0, thereby proving the above remark that P,,({) is given by (18.2) fi)r all ( 6 [ J - , J+]. The assertions we just proved immediately
145
imply the first part of the statement of Lemma 18.1. On the other hand, there exists a rotational invariant curve of rotation number co 9 R \ Q if and only if A~ = R. This follows from Theorem 14.1 and Theorem 17.4. Therefore the argument is completed. [] The dependence of the Peierls's barrier with respect to { 9 R is described in [Ma8, Lemma 6.3]: if h is a variational principle wich satisfies the conditions (Ho), introduced in w then P~({) is a Lipschitz function of { E R, whose Lipschitz constant can be taken to be 20. The argument applies estimates on the derivatives of the variational principle h, which are a consequence of (H0), and the Aubry Crossing Lemma. The fundamenta.1 fact about the Peierls's barrier is that it satisfies a modulus of continuity with respect to the rotation symbol. The proof of this property, contained in [Ma8]-[Mag], is rather technical and consists in a delicate compm'ison, heavily relying on Aubry theory of minimal configurations exposed in w167 between the "energy" of minimal configurations of rational and irrational rotation symbols. The result is the following: T h e o r e m 18.2. There exists a positive reaJ number C sud~ that the following holds. Suppose h satistles the conditions (Ho). If p/q is a rational nmnber (in lowest terms) and w is a rotation symbol, then
a) [P~(.~)- P~,/q(()[ _< C O ( q - l + [qTr(co)-Pl) b)
IP,,(~)
-
IP~(~) -
Pp/q+(()[
< CO]qTr(co) - p[ in the case co >_ p/q+ , and
Pp/,~- (()[ _< CO Iqzr(~) - pl iz~ the case co <_ p / q - ,
where zr(w) denotes the ,mderlying number associated to the rotation symbol w (see w Furthermore, in [F], it has been shown that, under certain conditions on the Peierls's barrier P~,/q(~), where p/q is a rational number in lowest terms, rela.ted to the hyperbolicity of the minimal periodic orbits of type (p, q), the modulus of contimfity in part b) of Theorem 18.2 can 1)e replaced by an exponential modulus of continuity. The proof combines the methods of [Ma,8-9] with Herman's approach to the destruction of invariant circles in the completely integrable case [He]. The result is essentially the following, although some fllrther technical difficulties are not discussed. T h e o r e m 18.3. There exist consta~ts C and C(O) (depending only on 0 > O) such that the tbllowing holds. Given a vari~,,tiona.1 principle h, satist~ying the conditions (go), and a rational number p/q (in lowest terms), assume that: a) there is a single minimal periodic orbit of" type (p, q);
b) there exists X(p,q) > 0 such that Pp/,l(~) >- A(p,,I)dist(~, A,,/,~) 2 , for any r 9 R , where dist(., Ap/,l ) denotes the euclidea1~ distance timction from the dosed set Ap/q on the r e d line.
146
Then, for any rotation ~'ymbol w whose underlying l m m b e r ~r(w) is' irrational ~nd ~ a t i s ~ IW("~) - "Pl < e(p,,,),
IP~(~) - G/,,~(~)I -< c(8) exp(-~(p,~,)/Iw(~) - p l ) , + or - sign according to whether w > p/q or w < p/q, where ~(p,,l) -- C/q(1 +
A--1/2 (1,,q) )
"
The conditions a) and b) in Theorem 18.3 are generic in any smooth topology and in the analytic topology. However this result can be applied to the destruction of invariant curves only when it is possible to achieve a good lower b o u n d for the constant A, which is related to the hyperbolicity of the (unique) minimal periodic orbit of type (p, q). This ha.s been done IF] in the case of the completely integrable map (17.6) and it is conjectured to be possible also in the case of a rotational invariant curve 3' such that the restriction f ] 7 of the twist diffeomorphism f to 7 is sufficiently smoothly conjugate to a rigid rotation. In the general case the problem of constructing a perturbation which yiekls the desired lower bounds for A in Theorem 18.3 is open. In the following we will apply the reduction of a periodic (minimal) orbit to a fixed (minimal) point. This is usually done in the theory of dynamical systems by considering compositions of the map with itself. In the case of exact area-preserving monotone twist mappings, generated by a. variational principle, composition corresponds at the level of variational principles to the operation of conjunction, introduced in [MaS]. Given two variational principles hi and h2, satisfying the condition (//2), their conjunction is defined in the following way: (18.4)
(hi * h2)(x, x') = ~ i ~ ( h l (x, ~) + h2((, x'))
(the m i n i m u m exists by condition (1t2)). In [Ma8] it is proved that, if both hq and h2 satisfy the conditions (H0), then so does the conjunction hi * h2 with the same 8. Notice that, even when both hi and h2 are smooth, the conjunction hi * h2 needs not to be smooth. Given a variational principle h aml a rational number p/q (in lowest terms), we will often consider in the following the variational principle H (depending on p/q) defined by (18.5)
H(x,x') = (h,...,h)(x,,x'
+ p ) ( q - times) + constant ,
where the additive constant will be chosen so that min H ( x , x) = 0. One has (18.5')
Ah
=
H Aq~_ v and
h
P:(0
=
pH
,,~-~(0 ,
as long as w is a rotation symbol which is not a rational munber or is a rational number whose denominator is divisible by q.
Th, e destruction of invariant curve,,~. The qualita.tive principle underlying the destruction results for invaria,nt curves can be described as follows. It is known since the work of Poincar6 that rotationM
147
invariant curves of rational rotation number can be destroyed by a perturbation as small as we wish in any smooth topology or in the analytic topology. On the other hand, it follows from Birkhoff invariant curve theorem, namely from the fact that rotational invariant curves are Lipschitz graphs whose Lipschitz constant is determined a priori from the map, that the set of rotational invariant curves is closed. Therefore, the destruction of an invariant curve with rationM rotation number p/q is accompanied by the destruction of an open set of nearby curves, including those with irratioim.1 rotation munbers contained in a certain open interval I(p, q) around p/q . To obtain a destruction theorem h n invariant curves of a fixed irrational rotation number, one has to provide lower e,~timates for the size of the interval I(p,q), in terms of (p, q) (in fact in terms only of the denominator q > 0). In view of Lemma 18.1, this cml be done using the modulus of continuity for the Peierls's barrier. The destruction of invariant curves of rational rotation number p/q is described, in terms of the Peierls's barriers Pv/v(~) a.nd Pp/,~+ (~), in the following. Details can be found, in the smooth case, in [Ma9], and in the analytic case, in [F]. L e m m a 18.4. There exists positive constants C,-(O) such that the tbllowing holds. Given a smooth variationM principle hi, , satisfying the conditions ( Ho ), associated to em exact area-preserving monotone twist mapping f, tbr any r E N and e > O, there exists a periodic smooth function w on R such that:
Ilwll,-+l < e a~d
maxPh, ~:~(~) > ,~ E I:L
--
P l q
where G(:~, x') = h i ( x , x') + ",,,(:~) a~d of real-valued fm~ctlons on S 1 .
C,.(e)e"+2/q (r+')~
--
I1" I[,
'
de~otes the C'" . o r m on the space
P r o o f . Choose a nfinima.1 configuration x = (xi)iEz of type (p, q). Then the set {xi + j [ i, j E Z} intersects each interval [a, a + 1), a E R, in exactly q points. By the pidgeon hole principle, there exists a complementary interval J of length > 1/q to this set. We choose a smooth non-negative function u, satisfying u(x + 1) = u(x), supported in J + i, i E Z, whose C "+1 norm is small. If IlUllr+l ~ e/2, then, since the length of J is > 1/q, u can be chosen such that (18.6)
maxu
>_ C,.e/q"+l .
On the other hand, since the support of u does not contain any point of the minimal configuration x = (xi)iEz and u is non-negative,
(18.7)
h.! p"~ r ~ = p,/,j(r p/q\"~]
+ ,u(r
where h~ is the variational principle h~(x,x') = h f ( x , x ' ) + u ( z ) . This follows immediately fl'om the definition of the Peierls's barrier Pp/v" We will now use the reduction to the case of a (minimal) fixed point. Let H~ be the variational principle associated to h,, as in (18.5). It follows fronl (18.5) and (18.5') tha.t
(18.8)
H.({,()
= phi(()
= p ~h/ uq ( ( ) ,
148
which, in view of (18.7) implies (18.8')
H , ( ( , () > u ( ( ) .
Let J = ( J - , J + ) be the complementary interval to the set {x~ + j l i, j E Z} containing the support of u. By a result in the Aubry theory of minimal configurations (Corollary 13.6), since, by (18.5'), A H~ = A p/q h~ there exist minimal configurations x i of rotation symbol 0 + such that :c~- ~ j i as i ~ Toe and x + --~ J • as i ~ +oe. We are interested in a lower bound for the m a x i n m m , over i E Z, of Ix~+1 - x ~ l . Let J' be the middle third of the interval J. If no x~ E J', then we m a y take the length of J' as a lower bound. Suppose x~ E J'. Removing :c~ from the configuration x • we obtain a new configuration y• of rotation symbol 0 • namely y]= = x~=, for j < i , a n d y ~ = ' c9 j+l, • for j > i. By Theorem 13.1, the quantity (13.3)
+c~ •
=
•
•
H~,(xi , :Ci+l)) i=-oo
exists and it is positive. On the other hand, clearly (18.9)
A(y•
•
= H,,(:r#_a,:~:~+,)-- H , , ( x # _ ~ , z # ) - - H , , ( : c # , z h i ) .
Since H.,, satisfies the conditions (H0,), for t9r = ()+ 1, provided that e is sufficiently small, the following estimate holds [Ma9, w (i8.9')
• H , , ( x i•_ l , x i +• i ) - H ~ ( x ~ _ l , x ~ ) - g , , ( . "ci• ,xi+i) <_ O' (x~+ 1 - x#_l) - H~(xi, :h) .
Therefore, by the positivity of A(y • x • and, since Hu(~, ~) >_ u ( ( ) > C" e/q "+a on J ' , as a consequence of (18.6) and (18.8'), and x~ E J ' , there exists i E Z such that
(18.10)
I:C~.{_I-- .T~I ~ Cr(O ) ~/2q T M .
Let v be a. non-negative snlooth function on R, satisfying v(x + 1) = v(x), whose i• ~ + J, J E Z. Since the estilnate (18.10) on the support is contained in rix •i ,::i+lj length of [x~• , Xi+l] • holds, if Nvll,.+l _< e/2, it is possible to choose v satisfying (18.11)
m a x v _> C~e(C,.(8) e/qr+~)~+l = Cr(O) e~+2/q (~+1)2 .
Since v vanishes outside the union of the intervals [x~, x#+l] + j, j E Z, and it is non-negative, (18.12)
P~'/~,,~(() >- PI'/~,~ (() + v ( ( ) .
where w = u + v and hg(x, x') = h i ( x , x') + w ( x ) = h,,(x, x') + v ( x ) . The inequalities (18.11) and (18.12) immediately give tile desired estimate. []
149
In the analytic case, the argument is slightly complicated because of the absence of compactly supported analytic functions. The appropriate substitutive tools are a version of the m a x i m u m principle for holomorphic functions (Hadamard 3-circle theorem) and the approximation theorem for real analytic functions by trigonometric polynomiMs (Jackson approximation theorem). The outline of the argument is, in any other respect, similar to tile smooth case. The result is the following [F]: L e m m a 18.5. There exist positive constaalts C, C(8) such that the following holds. Given a real analytic varia, tional princilfle h s, satisi~ving the conditions ( H e ) , / b r ally r > 0 aaad e > 0 there exists a trigonolnetric polynomial w such that:
Iwl, < ~ ~ld
,,,
(
max G/,,~(~) > (~/4) e~p -C(0)~, ~/~ e•
)
~ER
where h A x , x') = h• + ,,,(:~) , Iwl,. denotes the maxinmnl moduh, s on the iu~,~ite strip S, = {z ~ C I Ihn zl <_ ,'} of the hololnorphic extension o~ the trigonometric polynomiaJ w to the complex plane C, and el = rain (e, 1). Combining Lemma, 18.4 a,nd 18.5 with the modulus of continuity for the Peierls's barrier, we obta,in destruction results for invariant curves. T h e o r e m 18.6. [Ma9] Let 7 be a rotational invariazlt curve o f rotation n u m b e r P(7) = w 6 R \ Q, for an exact area-preserving m o n o t o n e twist C ~ diffeomorphisln f 6 j,oo o f the ammlus. A s s m n e w is a Liouville number, i.e. the DiophaJ~tine condition (16.1) does not hold. Then in ally neighborhood ldf o f f in the C ~ topology there exists an exact area-preserving m o n o t o n e twist diffeomorphism g C J,~ which does not ~dmit any invaria~t circle of rotation n u m b e r w. P r o o f . We argue by contradiction. Assume that there exists a neighbourhood Hf of f in the C" topology such tlmt ally y E Uj. has a rotationM invariant curve of rotation number w. In particular, if e > 0 is sufficiently small, the diffeomorphism g associated to the variational principle h,j, constructed in Lemma, 18.4, belongs t o / 4 f . Therefore, g has a rotational invariant curve of rotation number w and, by L e m m a 18.1,
p h,(~) ___0 , which implies, by Theorem 18.2 a.nd Lemma. 18.4,
(lS.13)
c,(0) ~"+"/,~('+')~ < c o l , ~ - p l ,
and this holds for a.ny p, q E Z, q > 0. Therefore w satisfies a Diophantine condition (16.1) with exponent (r + 1) 2 and we reached the desired contradiction. []
150
In fact the argument given in the proof of Theorem 18.6 proves that a necessary condition for the stability of a rotational invariant curve of rotation number w under perturbations sufficiently small in the C ~ topology is that w is Diophantine with exponent > (7'+ 1) 2. On tile other hand,the K.A.M theorem (Theorem 16.1) asserts that tile curve is in fact stable under C ~ small perturbations, if the Diophantine exponent of its rotation number is _< (r - 3)/2. Therefore, the stability and destruction results for inwi.riant curves in the C~'s topologies exhibit a gap which does not appear in the C ~ case. Namely, the dependence of the Diophantine exponent of the rotation number of the invariant curve is O(r) in the K.A.M theorem and 0(7 "2) in Theorem 18.6, as r --+ oo. This ga.p leaves open the possibility of n o n - K A M stability phenomena. We will see that the gap is even more significant in the analytic topology. In the particular case of the completely integrable map (17.6), the modulus of continuity given by Theorem 18.3 can be applied, thereby obtaining a destruction result, due to Herman [He], where the gap dishppears and the det)endence is 0(7") as in the K.A.M. theorem. In fact, any stationary configuration (:r:~)z of type (p, q) in the completely intergrable nm.p is eq7t,i.~paced mod. Z, i.e. the set {xi + j I i, j E Z} intersects any interval [a, a + 1), a E R, in q points exactly equispaced. Therefore, it is possible to choose the function u which gives the first step in the construction of the perturbation in Lemma 1.4 such that I[ull,.+a _< e/2 and
~"(,~,) >_ c ~ / , 1
(18.14)
~-~ ,
besides satisfying (18.6). To this purpose, it would be sufficient that the complementary set to {:,:~ + j I i, j E Z} contains two adjacent intervals of length >_ C/q, where C > 0 is a universal constant independent of q > 0. Since w"(xi) = u"(xi), i E Z, and 5h.q/,,,~) {.. = P~/,,(~) h.! + w(C), the constant A0,,<s) > 0 contained in Theorem 18.3 can be estimated as follows:
(18.15)
A(v,v ) >_ C ~ e / q " - '
.
h!
This holds in view of the fact that P/,~(~) --- 0 in the case f is the completely integrable map. Then Theorem 18.3, together with L e m m a 1.4, allow to replace the estimate (18.13) in the proof of Theorem 18.6 by the following: (18.16)
C~(O)e~+'21q (~+1)~ <_ C(O) exp(-C~ell2q -('+1)12 I<sw-pl -~) ,
which leads to (18.16')
Iqw - p[ >_ C,,(O, e)/q("+a)/2Log q ,
for any p, q E Z, q > 0. Therefore, in this case, we obtain a destruction result in the C ~ topology under the hypothesis that the Diophantine exponent of the rotation number w is > (r + 1)/2. This has to be compared with the condition given in the K.A.M. theorem, according to which we have stability in the C ~ topology provided that the Diophantine exponent is < (r - 3)/2. We see that there is no gap in the behaviour as r ~ +ec. In the general case the estimate (18.15) is not available, as
151
a. consequence of the fact that, if' a. (minimal) 1)eriodic orbit is not equispaced mod. Z, it is unclear whether it is possible to produce a sufficiently hyperbolic orbit (with estimates) by a perturbative construction. In the analytic case, similar arguments, relying on L e m m a 18.5 instead of L e m m a 18.4, lead to the following results contained in IF]: T h e o r e m 18.7. Let 7 be a rotational invariant curve of rotation n u m b e r P(7) = ~o E R \ Q, for an exact area-preserving monotone twist analytic diffeomorplaism f ot" the ammlus. A s s u m e ~o satistles the condition (I) where of w. exact admit
lira snl) ,.--.+~
Log Log,"q,,+ 1 qn
> 0 ,
(pn / q,, )neN is the sequm~ce of convergents of" the continued fraction expansion Then in any neighborhood DIS of f in the analytic topology there exists an area-preserving monotone twist analytic diffeomorphism g which does not any rotational invariant curve o~' rotation number w.
P r o o L Let U,.,, be the set of all real analytic periodic functions w on t t which extends to a holomorphie flmction W on the strip S~ = {z 9 c I Ilmzl < r}, in the complex plane C, and IWI, < e, where, as befl:,re, IWI,. denotes the m a x i m u m of IWl on S,.. N)r any flmction e : I t + ~ R +, let (18.17)
U~ =: U U~,~(~) . r>0
The family of sets described in (18.17), as the flmction e varies, is a basis of open sets for the analytic topology. Assume by contradiction that there exists a strictly positive function e : I t + ~ I t + such that, whenever h I - h,j E ld~, then g admits a rotational invaria,nt curve of rota,tion number w and therefore (18.18)
P2~(() = 0 .
Let (Pn/q,,),,eN be the sequence of convergents of the continued fraction expansion of 0J. Then, given any r > 0 and any e < e(r), by Lemnla 18.5 one can construct, for each n E N, an exact area-preserving monotone twist analytic diffeomorphism gn such that h I - h,,. E Ur, e C Me, where h,~ is a wu'iational principle for gn and (18.19)
h,, max P:,,/q,i (r
( -3/2 ) -> (e/4)exp - C ( 0 ) E 1 exp(C/'qn)
9
By the modulus of contimfity given in Theorem 18.2: (18.20)
(e/4)exl)(-C(8)e-13fZexl'(Crq,,))
<- COIq,,~- P,,I <- CO//qn+ 1
where the last inequality holds by tile known properties of tile continued fraction expansion.
152
By (18.20), taking the logarithnl, one gets (18.21)
Logqn+l < Co(O) -1- Log(e{-1) + C(O)el a/2 exp(Crqn) < < C 1(O)ff? 3/2 exp(Crq,~) ,
and, by taking the loga.rithm again in (18.21), (18.21')
Log Log qn+l _< Log C1 (0) - 3/2 Log el + Crqn ,
which, since q,, > 2n, leads to (18.22)
lim Slll) Log Log qn+l < Cr . n-*+~ q,,
Since r > 0 was arbitrarily chosen, (18.22) implies that condition (I) is not satisfied, which contradicts the initial assmnption and conchldes the proof of the theorem. [ ] In the completely integrahle ca.se, the double logarithm in condition (I) is replaced by the single logaritlnn of condition (J) since, in applying the exponential modulus of contimlity for the Peierls's barrier (Theorem 18.3), one of the two exponentials appearing in the estimate provided by Lemma 18.5 cancels out. In fact, it is possible to construct a small analytic perturbation, given by a trigonometric polynomial w satisfying Iw]~ _< e, thereby a.chieving: (18.23)
X(p,,~) _> Cel q2 e x p ( - r q) .
The estimate (18.23) should be compared with (18.15) which holds in the smooth ease. The estimate (18.20) is then ret)laced by the following: @/4) exp ( - C ( O ) e l 3/2 exp(C'r qn)) <-(18.24) < C(O)exp "
~(C'e[/2exp(-',' qn/2)lq,w - p,1-1)" ,
by applying the exponential moduhls of continuity given in Theorem 18.2 and Lemma 18.5. Taking twice the logarithm in (18.24), it is not difficult to obtain: (18.25)
lira sup Log q,+~ <_ C' r . n--.+oc qn
Therefore, the following holds: T h e o r e m 18.8. Under the stone hyl)otheses ot" Theorem 18.7, assume furthermore that 7 is a rotationaJ invariant curve of the completely integrable map (17.6). Then, if its rotation nmnber cv satisfies the condition (J)
limsup Log q,~+~ > 0 , n~+o. %
the same conclusion as in Theorem 18.7 holds. Theorem 18.8 provides a partial converse to the analytic K.A.M. theorem due to Riissmann, mentioned in w The gap between the Brjuno condition (16.2) of
153
Riissmann's theorem and the above condition (J) is probably due to limits of the destruction method employed. More significa.nt appears the gap between the conditions (I) a.nd (J). These conditions a.re rela.ted to ana.logous conditions obtained for the problem of linearizing a holoniorphic germ, having an elliptic fixed point with multiplier exp(2~riw) (Siegel center problem). This problem has been studied for many years as a simplified model for the "small divisors" problems which ai'ise in the stability problem for invaria.nt tori of real analytic hamiltonian systems. Thanks to the work of Yoccoz and Perez-Marco the picture in the case of the Siegel problem is fairly complete. We refer to the survey a.rticle [P-M]. The Rfissmann's K.A.M. theorem together with the above Theorems 18.7 and 18.8 suggest a. similar picture for the case of rotational invariant curves of exact area.-preserving monotone twist mappings, but here the description is incomplete and dit-ficult problems concerning especia.lly the beha.viour of non-sin(x,th curves a.re open.
154
w
Dynamics in the Sthocastic Regions.
As mentioned above, Birkhoff inva,riant curve theorem implies that the union of the set of all rotational inwtria.nt curves is a. closed subset of the annulus. A connected component of its complementary set, which is open and homemorphic to an annulus, is cMled a Birkhoff region of instability. This is characterized by the fact that it contains no rotational invariant curves. Nevertheless, for any rotation number ca E R \ Q, there exists an action minimizing invariant set E~, described in w which is given by the union of all minimal orbits of rotation symbol ca, replacing the invariant curve of rotation number ca. The Aubry-Mather set E* is the unique mininml set (in the sense of topological dynamics) contained in the set E~. Furthermore, E~ is a. Denjoy invariant set, i.e. flom the point of view of ergodic theory, the dynamics of the map on E* is isomorphic to the dynamics of the rigid rotation R~ of the circle. This also implies that E* is uniquely ergodic, i.e. it supports a unique Borel probability invariant measure a~. In [Ma6], the first author has shown that, when the minimizing set E~ is not an invariant curve, as it is the case in a. Birkhoff region of instability, then an abundance of Borel probability invariant measure of Denjoy type arise. In fact, for any rotation number w E R \ Q, there is a topological disk of any given dimension of Denjoy invariant measures of rotation nulnber ca (i.e. isomorphic to the rotation R~), with respect to the weak topology on the space of Borel probability measures. The support of these measures is contained in the set of orbits which minimize the action locally, while the action mininlizing sets E,~ corresponds to global minima.. This construction has been generalized in IF], where ahnost periodic invariant measures, having arbitrary set of frequencies, and positive entropy invariant measures are constructed for any rotation number ca E R \ Q, in case the corresponding minimizing set E~ is not an invariant curve. This is the first variational result on the dynamics in the stochastic regions which will be described below. The second result, due entirely to the first author [Mall], consists in the variational construction of wandering orbits in a Birkhoff region of instability R. This orbits can approach infinitely m a n y times, in the past a,nd in the fllture, each Aubry-Mather set contained in the stochastic region R. We fina,lly recall that, since the union of all minimal and locally minimal configurations in R is a closed set of zero Lebesgue measure, one cannot hope to obtain by this method positive Lyapunov exponents on a set of positive area (or equivalently positive entropy for the Lebesgue measure) in the Birkhoff regions of instability. Thus, in spite of the numerical evidences, the problem of proving the stochastic behaviour in these regions, in the sense of smooth ergodic theory, remains open.
Invariant mea.~ures .~npported within the gap.~ of minimizing .~et~. We will outline below the proof of the following theorems: T h e o r e m 19,1. Suppose that f E j1 is an exact area-preserving monotone twist diffeomorphism of the annulus S 1 x R which does not admit zmy rotationM invariaaat curve of rotation n u m b e r ca E R \ Q . Let ~2 be any finite or countable set of rationally independent frequencies mad let An be the free group generated by ft. If ~2 satisfies the condition fit)
~ E A.
,
155
then there exists an f-invariaalt erg'odic ahnost periodic Bord probability measure #n on S 1 x R, of angular rotation n u m b e r w, having f~ as set of its frequencies, i.e. the set of eigenva/ues {exp(27riwe)/we 6 ft} generates the spectrum ot'#n. Furthermore the following localization property holds: if ~ is the unique f-invariaa~t ergodic Borel probability measure supported on the Aubry-Mather set E*, then #n cm~ be chosen to be arbitrarily dose to a,,, with respect to the weak topology on the space of Borel probability measures with compact support on S a x R. T h e o r e m 19.2. Under the stone hypotheses of Theorem 19.1, there exists an f invariant ergodic Bore1 probability measure #h, 01("angulax rotation n u m b e r w, having positive entropy. ~-~rthermore the measure #h can be chosen arbitrarily close to the maique f-invariant ergodic measure c~ supported on the Aubry-Mather set E~, with respect to the weak topology on the space of compactly supported Bore/ probability measures on S 1 x R. The a.~l,gM~Lr rot~ttion ~tmber of a f-invm'iant ergodic Borel probability measure # is defined as the rotation number of almost every orbit (with respect to the measure #) contained in the support of #. It is well defined as a consequence of Birkhoff ergodic theorem. Theorem 19.1 contains, as a particular case, the Denjoy type invariant measures constructed and studied by tile first author in [Ma6]. They correspond to the case ~2 = {w/t~}, k e Z \ {0}. For each /c e Z \ {0}, there exists a [k[-dimensional topological disk of such Denjoy measm'es (with respect to the weak topology on the space of Borel probability measures), as it is shown in [Ma6]. A previous result on the entropy of twist maps was obtained by S.Angenent [An1-2], who proved that, if R is a Birkhoff region of instability, then f i r has positive topological entropy. This result implies, via the variational principle for the topological entropy, the existence of positive entropy Borel probability measures whose support is contained in R. The main advantage of Theorem 19.2 consists in a finer localization of the supports of positive entropy measures. Tile basic idea, contained in [Ma6], underlying tile proof of Theorems 19.1 a,nd 19.2, is to minimize the Percival's Lagrangian (6.1) subject to constraints, thereby constructing configurations which minimizes locally but are not minimal in general. The difficulty in realizing this plan is due to the possibility that the minimizing element does not lie in the "interior" of the constraint, so that it does not produce, as it does in w stationary configurations, i.e. the Euler-Lagrange equation may fail. This can be avoided when the Peierls's barrier P2 is striclty positive at some point ( E R, i.e. when no rotationa,1 invariant curve of rotation number w exists, according to Lemma 18.1. The constraints will be given by biinfinite sequences of real numbers, i.e. by elements A C tl. z. Therefore the space of constraints will be a subspace of R z,
156
which will be endowed with the product topology. Let in fact consider the following definition: 790o = {A E Rz/[[A[[0O < oo and j + A ( j ) < j + 1 + A ( j + 1), for all j C Z}, where ]]. ]10o denotes the g0o norm on R z. The constraints themselves will be given by tile following spaces: D e f i n i t i o n 19.3. Let A~ be the set of M1 Borel measurable functions r : 790o x R --+
R satisfying the tbllowing conditions: 1)
r
+ 1) = Ca(t) + 1 ,
wl, ere ~ : 7900 --+ 7900 i~ the t ~ r w ~ i d s h i a , i.e. ~ a ( . i ) = a ( . i - 1), for ~ i j 9 Z.
2)
f fa(t) s { + j , ift G j + n(j) lr ift>j+A(j),
/bra,llj 9
D e f i n i t i o n 19.4. Let Bf he t,he subset, of Af consisting of those elements which are weakly order preserving with respect to t 9 R. Notice that 790o is chosen to assure that Definitions 19.3 and 19.4 are well posed. In order to better understand the meaning of the definitions just given~ it is useful to consider the following fact: it is possible to normalize the minimizer r whose existence is the content of Theorem 6.1, in such a way that it satisfies the following condition: .)
{ r r
This can be done since, by minimizes F~ over Y*, if r Choosing a = s u p { t / r for t > 0. Since r + 1) =
(+j,
ift<j ift>j.
the translation invariance of F.,, for any a 9 R, r does. Here T,, denotes the translation t --+ t + a, t 9 R. < {}, one obtains r < ~, for t <<_O, and r > 0, r + 1, r satisfies *).
In view of *) the constraints introduced in Definitions 19.3 and 19.4 can be interpreted as a sort of noise around the minimal configurations associated to r Therefore it is natural to introduce an a.veraged version of the Percival's La.grangian. Let ,4* be the space consisting of all Borel measurable functions r : 790o x R ---* R in L0o(790o,L~o~(R)) satisfying the condition r + 1) = CA(t) + 1, for all ( A , t ) 9 79m x R. Let # be a shift-invariant ergodic Borel probability measure on 790o, then the averaged Percival-Zagrange functional is defined on ,4* as follows:
,,s
F2
(r =
h(r
Ca (t + ~))dt d , ( A )
.
.let
Notice that the definition is independent of a 9 R and therefore F~ is tranalation invariant, i.e. F~'(r = F~'(r where T,, : ( A , t ) --, ( A , t + a). The functional F~' coincides with the Percival-Lagrange functional considered in [Ma6], in case # is an atomic measure. In this case, the vaxiational problem for F~' yields in fact the Denjoy invariant measures constructed there.
157
T h e o r e m 19.5. There exists r E B(, continuous from the le~ with respect to t E R , which m i n i m i z e s the averaged Percived-Lagrmlge fimctionM F$' over .A b P r o o f . The functional F~' is bounded flom below and lower semicontinuous, since the variational principle h is bounded from below by properties (hi) and (H2) and Fatou lemma can be applied. Furthermore, F~' is ~ubmodular, i.e. it satisfies (i9.i)
F2( ~ V r
+ Fg(r
r
_< F 2 ( r
+ Fg(r
,
for any r r E A~. This property is an immediate consequence of the inequality (9.1) verified by the variational principle h and it is related to the Aubry's Crossing Lemma proved in w Since the spaces A~ and B~, considered with the natural order induced by die real lines and the standard lattice operations V, A, are complete lattices (i.e. each subset X has a. least upper bound sup X and a greatest lower bound inf X), F~' has a minimum on ./t~ and one on B~. Finally, the minimum on B~ is also minimum on .AO since, if r minimizes over .A~, there exists r E B~ such that F~'(r < F~(r This is a, consequence of the submodularity and of the tra,nslation invariance of F~'. The existence of the minima depends on the following: 1 9 . 6 . L e t F : A ---* R be a real valued fimctional on a complete lattice A. A s s m n e F is s,,bmodular, lower semicontinuous a~ld botmded from below. Tlaen F has a m i n i m u m on A. Lemma
P r o o f . Since F is bounded floin below, g = infA F is a real number. Therefore, given any sequence of positive real numbers (ei)ieN, converging to zero, there exists a sequence (r of elements of A such that:
(19.2)
e _< F ( b , ) < e + e, .
By the submodularity, since F ( r
F(r
Vr A r
_> g, _~ 1~.-I- s "~- ~ i + l ,
and iterating
F(4, A ... A r
(19.3)
_< e + ei + ... + ek .
Define '~/)i,k = 6i A ... A r fi:,r all i , k _> 1. Then by construction ('~bi,k)keN is a non-increasing sequence of elenients of A and, since A is complete, it has a limit '~/Ji = infkeN'(Ji,k. As a consequence of the inequality 2) and of the lower semicontinuity of F , one gets:
(io.4)
where r i
F(r = 2k>i
~k
_< e + , ~ ,
is finite and converges to zero, as { ---* +o% if @i)iEN is chosen
such that ~--~ei < + o c . i>0
158
On the other hand, since r _< r for all i, k > 1, then (r is a nondecreasing sequence, hence it has a limit ~ E A. By the lower semicontinuity and the choice of the sequence (ei)~eN, (19.4) implies: (19.5)
F('4,) _< e + l i m i n f r i = e , i~+oo
which concludes the proof, by showing that ~b is a mininmm point for F.
[]
Given $ E ~4" and t E R we define configurations x a = x,,~at by (xa)i=Ca(t+wi),
for a l l i E Z
and, if r is also weakly order preserving with respect to t E R, we define configurations x ~ = x r by (:c~)i = Ca(t +coi+) , for a,lli E Z . If r E A~ a.nd x = xr
then xi<_~+j,
if t + w i < _ j + A ( j ) ,
xi>_~+j,
if t + w > j + A ( j ) .
We let 7-/,oz~t- denote the set of configurations x E R z which satisfy these conditions. If r C A~ and x = z e t a , + , then zi<_(+j,
if t + w i < j + A ( j ) ,
xi>_~+j,
if t + w i > _ j + A ( j ) .
We let 7-/~ar denote the set of configurations x E R z which satisfy the latter conditions. Clearly the sets of configurations just defined also depend on ~ E It, but we prefer to drop this dependence, since ~ will be a fixed real number to be chosen appropriately, i.e. such that P~0(~) > 0. Notice that, if r E B~ is continuous from the left with respect to t E It, then x4,~oAt = X4,~A~-- E "]-{,~A't-- 9
A configuration x E ~ , t •
is sa.id to be "minimal relative to ~.~A,+ if:
for any pair of integers rn < n and any configuration x I E "H~.xt:t: such that ,! 2L m
~-
X m
and
I X n
~--- X n ~ .....
....
where h .... is the function defined as h)llows
L e m m a 19.7. Let r E B~ emd suppose r m i n i m i z e s F~~ over B(. Then, tbr a,ny given t E R, there exists a fiHl measure set :Doo(t) C :Doo, with respect to the
159
mea~ure #, such that the configuration xr
is minimM relative to 7"l~nt+, for a11
a 9 9~(t). Thus a minimizing element r 9 B~ for the averaged Percival's Lagrangian can be used to produce an abundance of relatively minimal configurations. Assuming that P~o(~) > O, it is plausible that the minimizer q~ can be chosen satisfying strictly the inequalities defining .Ar In fa.ct, if it were not so, we could produce relatively minimal configurations x = xr177 satisfying x0 = ( + j, which is impossible since P~o(~) > O. This fact is made precise in the following: 19.8. Suppose the generating function h satisfies the conditions (H1) ( H~o ). If w is irrationM, P~( ~) > 0 and # is a shift-invariant ergodic Borel probability measure supported on 7:)r 0 < c < b0(co,P,~(~)), where 7)r = {A 9 / ) ~ I IIAII~ _< c}, then there exists r 9 B~ minimizing F~' over .A~, which satisfies strictly the inequMities defining .A(, i.e. Lemma
{r 2')
if t _ < j + A ( j ) CA(t)>~+j,
if t > j + A ( j ) ,
tbrallj 9
fbr ahnost M1 (A, t) 9 7)~ x R, with respect to the measure # c = # x Lebesgue. In view of L e m m a 19.8, the proof of the Euler-La.grange equation is immediate and it does not deviate fl'om the usual argmnent. L e m m a 19.9. Suppose the smne conditions as in L e m m a 19.8 hold. Then r satisfies the Euler-Lagrm~ge equation, i.e. h2(r
- ~o), Ca(t)) + h~(r
Ca(t + w)) = 0 ,
for aJmost MI (A, t) 9 Z)~ x R , with respect to #c = # x Lebesgue. Let f be an exact area-preserving monotone twist map of the annulus S; x R. Let h be a variational principle associated to f. The conditions imposed on f imply that h satisfies the conditions (H1) - (H6o), for some 0 > 0. Let a~ E R \ Q and assume that there is no rotational f-invariant curve of rotation number w, hence P~(~) > 0, for some ~ E R, by L e m m a 18.1, and consequently L e m m a 19.8 and 19.9 hold. Let 0 < c < 5o(w,P~o(~)) and let # be a shift-invariant ergodic Borel probability measure on T)c. We will introduce a dynamicM system (7~, #L:), which is the suspension of (~r, #) over the rotation R~, where a is, as before, the forward shift on R z and R,~ is the rigid rotation t ~ t + w (rood. Z). Let Sc be the quotient space S~ = (7)~ x R ) / T , where T : T)~ x R ~ :Dc x R is the transformation T : ( A , t ) ~ ( a A , t + 1). The measure # c induces on S c a Borel probability measure, still denoted by the same symbol. The Euler-Lagrange equation suggests to consider on the measure space ($~,#c) the transformation ~ : (A, t) ---* ( A t + w) (mod. T), induced by R~, which preserves the measure #L. We formalize this construction in the folk~wing:
160
D e f i n i t i o n 19.10. If (X, T , #) is a dynamical system on a measure space (X, 13), leaving the probabiBty measure # invariant, its ~uspen~ion over the rotation R~ is the system ($, 74, #c ) de~ned as tbllows:
1) S = (X x R ) / T , where T ( x , t ) = (7-x,t + 1), for any ( x , t ) e X x R and the measure #z; is the probability measure induced by the product measure # x s where s is the Lebesgue measure on R ; 2) T4 is the transtbmnation induced on $ by the rotation R~o, i.e. 7"4 : ( x , t ) --* ( x , t + co) Onod.T). The minimizing element r 6 B( yielded by the previous m-guments induces a Borel m a p ~5 : $~ ~ S 1 x R which semi-conjugates f to 7-4 on S~. In fact, let r]a = - h i ( C a ( t ) , Ca(t + co)) = h 2 ( r
- co), Ca(t))
and
gS:(A,t)--,(Czx(t),qa(t))
(rood. Z x { i d } ) ,
then f o (I~ = ~ o T4 , ahnost everywhere on N~:, with respect to #c. Thelefin'e the measure #L induces by push-forward an f - i n v a r i a n t Borel proba.bility measure #~ on S 1 x N, i.e. #,o = 9 ,(#~;). The condition for the measure pc to be ergodic, with respect to the transfi)rmation 7~, and consequently for the measure #~o to be ergodic with respect to the
diffeomorphism f (since the push-forward of an ergodic measure is still ergodic), is the following: (E~)
exp (2rrik,/co) r E V ( p ) , for all k E Z \
{0},
where E V ( # ) is the eigenva.lues spectrum of (a,//,). This ca.n be seen by a spectral theory a.rgmnent which shows tha.t the only eigenfimctions corresponding to the eigenvalue 1 are the constant functions. The previous construction can be summarized as follows: T h e o r e m 19.11. Let w E R \ Q and suppose tha, t the diffeomorphism f does not a&nit a n y rotational invariant cm've of rotation n u m b e r co. Then any shift-invariant ergoclic Bore/ probability measure # on ~9~, 0 < c < 60(w, P,o(~)), satisfying the condition (E~o), induces an .f-inw, riant ergodic Borel probability measure #~ on S 1 x R , of angular rotation n u m b e r co. The resulting dynamicM system (f, #~) caa2 be described as a f~ctor of the suspension (74, #s of (or, it) over the rigid rotation
R~. The thctor m a p 9 : S~ ~ S 1 x R is described as:
@(A,t) = ( r
(rood.
Z x {id}) ,
where r : 59c x R --~ R is weakly order preserving, with respect to t E It, it satisfies the following localization property:
r
- c) _< Ca(t) < r
+ c),
161
and the M( constraints strictly:
1)
~b~,zx(t + 1) = ~bLx(t) + 1 ,
&n d
{r 2')
CA(t)>(+j,
if t<j+A(j) it" t > j + A ( j ) ,
forallj EZ,
tbr ahnost ali (A, t) E Z)~ x R , with respect to the m e a s u r e #L.
As betbre, a denotes the tbrward shift on R z and r E Y1 minimizes F~ according to Theorem 6.1 a n d it is normMized as in *). Finally ~ E R is chosen such that P~(() > O. According to Theoreni 19.11, it is possible to associate to any shift-invariant Borel p r o b a b i l i t y measure # on 7Pc a f - i n v a r i a n t Borel p r o b a b i l i t y measure #,o on S i x R . T h e question n a t u r a l l y raised by the previous construction concerns the ergodic-theoretical properties or classification of the mea.sures #~, which can be o b t a i n e d t h r o u g h it. Since #~ is given as a factor, it is not s t r a i g t h f o r w a r d to u n d e r s t a n d its n a t u r e in general. However, the factor nlap is p a r t i a l l y controlled by the information provided by the constraints. This is enough to conclude t h a t if # is an almost periodic measure, satisfying condition (E~), then the factor m a p is an isomorphisni, therehy obtaining T h e o r e m 19.1. This case includes the case when # is a.n atomic sliift-inva.riant p r o b a b i l i t y measure, which give the Denjoy f invariant p r o b a b i l i t y nleasures constructed in [Ma6]. Furtherniore, it is possible to choose (a, #) isomorphic to a Bernoulli shift on two symbol, in such a way t h a t the e n t r o p y of the associated f - i n v a r i a n t measure #~ is also positive. This is the content of T h e o r e m 19.2. T h e details of these arguments are contained in [F]. We finally r e m a r k t h a t the f l e e d o m in the choice of c > 0 in T h e o r e m 19.11 can be used to locMize the measure #~ constructed there, as claimed in T h e o r e m s 19.1 and 19.2. We will briefly sketch the basic idea underlying the construction of positive e n t r o p y measures. According to Lenima 19.8 and 19.9, there exists r : :De x R ---* R which satisfies the constraints strictly and therefore the Euler-Lagrange equation, if c > 0 is chosen sutficiently small. Let A_ and A+ be the subset of the cylinder S 1 x R defined as follows: A _ = {(0, y ) [ ( - 1/2 _< 0 < ( ( n i o d . Z ) } , A+ = {(0, y ) [ ( < 0 _< ( + 1 / 2 ( m o d . Z ) } . Consider relatively m i n i m a l configurations :cA i defined by (:c~A)i = CA(t + iw+) and the corresponding orbits ((z~)i,(~J~)i), i E Z, for the lift of f to the universal cover. If r E Z is such t h a t t + 'rw E ( - c , c) (rood. Z), i.e. there exists j~ E Z such t h a t t + r,~ - j,. E ( - c , c), then it possible to choose A(j~) such t h a t I/x(j~)l < c and either t + r w < j , . + A ( j , . ) or t + , w > j r + A ( J ~ ) is a.chieved. In the first case, since r satisfies the constraints, ((x~),.,(yg)~) belongs to the lift of A _ to \
;
the universal cover. In the second case it belongs to the lift of A+. Thus, we are able to construct, by a p p r o p r i a t e choices of the constraint A E ~Dc, orbits for the d i i ~ o m o r p h i s m f which belongs a'rbitra.rily to A _ or A+ each time the orbit R~(t),
162
i E Z, of the rigid rotation R~ : t --* t + w (rood. Z) belongs to the interval ( - c , c). By the irrationality of ~o, R~o is ergodic with respect to the Lebesgue measure. Thus, R i ( t ) E (--c, c) with frequency equal to 2c. Therefore the entropy of the m a p f is positive and proportional to c.
Chaotic orbits in a Birkhoff region of instability. It is a consequence of Birkhoff invariant curve theorem that, in a Birkhoff region of instability R, for an exact area-preserving monotone twist dili~omorphism of the annulus f , there are orbits which connect two preassigned open neighborhoods of the two connected components of OR. This fact has ah'eady been noticed in the proof of the existence, for any e > 0, of e-glancing trajectories of convex plane billiards in a domain whose b o u n d a r y has zero curvature at some point. We are refering to L e m m a 17.2. This statement can be significantly strengthened by a variational construction of orbits based on the positivity of the Peierls's barriers p h ( ( ) , corresponding to rotation symbols w E (p(r_),p(r§ where F+ are the two rotational invariant curves giving the b o u n d a r y of the Birkhoff region of instability. The main idea consists, as in the previous construction of invariant measures, in minimizing the "energy" associated to the variational principle h over configurations subject to constraints. The positivity of all Peierls's barriers in the region R will assure that, for a wide but appropriate choice of constra.ints, the resulting nlinimizing configurations will be contained in the interior of the constraint, thereby satifying the stationarity condition. We recall that stationary configurations axe in one-to-one correspondence with orbits of the diffeomorphism f associated to the varia.tiona.1 principle h. The results which we will describe are contained in [Mall]. We recall that a Birkhoff region o/instability is a compact f-invariant subset of the infinite cylinder, satisfying the following properties: (1) OR consists of two connected components F_ and P+, which are rotational f-invariant curves; (2) if F is a rotational invariant curve contained in R, tlmn F = F_ or F = F+. 19.12. If p(F_) < w_, w+ < p(F+), then there is a~ f-orbit (9 in R such that 0 is a-asymptotic (i.e. asymptotic in the past) to E*~_ and w-asymptotic (i.e. asymptotic in the ti~ture) to E* provided that if w_ (resp. w+) = p ( P _ ) (resp. p(P+)), then w_ (resp. w+) is irrationM. Here ~,*~ denotes, as be~bre, the Aubry-Mather set, of rotation symbol w, described in w Theorem
The statement that O = {(8i,y~)}~ez is a-asymptotic in R to E*_ means that dist ((0~ y i ) , E * _ ) --~ 0 as i --* -r The statement that O is w-asymptotic to E* means that (list ((0~, yi), E*+) --~ 0 as i ~ +z~. T h e o r e m 19.13. Consider for each i E Z, a real number p(F_) < w~ __ p(P+) and a positive number ei. There exists an f-orbit O = {(0j, yj ) } j e z in R and m~ increasing bi-in~nite sequence..., j ( i ) .... ot" integers such that dist ((0j(0, YJ(0), E*, ) < ei. In other words, O approaches within e~ > 0 of E~, at the j(i) *h iteration. By a. con.*traint ,7, we will take in this case a bi-infinite sequence (..., Ji, ...), where each Ji is a closed, connected, non-empty subset of R. A ,]-configuration will be a biinfinite sequence (xi)iez, with xi E Ji- A ~eg'm.ent of a J-configuration will be a finite sequence (xi , ..., xk) such that xi E J,, for each j < i < k. Let h be
163
a variational principle associated to an exact area-preserving monotone twist map f. A segment (x j, ..., Xk) of a ,Y-configuration will be said to be J - m i n i m a l (with respect to h) if h(xj, ..., x~) < h(x;, ..., x ~ ) , . . Xk). such . . that xj* = xj a n d x ~ = x k . for every segment of a ,:7-configuration (x. j,..., We will say that a J - c o n f i g u r a t i o n (xi)iez is J - m i n i m a l if for every j < k, the corresponding segment (x j, ..., xk) of it is J - m i n i m M . It is not difficult to specify conditions on the constraint J which assures the existence of J - m i n i m a l configurations. L e m m a 19.14. Let (..., Ji, ...) he a, constraint sl,ch that there exist arbitrarily smadl and arbitrarily large i tbr which Ji is bom~ded. T h e n there exists a J - m i n i m M contlguratiou, P r o o f . By properties (H1) and (H2), the function h ( x - N , X - N + l , . . . , X N ) is proper, continuous and bounded below on J - N x J-N+1 x ... x JN. It follows that there is a sequence (x(NN), X -(N) N + 1 , ..., X(NN)) which minimizes this function over J - N X J - N + I X ... • JN. Furthermore, using (H2), one nlay find, for each integer j, a compact set K j such that x(.J N) E K i for a,ll N: if Jj is bounded one takes K j = Jj. Otherwise, it follows fl'om the fact that there exist j ' < j < j " for wich Jj, and Jj,, are bounded and from property (/-/2) of the variational principle h. By a compactness argument combined with the Cantor diagonal process, it is possible to choose a sequence (Ni)iEN such that the subsequence xj.(ND ~ xj C K j , as i --* +oo. It is straigthforward to verify that the limiting configuration (..., x j, ...) is a J - m i n i m a l configuration. []
,:?'-minimal configurations are not always stationary, i.e. they do not yield always orbits of the diffeomorphism f. This will happen in case the J - c o n f i g u r a t i o n is also J-free, i.e. xi E int Ji, for each i. L e m m a 19.15. Let x = (...,xi,,..) be a J - m i n i m a d contlguration. If x is J - t r e e , then it is stationary, i.e. - a l h( xi, xi+ l ) = O~h( xi-1, :ci ). In particular, .f( xi, yi ) = (Zi-I-1, ~i-I-1), wliere ?,]i = --Ol]l(:l;i, :l;i-t-1), i.e. (Xi, ~i)ieZ iS ~n f - o r 6 i t . P r o o f . Since x is J-fi'ee, any sufficiently small variation of the configuration x will still be a J-configuration. Thus, the J - m i n i m a l i t y of x implies its stationarity by a simple standaxd argument. []
164
T h e orbits whose existence is asserted in Theorems 19.12 a n d 19.13 will be constructed as the orbits associated to f f - m i n i n m l a n d `y-fl'ee configurations. T h e m e t h o d of proof consists in using tile positivity of the Peierls's barriers, for any r o t a t i o n symbol in a given range, in such a way t h a t , if `y is a p r o p e r l y chosen constraint, each `y-ininimM configuration will be `y-free, and fllrthermore the corr e s p o n d i n g orbits ha~e the properties required in T h e o r e m s 19.13 a n d 19.14. T h e specifications on 27 which produce at least parti~,ll~1 ,Y-free configurations, i.e. i f free on some subinterval of Z, are quite complicated. T h e following holds. Let ~o be a real n u m b e r and let ~b~ E Y1 be a flmction which minimizes the Percival's Lgrangian, according to T h e o r e m 6.1. Let z be a m i n i m a l configuration of r o t a t i o n n u m b e r w, which corresponds to a rec'~Lrre'r~torbit of f , i.e. xi = r + a~i• for some t E R . We also choose a real n u m b e r a such t h a t P 2 ( ( 0 > 0, which is possible by L e m m a 18.1, since there is no r o t a t i o n a l i n w u i a n t curve of r o t a t i o n n u m b e r w. For each integer i, we let ai be the unique real n u m b e r such t h a t ai - a 6 Z and xi E (ai, ai q- 1). This is possible, since :ci - a is never an integer because P2(a) > 0 and the Peierls's barrier is a periodic flmction. F u r t h e r m o r e , we will associate to certain pairs (w, a) of real numbers an integer K ( w , a). L e r n m a 19.16. L e t `y = (..., Ji, ...) be a constraint. Let jo <_ j l be integers, aa~d let w, a be r e a / n u m b e r s . Suppose that K(w, a) is defined a n d Yi = [ai, ai + 1] for jo - K(a~,a) < i <_ jl Jr" If(w, a), with ai defined as above. Let ( = (~i)iEZ be a Y-minimad con~guration. Then
~ti < ~i < ai + l ,
ibr jo <_ i <_.'Jl 9
This s t a t e m e n t will represent the typical situation, i.e. we define a constraint ,7 over a certain set of integers in t e r m of a recurrent m i n i m a l configuration (in this case z) a n d we prove t h a t a `y-minimal configuration is p a r t i a l l y if-free, meaning t h a t it is `y-free on a smaller range of integers. T h e integer K ( w , a) is defined as follows. If P 2 ( a ) > 0, we let k be the smallest integer k > 2 0 / P 2 ( a ). Let n be the sma.llest integer such t h a t k < q,,, where (P,,/q,,),,6N are the convergents of the continued fraction expansion of the real n u m b e r w. If such integer n exists (i.e. if w is i r r a t i o n a l or w = p/q, in lowest terms, with k < q), we set K ( w , a) = ~t,~-1 + ~1,~and it will be undefined otherwise. We give below the proof of L e m m a 19.16. P r o o f . The a r g u m e n t can be reduced to the case j0 -= j l , without restriction. We write j for the c o m m o n value j0 --- j l . F u r t h e r m o r e we will prove only t h a t ~j < aj + 1, since the other inequality (.i > (~.i can be o b t a i n e d in a similar way. Let s be the unique real n u m b e r such t h a t c p ~ ( s + w j - ) < a j + l < ~ b ~ ( s + w j + ) . T h e real n u m b e r s exists since P 2 ( ( g + 1) = P2(a) > 0, ~ is order preserving a n d satisfies r + 1) = ~b~(.s)+ 1. Let ~/i = ~ ( s + w i - ) . Thus 'g defines a. m i n i m a l configuration of r o t a t i o n n u m b e r ~. F u r t h e r m o r e , since zi = q~(t + w i • and zi 6 (ai, ai q- 1), clea.rly t _< s and :c < y, a.s configurations. T h e first step in the proof consists in the case when there exist i0 satisfying j - K _< i0 < j and il satisfying j < il _< j + K such t h a t ai q- 1 < Yi for i = i0, il. Here we d e n o t e d by K the real n u m b e r K ( w , a ) . In this case ~i _< ai q- 1 < yi, for i = i0, i l , a n d it follows from an a d a p t e d version of A u b r y ' s Crossing L e m m a , which states t h a t the A u b r y graphs of m i n i m a l configurations cross at most once,
165
that ~i < yi, for i0 _< i _< il, and in i)articular ~j < yj < aj + 1, which is what it was to be proved. The ditt~rence with the standard Aubry's Crossing L e m m a established in w consists in the fact that the configurations y and ~ are not both minimal, since ~ is in fact J - m i n i m a l , i.e. minimal among configurations subject to the J - c o n s t r a i n t . However, it is not difficult to adapt the proof given in w to deal with this slightly more general case. In the general situation, we can assume yi < ai "Jr 1 for j < i <_ j + K or for j - K < i < j. In fact, if it is not so we can apply the previous argument. It is not restrictive to assume that the first alternative holds, since the other is similar. We set
(19.6)
z,
= r
+ IIq.-~ll + ~i+),
where, as before, p,,/q,, denotes the 'na' approximant of the continued fraction expansion of the real number w and flAIl denotes in this context the distance of the real number A from the closest integer. By its definition, z = (zi)ieN is a minimal configuration. Since, by the standard properties of the continued fraction expansion, q, is defined as the smallest integer q > 0 such that llqwll < llq,,_lWll, the projections in R / Z of the intervals [(i - j ) ~ , (i - .i)~ + II~--1~II], i = j, . . . , j + q~ - 1 do not overlap. Since r is injective and satisfies r + 1) = r + 1, this implies that the projections in R / Z of the intervals [y,, z,], i = j, ..., j + q,, - 1, do not overlap. Let k be the smallest integer > 2 0 / p h ( a ) in the definition of K ( w , a ) . Since q,, is chosen so that k _< q,, - 1, it follows that there exists il satisfying j < il _< j + k such that z~ - y~ < k -~. Since the projection in R / Z of [y~, zi] does not overlap that of [yj,zj], for j < i < i1, and yj < aj + 1 < zj, by definition of yi and zi, it follows that ai + 1 does not belong to [yi, zi]. Since, as we assumed, Yi < ai + 1, we then obtain zi < ai + 1, for j < i < il. Let i2 be the smallest wdue of i > j such that the interval [(i - j ) w , (i - j ) w + Ilqn_lWl[] contains an integer. We have il < i2 _~ j -~ If, in fact , i2 = j + q,, or i2 = j + q,, + q,~-l, as a consequence of the fact that q,, is defined as the smallest integer (1 > 0 such that II(lWll < I]q,,_lwll. Therefore z is a minimal configuration satisfying aj .-}- 1 < zj, ai2 + 1 < zi2 and a, < zi < ai + 1, for j < i < i2. This is because ai - a (mod. Z), for all i E Z, by definition of the configurations y and z, and by the assumption that yi < ai + 1, for j < i < j + K. Since ~ is J - m i n i m a l , we obtain (using as before the adapted version of the Aubry's Crossing Lemma) that ~i < zi for j _< i < i2. In particular ~i~ < zi~ < yi~ + k -1- To summarize, we have obtained the following. There exists j < il _< j + K such that (19.7)
(it
and
Yi < a i + l ,
for j < i < i s
.
Similarly, there exists j - K < i0 < j such that (19.7')
~i0 < Yi0 + k - s
and
yi < ai Jr 1
,
for i, < i < j .
(If there exists j - K < i0 < j such that aio + 1 < yio, then clearly ~i0 < Yi0; otherwise the argument we have just given applies).
166
We will argue by contradiction. We set wi = {i, i = io, il; wi = min(yi,{i), i0 < i < il. Assuming {j = aj + 1, we will obtain h(w~o, ...,'wi,) < h({io, ..., {i~), contradicting the ,]-minimality of the configuration {. We also introduce auxiliary configurations by vi = yi, for i = i0, il, vi = max(yi,{i), i0 < i < il, and ~ = y A { , ~3 = y V ~. Note that vi and /)i are the same, except at the endpoints i --= io and i = il, where they differ by an error of at most k -1. The same holds for z~i and wi. Using the properties (Ho) of the variational principle h, it. is not difficult to prove that
(19.8)
h ( w ) - h ( e ) + h(v) -
<
20k-'.
Thus, we replaced the segment of configurations w and v, which have the same endpoints as resp. ( and y, by the segments u5 and ~ for which the formula. (9.1) (Aubry's Crossing Lemma) holds:
(19.9)
h(,a,) + ,'~(,~) _< h(r + h(~,,).
Thus, h(w) < h(,3) + h@) h(v) + 20k -1 < h(~) + h(y) h(v) + 20k -1, where the first inequality is a consequence of (19.8) and the second follows from (19.9). Furthermore, the assumption ~j = aj + 1 clearly implies vj = aj + 1, and we also have vi = Yi, for i = i0, il. Since y is a minimal configuration of rotation symbol w defined by Yi = r + a J i - ) , where r + w j - ) < aj + 1 < r + w j + ) , it follows that h(v) - h(y) >_ Ph(aj + 1) = p h ( a ) > 20 k -1, by the choice of the integer k. Therefore, -
(19.10)
-
h(w) < h(()
-
-
-
-
Ph(a) -b 20k,-1 < h,(() ,
which gives the announced contradiction with the ff-minimality of the configuration
r
[]
The proof of L e m m a 19.16 illustrates tile use of the Peierls's barrier in proving that configurations which minimize, subject to appropriate constraints, are in fact locally minimal. In particular, a similar argument yields the proof of L e m m a 19.8. This method is based on the fact that minimizing configurations (subject to constraint), which are localized nearby minimal configurations of rotation symbol ~o, tend to aw:,id the regions where the Peierls's barrier P~(() is sufficiently positive, It is then sufficient to choose constraints in such a way that they produce minimizing configurations localized nearby a minimal configuration and whose b o u n d a r y lie in the region where the Peierls's barrier is sufficiently positive. In the choice of the appropriate constraints for which L e m m a 19.16 holds, a very important role is played by the choice of a recurrent minimal configuration x of rotation number w and by a "barrier" a E R such that P~(a) > 0. In fact, the constraint ff is completely specified in the range j0 - K < i < ji + K by the configuration x and the number a. One may say that the Y/s "follow" the recto'rent configuration :c in that range. Nevertheless, there are other recurrent minimM configurations y that the intervals Yi also follow. There is, in fact, an open interval ~ such that, for any w E ~, there is a recurrent configuration y of rotation number w such that the Ji's follow the configuration y in the range
167
J0 - K < i < jl + K. Thus it is possible to have constraints which follow recurrent minimal configurations with different rotation numbers in overlapping subintervals of Z. The definition of constraints J depend on the choice of a "barrier" a E l:t and of a bi-infinite sequence of integers ( n i ) , ~ z , by setting ai = a + hi. We would like to specify conditions on the sequence (..., hi, ...) which assures that, by appropriately chosing a so that P ~ ( a ) > 0, for any co E f~, then any J - m i n i m a l configuration is J-free. Let e > 0. A bi-infinite sequence of integers (ni)iez is said to be e-restrained, if for each j E Z, there exist real numbers coj and sj such that the following holds. Let (P.i,~/qjn)neN be the sequence of convergents of the continued fraction expansion of the real number wj. We suppose tha.t co is irrational or it i~* rational with denominator (in lowe,,t term.~) > e -1. Under this conditions, it is possible to introduce the smallest integer t~(j) such that q.i,e(j) > e-1. We let I(.i = qj,e(j)-i + qj;e(j). We require that the sequence (...,hi,..,) satisfies ni < w.ii + sj < ni + 1, for j - I ( j < i < j + K j . If, furthernlore, wj E f~, where ft is some open interval, then (rti)ieZ is said to be (e, f~)-re.,trained. Clearly, if (..., hi, ...) is e-restra.ined and a E R, the constraint J = (..., Ji, ...), given as before by Ji = (ai,ai + 1), where ai = a + ni follows a recurrent minimal configuration x (j) of rotation number co.i in the interval j - I ( i <_ i <_ j + I ( i , for + i w j + ) , where r any j E Z. These configurations are given by x i(J) = r is normalized by the appropriate translation to satisfy r ( 0 - ) < a < r (0+), so that x i(J) E ( a i , a i + l ) . It is then a. stra.igthforward application of Lemma 19.16 to obtain the following: L e m m a 19.17. Let n = ( . . . , h i , . . . ) E Z z. Let e = k -1, where k is a p o s i t i v e integer. L e t fl be em open interval. Let a E R be chosma such that p h ( a ) > 20 k -1 ,
tbr MI co E ft. S u p p o s e that n is (e, f~)-restrained. Let ai = a + hi, Ji = [ai,ai+l] a~ld J = (..., Ji, ...). T h e n every ] - m i n i m a / c o n t ~ g u r a t i o n is ,7-~i-ee and theretbre it satistles the s t a t i o n a I i t y condition ~ d corresponds to m~ orbit of the d i f f e o m o r p h i s m
f. Thus, the problem of finding appropriate constraints is reduced, in view of L e m m a 19.17, to the purely number theoretical problem of finding e-restrained hiinfinite sequences of integers. If M is a real number, we let F ( M ) be the set of rational numbers whose denominator (in lowest terms ) is less o1' equal to M. We will call, following [Mall], a connected component of the complement of F ( M ) an open Farey interval of heigth M , in analogy with Farey series of number theory. If n = (...,hi,...) is e-restrained, an open Farey interval ( p / q , p ' / q ' ) of heigth e -~ is said to be the Farey interval o..*.~ociated to n if the intervals (19.11)
A,L[j, k] = ((nk -- n j -- 1)/(k - j ) ,
(nk - n j + 1)/(k - j))
168
intersects ( p / q , t / / q ' ) criterion. Let (19.12)
fin all j < k. This definition axe motivated by the following
Bn[j,k] =
N A,~[y, k'] . .i<_J'
L e m m a 19.18. If'co 9 B , [ j , Ic], there exists s 9 R such that 'rzi < coi q- s < ni q - 1 ,
tbr j < i < k .
L e m m a 1 9 . 1 8 ' . Let n E Z z T h e u n is e-restrained it; tbz eadl j E Z, there exists w = wj such tha.t K ( w , ~) is defined and co 9 B,,[.i - ~c(co, e),.i + K(co, e)]. We recall once more that K(co, e) is defined if co is irrational or it is rational with d e n o m i n a t o r (in lowest t e r m s ) > e -~, as K(co, e) = qe-~ + qe, where g is the smallest integer such that qt > e -1 a.nd P l / q l , P2/q2, ... a.re the convergents of the continued fraction expa.nsion of co. The proofs of Lemma. 19.18 and Lemma 19.18' are elementary and follow from the definitions of A,,.[j, k], Bn[j, k] and from what it means for n to be e-restrained. Details can be found in [Mall, L e m m a 7.2-7.3]. It can also be shown that every erestrained biinfinite sequence of integers has a unique Farey interval associated to it in the above sense. However, what it reaJly matters for our purposes is the possibility of constructing e-restrained sequences which permit us to achieve arbitrary sequences of rotation numbers (...,coj, ...), as long a.~ they are contained in a Farey interval fl. We skip the details of this construction which can be found in [Mall, w The solution of the munber theoretical problem lea.ds, in view of L e m m a 19.17, to the possibility of constructing appropriate constraints J (i.e. constraints for which every J - m i n i m a l configuration is J-free) "following" an arbitrary sequence of rotation numbers (..., w j, ...), as long as these rotation numbers ave contained in a Farey interval, whose heigth is determined by a lower bound for the PeierIs's barriers corresponding to rotation symbols in that interval. Furthermore, it can be proved, essentially as a consequence of the Aubry's Crossing Lemma, that, given e > 0, if the constraint J "follows" a minimal configuration x of rotation number co 9 R \ Q for a sufficiently long segment (Jio,.-., Ji~ ), then the f-orbits corresponding to x and to any J - m i n i m a l and J - f r e e configura.tion ( approach within e on a subsegment i0 + K < i < il - K 1 , i.e.
where (..., (xi, yi), ...) and (..., (~i, r/i), ...) are f-orbits corresponding to x = (..., xi, ...) and ~ = (...,~i,...). An analogous result holds in case co 9 Q. We refer to [MM1, w for details. The above arguments lead to the following partial results in the direction of Theorems 19.12 a.nd 19.13. Let a 9 R a.nd P > 0. Let f~ C R be an open interval. Let k be tile snmllest integer > 2 O / P and suppose that for any rational n u m b e r p / q 9 f~ (in lowest terms), we have q > k. The positive number P has to be chosen
169
to be a lower bound for the Peierls's barriers associated to rotation symbols in ft, i.e. P h ( a ) > P , for any w E ft.
Proposition 19.19. Consider w_, w+ E ft. There exists m~ f-orbit 0 which is a-asymptotic to E * aad w-asyn~ptotic to E*r P r o p o s i t i o n 19.20. Consider tbr each i 6 Z, a reM mm2ber coi E f t emd a, positive n m n b e r ei. There exists an f-orbit 0 = (..., (Oi, yi), ...) and aa~ increasing bi-in/~nite sequence ..., j(i), ... of integers sud~ that
dist ((O.i(i), YJ(O), E*, ) < ei . In Proposition 19.19 and 19.20 it is summarized the construction of orbits which approach Aubry-Mather sets whose rotation numbers lie in an interval ft. The limitation of these results consists in the fa,ct that the interval ft cannot contain any rational numbers p/q with q <_ 2O/P, where P = sup,L6 R inf~en {P~'(a)}. If ft contains a rational number with such a. small denomina.tor than the above results do not apply. However, it is possible to prove a.n analogous of L e m m a 19.16 which allows to construct appropriate constraints J which "follow" sequences (...,w j, ...) of rotation numbers across any rational rotation number, independently of the size of its denominator, thereby completing the ingredients for the proof of Theorems 19.12 and 19.13. Let p/q be a. rational number, in lowest terms and with q > 0. It is possible to construct a segment of an orbit which first follows a minimal orbit of rotation number slightly less than (resp. slightly greater than) p/q, then follows a minimal orbit of rotation symbol t,/(t- (resp. p/q+), then follows a minimal orbit of rotation symbol p/q+ (resp. p / q - ) , and fnMly follows a minimal orbit of rotation symbol slightly greater than (resp. less than) p/q. Furthermore, it is possible to construct such an orbit so that it approaches m'bitrarily closely the minimal periodic orbit of rotation symbol p/q. This results completes Lemma 19.16, which provides a segment of an orbit following a minimal orbit whose rotation symbol is irrational or is a rational number of large denominator. The proof of the result just described is long and quite technical, therefore we will omit it. We simply observe that it is based on the positivity of the Peierls's bm'riers corresponding to the rotation symbols p/q:t:, in the same spirit of Lemma 19.16. Details can be found in [Ma.11, Proposition 8.1].
170
w
Action Minimizing Invariant Measures for Positive Definite Lagrangian Systems.
In this section we will discuss a generalization to more degrees of freedom of the variational approach to area-preserving mappings which represent the case of 2 degrees of freedom. We will not generalize the notion of minimal orbit, introduced in w and w but the related notion of m i n i m a l meaL~ure, which is a measure minimizing the action functional in sense to be specified (see [Mal0]). In generalizing to more degrees of freedom a major diff-iculty consists in finding the right setting. The setting proposed is inspired by a result due to Moser [Mo2], already mentioned in w According to this result, any exact area-preserving monotone twist map f of the annulus (or any finite composition of them) can be interpolated by a time dependent (non-autonomous) periodic Hamiltonian flow on T*$1, induced by a Hamiltonian H : T*S 1 x R x R ~ R, satisfying the Legendre condition H~:j(O,y,t) > 0 ,
i.e. f coincides with the time-l-map of the Hamiltonian flow associated to H. Since H satisfies the Legendre condition, f can also be interpolated by the timel-map associated to the La.grangian flow which can be obtained fl'om the previous Hamiltonian flow by the usual Legendre transformation. Thus, periodic positive definite Lagrangian systems provide a setting which generalize at once exact area-preserving twist mappings of the annulus and the geodesic flow on Riemannian manifohts diffeomorphic to the 2-torus. For these problems (2 degrees of freedom) a series of related results are known, namely the first author's results, exposed in w (and the closely related results by Aubry-Le D ~ r o n [Au-LeD]) for area-preserving mappings and Hedlund results concerning "class A" geodesics of Riemannian metrics on the 2-torus [Hd]. The reader can consult on these subjects and their nmtual connections the survey paper by V.Bangert [Ba]. In this section we will describe a generalization of these results, due to the first author [Ma.12], to more degrees of fi-eedom: an existence theorem for minimal measures, and a regularity theorem which asserts that the support of minimal measures can be expressed as a (partially defined) Lipschitz section of the tangent bundle. The first result generalizes Theorem 6.1, while the second extends to more degrees of freedom Theorem 14.1. The set of ergodic minimal invariant measures is also (partially) described for generic Lagrangians, following Mafid [Mfi]. All his results have been obtained by a slight modification of the setting proposed by the first attthor in [Ma12], which we will now describe. Let M be a compact, connected C ~ manifoht, and T M be its tangent bundle. Let L : T M x R --+ R be a C 2 flmction, called the "Lagrangian". The typical situation is when M is a torus. In particular, when M = S 1, then T M = S 1 x R is an infinite cylinder, which is familiar in the theory of twist maps. We impose the following conditions on the Lagrangian L. We suppose that L is periodic with respect to the R coordinate, i.e. L(~,t) = L(~,t + 1), ~ 6 T M and t G R, where the period is, for convenience, normalized to be 1. We suppose that L has po.,itive
171
definite fiberwise Hessian everywhere, i.e. L I T M , has positive definite Hessian, for any x E M . We suppose t h a t L has fiberwise xaperlinear growth, i.e. L(~,t)/ll~l I ---* +oo ,
a.s I1(11 --~ + o c ,
for ~ E T M ,
t ER .
Here N" H denotes the n o r m associated to a R i e m a n n i a n metric on M. Since M is compact, this condition does not del)end on the choice of the R i e m a n n i a n metric. T h e last two conditions imply t h a t the Legendre t r a n s f o r m a t i o n 1: is defined: if x E M , v E TM:,., t E R , then (20.1)
E ( x , v , t ) = ( x , d v ( L I T M , x {t}),t) .
If L is C ~ (r > 2), then s is a C ~-1 diffeomorphism of T M x R onto T M * x R , where T M * denotes the cotangent bundle of M. T h e fourth condition regards the completeness of the Euler-Lagrange flow, associated to L. T h e Euler-Lagrange flow can be o b t a i n e d by the the first variation of the action functional in the following way. We pose the variational p r o b l e m for the functional (20.2)
A(~/) =
L(dT(t), t) dt
over C i curves 7 : [t0,ti] --~ M with the fixed e n d p o i n t s constraint. Here, d 7 denotes the differential of the m a p "7. The trajectories of the Euler-Lagrange flow correspond to the solution of the va.riational equation (20.3)
~d(7) = 0 ,
associated to the variational probleni for A(~/) (with fixed endpoints). In other words, a C 1 curve in T M x S 1 is a t r a j e c t o r y of the Euler-Lagrange flow if a n d only if it is of the i b r m (d~f(t),t(mod. 1)), where 7 is a curve on M which satisfies the variational equation (20.3). The first varia.tion of the functional A(7) over the space of curves with fixed endpoints can be c o m p u t e d as:
(20.3')
~A(~)(r) = ~
St.
'
for any C 1 m a p p i n g D : [-e,e] x [t0,tl] ---* M such t h a t F ( 0 , t ) = 7(t), for all t E [t0,tl] and F ( s , t 0 ) = "/(to), F(.s, t l ) = 7 ( t i ) , for all .~ E [-e,e]. It is well known t h a t (20.3), with resl)ect to a system of C ~ coordinates ( x i , . . . , x,,), takes the form: (20.3")
d ~ L~, = L~: .
Therefore, the Euler-Lagrange flow is associated to the vectorfield EL described by (20.4)
dx d--/, = :~ '
d ~L~
=L,:.
T h e Euler-Lagrange vectorfield corresponds, t h r o u g h the Legendre t r a n s f o r m a t i o n , to a H a m i l t o n i a n vectorfield on T M * . It is not difficult to show t h a t , if the Lagrangian L is C", the corresponding H a m i l t o n i a n flmction is also C ~, thus the
172
Halniltonian vectorfield is C r - 1 . Consequently, since the Legendre t r a n s f o r m a t i o n is C ~-1, the E u l e l - L a g r a n g e flow is C ~-1, although the Euler-Lagrange veetorfield (20.4) m a y be only C r-2. Since r >_ 2, we o b t a i n t h a t even t h o u g h the vectorfield EL m a y be only C ~ it satisfies the conclusion of the f u n d a m e n t a l existence and uniqueness t h e o r e m for o r d i n a r y differential equations. We now state the fourth condition. The Euler-Lagrange flow is coTnylete, i.e. every m a x i m a l integral curve of the vectorfield EL has all of R a.s its d o m a i n of definition. In the classical calculus of variations tile following basic result, concerning the above b o u n d a r y value problem, holds: T o n e l l i ' s T h e o r e m . Let a < b E R, and let x,,, z~ ~ M. I l L : T M x R ~ R, periodic with respect to the R coordinate, is fiberwise positive definite and has superlineaz growth, then, eanong the absolutely continuous curves 7 : [a, b] --* M such that 7 ( a ) = :~:, a n d 3,(b) = z~,, there is one M~ich minimizes the action I" h
A(~) = ]. L(~t~(t), t) ~It
m
As p o i n t e d out by Mafi~ in [Mfi], it is not necessary to assume compactness of M for the Tonelli's theorem to hold, if the superlinear growth condition is satisfied with respect to some complete R i e m a n n i a n metric on M. A curve which minimizes in the sense of Tonelli's theorem is called a Tonelli minimizer. Ball and Mizel [B-M] have constructed examples of Tonelli minimizers which are not C 1, under the hypotheses of Tonelli's Theorem. However, under the a d d i t i o n a l hypothesis of completeness of the Euler-Lagrange flow, a Tonelli minimizer 3' m u s t be C 1, and therefore it satifies the Euler-Lagrange equation. In case L is C", we have seen t h a t a t r a j e c t o r y t --* (d3"(t), t) of the Euler-Lagrange flow is C "-1, thus 7 is C". T h e role of the completeness hypothesis can be explained as follows. It is possible to prove that, under the hypotheses of Tonelli's Theorem, a minimizer 3, not only exists and belongs to the space of absolutely continuous fimctions, b u t it is C 1 on an open and (lense set of full measure in the interval in which it is defined a n d its velocity goes to the infinity on the exceptional set. Consequently, the completeness hypothesis implies tha.t a Tonelli minimizer is C 1 ( a n d hence C~). Let - / ~ L be the space of EL-invariant p r o b a b i l i t y measures on P = T M x S 1. To every # E -/~L~ w e m a y associate its average action (20.5)
A(#) = / p L
d#
p
Since L is b o u n d e d below, the integral exists although it m a y be +oo. In case A(#) < +oo, we m a y associate to # its rotation vector pot) C H ~ ( M , R ) , which can be uniquely cha.raeterized as follows. Let c E H I ( M , R ) be a cohomology class. By the de R h a m Theorem, c can be represented by a closed 1-form ~. A differential 1-form is defined as a section of the cotangent b u n d l e T M * , b u t it can be considered also as a function on T M , linear on fibers, hence as a function on P .
173
Then, the integral on tile right in tile equation below is defined and it is is finite, since A(#) < + o z a.nd L satisfies the superlinear growth condition (along fibers):
(c, p(#)) = IF A elt,
(20.6)
6
The bracket on the left denotes the canonical pairing between the cohomology group H 1(M, R ) and the homology group Hj (M, R). It is elementary to show that, since # is En-invariant, if A is an exact foml, then the integrM on the rigth in (20.6) vanishes [Mal2, w Lemnla]. Since this integral is linear with respect to c E H i ( M , R), (20.6) defines a homology class p(#) E H i ( M , R). The basic idea of rotation vector goes back to Schwartzman's a.*ymptotlc cycle., [Sw]. It is not difficult to show the existence of invariant probability measures # such that A(#) < + ~ , for which consequently the rotation vector p(#) is defined. The argument is essentially based on the Kryloff-Bogoliuboff procedure to construct probability invariant measures for continuous flows on compact spaces. However, the space P = T M x S ~ is not compact. Therefore, we will consider the one point compactification J~* = P U {oz}. The Euler-Lagrange flow easily extends to P* to a flow which fixes oz and tile Lagrangian L can be extended by L ( ~ ) = oz to a flmction L : P* --~ R. 20.1. A(#) f L d # is a lower-semicontinuous timctionM on the space of Bore1 probability mea~qure on P* with the vague (weak) topology. Furthermore, there exists it E ~ n S~lC~ that A(#) < +oz.
Lemma
-=
P r o o f . Let AK(#) = f m i n ( L , K ) d # , for K E R. Then AK is continuous, since m i n ( L , K ) is a bounded fimction, and AK(F) //, A(#), as K /2 +oz. This implies the lower semicontinuity of A(p). We now apply the Kryloff-Bogoliuboff argument. Let a , be an absoulute raininlizer (i.e. with free boundaries) defined on a time interva.1 of length n. Let 7,,.(t) = (dc~,(t),t). By the previous remarks, %, is a. trajectory of the EulerLagrange flow. Let g., be the probability nleasure evenly distributed ahmg 7n and let # be an aecumula.tion point of the set {/t,,}neN, with respect to the vague topology on the space of the Borel probability measures on P*. This exists because P* is a compact space. An elementary argument, which we will omit, shows that # is n.n invariant measure for the extended Euler-Lagrange flow. On the other hand, it clearly exists for each n E N some curve fl,~, defined on an interval of length n, such that A(fl,~) < C a . Hence, (20.7)
A ( # , ) = 'n.-lA(c~,) < A(fl,,) < C .
Therefore A(t L) ~ C, by the lower semicontinuity of the action functional on the spa,ce of probability nleasures. Finally, since L(oz) = oz and A(#) < oz, tile measure # just constructed has no atomic part supported at the fixed point oo. Hence its restriction to P is a probability measure on P and tt E M L. []
174
T h e l e m m a we just proved has the following i m m e d i a t e consequence: 6 ~ L "which minimize.~ A o v e r .MIL.
there
exi~t.~ #
A refinement of the previous Kryloff-Bogoliuboff a r g u m e n t gives the following: 2 0 . 2 . Let +oc ~ d p(#) = h.
Lemma
h 6 HI(M,R).
Then there exists
# 6
J~L satisfying A(#) <
P r o o f . We will a p p l y Tonelli's T h e o r e m to the covering s p a c e / ~ / o f M , determ i n e d by 7q(.~/) = ker (7"/ : 7r,(M) --* H I ( M , R ) ) , where 7-~ denotes t h e Uurewicz h o m o m o r p h i s m . In the model case M = T " , then 2~/ = R " . The group of deck t r a n s f o r m a t i o n s of this covering space can be identified to (20.8)
7) = I m (7-/: Try(M) --, H i ( M , R ) ) = h n (7-/: H~(M, Z) + H i ( M , R ) ) ,
which clearly is a lattice in the finite dimensional vector space H I ( M , R ) . example, if M = T " , then D = Z".
For
Therefore, for any h 6 Hi(M, R ) , there exists a sequence of deck transformations T1,...,T,,,... such t h a t n-iT,,. ~ h 6 H I ( M , R ) , as n ---* +oo. F i x ~0 6 _~r and let 2,, = Tn20. Let 0 , , : [0, n] --+ 2~/minimize J0" n(da,,(t),t)dr, subject to the b o u n d a r y conditions 5~,(0) = 20 a n d &,,(n) = ~,,, where a,, is the p r o j e c t i o n of &n on M . T h e existence of 5~,, follows from an a d a p t e d version of Tonelli's T h e o r e m (cf. [Mal2, w To o b t a i n a E n - i n v a r i a n t measure, we a p p l y again the K r y l o f f Bogoliuboff argument. By the completeness hypothesis of the Euler-Lagrange flow &n is C 1. We let 7,, = (da,,(t), t) E T M x S ~. We let #~ be the p r o b a b i l i t y m e a s u r e evenly d i s t r i b u t e d along %,. a a d let # be an accumulation point of the set {/L,,},,eN , with respect to the weak topoh:)gy on the space of Borel p r o b a b i l i y measures on P*. As before, it is e l e m e n t a r y to show t h a t # is an invariant m e a s u r e for the E u l e r - L a g r a n g e flow. This is the core of the Kryloff-Bogoliuboff argument. On the other h a n d it is e ~ y to see t h a t there exists C > 0 and, for each n 6 N , a curve fl,~ : [0, n] --~ M such t h a t fin(0) = i:0, fl,~(n) = 2n and J'o~ L(rlfl,,(t), t)tlt < Cn, where fl,~ is the p r o j e c t i o n of ~,, on ~r. Consequently, (20.9)
A ( # , ) = ',~-]A(~,,) < A(/3,,) <_ C ,
which clearly implies A(#) < + o c . In p a r t i c u l a r the point at infinity has zero m e a s u r e with respect to #, thus t* can 1)e viewed as a p r o b a b i l i t y measure on T M x S 1, invariant with respect to the Euler-Lagrange flow. Finally, by construction (20.10)
p(#) =
lim
n-~T,, = h 6 H ~ ( M , R ) .
This complete the outline of the proof of L e m m a 20.2.
[]
175
It follows i m m e d i a t e l y fl'om tile conditions imposed on L t h a t L is b o u n d e d from below, i.e. there exists B E R such t h a t L > B. Therefore the set UL = {(p(#),z) e H l ( - ~ r , R ) x R [ A ( # ) <_ z, # E M L } is contained in H I ( M , R ) x [B, +oo). Furtherniore, since # ---* p(#) is continuous on {it E .AdL [ A ( # ) _< C}, as it is not ditficult to prove, and A is lower semicontinuous ( L e m m a 20.1), UL is a closed set. Clearly, UL is also convex and, by L e l n m a 20.2, its p r o j e c t i o n on H i ( M , R ) is surjective. Consequently, UL is the epigraph of a convex function f l = f i L : H I ( M , R ) - - + R , i . e . UL = { ( h , z ) E H I ( M , R ) x R I f l ( h ) _ < z } . For any h E H i ( M , R ) , we will call fl(h) the minimal average action, of the r o t a t i o n vector h. T h e value fl(h) clearly represents the m i n i m m n of the average action A over the p r o b a b i l i t y nieasures 1l E -AdL satisfying p(#) = h.. This is a niinimization problenl tbr the flnictiona.1 A, subject to the contra.int fl(t t) : h, and can be t r e a t e d by the m e t h o d of Lagrange multipliers. This remm-k motivates the i n t r o d u c t i o n of the m i n i m i z a t i o n p r o b l e m {or the following family of functionals. Let c E H i ( M , R ) , we set (20.11)
A~(#) = A(#) - ( c , p ( ~ ) ) = I p ( L - A)d# ,
where A is a closed l - f o r m in the de Rhaui eohomology class corresponding to c. This is defined for any # E A4L such t h a t A(it) < +oo and it can be e x t e n d e d to the ease A(#) = +0% by setting At(#) = + ~ . The cohomology class c plays the role of the La.gra.nge inultiplier. L - A satisfies the same conditions imposed on L, in p a r t i c u l a r the Euler-Lagrange flow associated to L - A is the same as t h a t of L. Consequently, Lemnia 20.1 applies and Ac takes a m i n i m u m vahle, which will be d e n o t e d by - a ( c ) . It is not difficult to realize t h a t o~ : H~(M, R ) ~ R is a convex function a n d its e p i g r a p h {(c,z) E H I ( M , R ) x R ] z _> o~(c)} is a convex subset of H 1(M, R) x R. Let a* : H i ( M , R ) convex analysis [Rc]: (20.12)
---* R denote the conjugate flmction of a in the sense of
- c C ( h ) = rain {a(c) - (c, h)} ,
where c varies over H i ( M , R). A priori, a* takes values in R = R U {+oz}, b u t in fact it takes vahles in R . The reason is the following: if # E JML and A ( # ) < + o z , then a * ( p ( # ) ) _< A(#), as it follows i m m e d i a t e l y fi-om the definitions. On the other hand, by L e m m a 20.2, for any h E H I ( M , R ) , there exists # E .AdL such t h a t A(#) < + o o and p(#) = h. F u r t h e r m o r e , by its definition, a = /~*. T h e basic convex analysis then implies: T h e o r e m 20.3. [Ma12, Theorem 1] The thnctions a : H ~(M, R ) ~ R a n d fl : H i ( M , R ) ~ R are co2~.iugate convex timctions and have superllnear growth. For h E H i ( M , R ) , we have
/~(h) = m i n { A ( # ) I # e M L
an(1
p(#) ---- h } .
For c E H 1( M , R), we have - ~ ( ~ ) = rain {A~(#) I# ~ M L } 9
176
The outcome of the above discussion can be summarized in the following terms. Let 3,t c, c E H I ( M , R ) , be the set of all probability measures # E AlL which minimize Ac and Alh the set ot all # E fl4L which minimize A and satisfy p(#) = h. Then
(20.13)
U cEHI(M,tt)
Al~ =
U
Alh,
hEHI ( M,tt)
in fact, if # E AlL, then A(#) = fl(p(#)) if and only if there exists c E H I ( M , R ) such that # minimizes Ac. Furthermore, c is the subderivative of the convex function fl at p(#), i.e. the slope of a supporting hyperplane of the epigraph of fl at p(#). A probability measure it satisfying these conditions is called a 7r~i'rdmM meo, s',r'e. The invariant probability mea.sures which are relewmt for the dynamics are the ergodic ones, i.e. those satisfying the condition tha.t every invariant set has measure 0 or 1. It is a standard elementary result of topological dynamics that the extremal points of AlL are the ergodic measures invariant with respect to the EulerLagrange flow. Since the convex fimction fl has superlinear growth, according to Theorem 20.3, its epigraph has infinitely many extremal points by a standard result in convex ana.lysis [R.c]. Let (h,~ fl(h)) denote an extremal point of the epigraph of ft. The extremal points of the set of iL E AlL for which p(#) = h and A(#) = fl(h) are ergodic measures, since they also are extremal points of AlL. Since this set is compact and convex such extremal points do exist. Thus, we have proved the existence of ~t ler o~,e invariant ergodie minimal measure # with rotation vector h, such that (h, fl(h)) is an extremal point of the epigraph of ft. This result opens the problem of describing the set of rotation vectors h for which A4h contains an ergodic measure. The simplest ease to understand is when M = S 1 . It fi:Jllows fi'om a theorem due t o Moser [Mo2] that this case is related to twist maps or, more generally, to finite
compositions of twist map. In fact, as we already mentioned at the beginning of this section, according to this result, any finite composition of exact area-preserving monotone twist maps f of the annulus can be interpolated by a time dependent (nonautonomous) periodic Lagrangian flow on TS 1, induced by a Lagrangian L : T S 1 x R --* t t which satisfies the conditions imposed here, i.e. the positive definiteness, the superlinear growth (along fibers) and the completeness of the Euler-La.grange flow. In this new setting the existence theorem of Aubry-Mather sets (Theorem 6.1) fi)r twist maps (or finite compositions of them) is a consequence of the following: P r o p o s i t i o n 20.4. In the case M = S 1 , the fimctionfl : H i ( M , R ) -* R is strictly convex, i.e. every point on the g'raph o[ [~ is an extremal point ot" the epigraph of ft. Therefore, Alh contaJns an ergodic measure, tbr any h E H i ( M , It). This result depends on the Lipschitz property of the support of minimizing measures, which will be described later. We will outline below some results obtained by Mafid [Mill in the direction of giving a satisfactory description of the set of ergodic minimal measures in the general ease, at lea.st for generic Lagrangians. These results leave many natural questions unsolved and the situation is far from being as simple and well understood as in the
177
case M ---- S 1. Marl6 approach is inspired to the first author's approach described above, but it differs slightly under the following respect. The minimizing measures are obtained through a variational principle not requiring the invariance a priori, but the invariance property is Inoved a.s a consequence of the minimization property over a,n a,pl)ropria,te sl)ace. Let 2Id he the set of probability measures on the Borel ry-algebra of T M x S 1 such that:
/pl1411~z# <
(20.14)
+~
Q
It is not difficult to show (cf. [Mil D that there exists a unique metrizable topology on .Ad such that (20.15)
/i.,,-~/te:=v/,~bd#,,--,ffCd#,
for every continuous function ~ : T M x S 1 ---* R growing (fiberwise) at most linearly, i.e.
(20.16)
sup ,~(~-,~,t) < + ~ . (~,c,) 1 + II~II
Tile space of probability ineasures which will be considered is a closed subset of M (endowed with the induced topology) defined as follows. Given a periodic absolutely continuous curve 7 : R ~ M, with period N E Z, define the probability lt~ on the Borel cy-a.lgebra, of T M x S 1 by posing (20.17)
/
~bd#~ =
1/0
~
~b(dT(t),t)dt,
for every continuous flmction ~b : T M x S 1 --~ R with compact support. It is an immediate consequence of the absolute continuity of 7 that #~ E 3d, since (20.14) clearly holds if 7 is absolutely continuous. Furthermore, observe that #-t~ = # ~ if aad only if 3'1 = 72. A probability measure #'r has a naturally associated homology class t5(#-~) defined as 1 fi(#~) = ~ [7] e H I ( M , R ) , where [7] denotes the homology class of the curve 7. Let C C M be the set of probability measures of the form #-r, a.nd let C be its closure. The space C satisfies the following three properties: (I) For every Lagrangian L on M, satisfying the conditions of positive definiteness, superlinear growth (along fibers) and completeness of the associated EulerLagrange flow, all the probahility measures # which are inw~.riant with respect to the Euler-Lagrange flow such that
pL(Iit < +oc are contained in C.
178
(II) Probabilities t t in C have a natura.lly associated homology ~(#) E H1 (M, R). The m a p f5 : C ---, H i ( M , R) is the continuous extension of the map /5 : C H1 (M, R), defined above. Fmthernmre, the map p, extended to C, is surjective. (III) For every Lagrangian L on M, satisfying the above conditions, the set
{# E M I J L d,# < +oo}
compact.
These properties are proved ill [Mn, w It Mlows that the variational problem of finding, given a rotation vector h 6 H I ( M , R ) , a probability measure # 6 satisfying ~Ld#=min{~Ld,~l,~EC,
/5(,,) = h}
has at least one solution. This is a consequence of the coml)actness p r o p e r t y (III) and the continuity of the map/5. We will denote the set ()f such # by 3dh, in analogy with the previous notation. As befi:)re, if A is a closed 1-h)rm, the above assertions hold true for the Lagrangian L - A. Thus, it is possible to consider the set 35t ~ of measures # 6 C which minimizes over C the average action fimctional associated to the Lagrangian L - A. Here, as before, c denotes the (de R h a m ) cohomology class of the closed 1-form A. Standard convex analysis [Re] gives:
c6HI(M,R)
h,e H t ( M , R )
A proba.bility mea,sure p, E C, belonging to the set in (IV), will be called a C9rr~iT~{mal m.eas~t.re for the Lagra.ngia.n L. The foIIowing result unifies the first author's approach and MafiCs approach, which we just described. T h e o r e m 20.5. [Mfi, Theorem A] A n y 0--minimal measure t L /br a Lagromgim~ L, satisfying the stated conditions o f positive definitm2ess, superlinear growth and completeness, is invariant under the Euler-Lagrm~ge ttow associated to L. f h r t h e r m o r e , if # E 3d ~ and A is a 1-/brm in the c o h o m d o g y class c E H i ( M , R), the function L - 1 is honmlogous to a constant on the support of/t, in the sense that there exists a Lipschitz fimction V : s u p p ( # ) ---, R ~ d a c o s t ~ t C > 0 sud~ that on s u p p ( # )
L-A=C+ELV, where EL V the directional derivative o f the function V with repect to the vectortleld
EL, which generates the Euler-Lagra.nge flow. In view of this result, Adh = 3Ah, h 6 H~(M,R)__, 2Q ~ = 3d ~, c 6 H ~ ( M , R ) , and the identities (20.13) and (IV) coincide. Thus, a C-m.iTdrn.al meas~tre in Mafid's sense is a minimal meas~tre in the first author's sense. Furthermore, the function /5 : -C ~ H i ( M , R), which associate to each probability measure # 6 C its rotation vector, coincides on the space 2%4L of probability measures inva.riant with respect to the Euler-Lagrange flow, with the map p introduced by the first author, chaxacterized in (20.6). In fact, if # is an ergodic invariant measures, it is a consequence of Birkhoff's ergodic theorem and of the continuity of/5 that /5(#) = p(#) and it is uniquely determined as follows: let 7 : R --* M be a #-generic trajectory of the Euler-Lagrange flow (in the sense of the ergodic theorem). For T > 0, let ZT be
179
the closed curve defined by 71[-T, T] and a "short" curve (of length b o u n d e d by d i a m ( M ) ) joining 7 ( - T ) with 7(T). Then (20.18)
fJ(#) =
p(#) =
~ -l-i+mo 0
~ 1 [zv] .
We will state below the basic properties of m i n i m a l measures which have been established in [Mal2, w Let AL C T M x S 1 be the union of all minimizing trajectories of the Euler-Lagrange flow, i.e. the set of all absolutely continuous (in fact s m o o t h ) curves 7 : [a, b] --* M which minimize the action /* b
A(-f) = ~, L(dT(t), t) dt over all absolutely continuous curves having the .~amc cndpoints 7(a) and 7(b) and belonging to the .~arne homology clas., as "f in H I ( M , R ) . A t r a j e c t o r y contained in An will also be called a minimizer for the L a g r a n g i a n L. An is clearly a closed invariant set. As before, M ~, c E H I ( M , R ) , will denote the set of p r o b a b i l i t y invariant measures minimizing Ac (20.11). As we remarked, M c = M c, where the l a t t e r is the set of measures in C minimizing A~ c o n s t r u c t e d in the Mafi~'s approach. F u r t h e r m o r e , M ~ is a compact, convex set a n d its e x t r e m a l points are ergodic measures. We will denote by s u p p M c the support of M c, i.e. the set of (x, ~, t) E T M x S 1 such t h a t every n e i g h b o r h o o d of (x, (, t) has positive measure with respect to some measure # E M e .
Proposition 20.6. [Ma12, Prop. 3] For any c E H i ( M , R ) , every trajectory of the Eu]er-Lagrange {Jow in supp./Pf c is a minimizer, i.e. supp.A4 c is cont,~ned in AL.
Proposition 20.7. [Ma12, Prop. 2] I f # E M L is ergodic and supp(#) C AL, then # is a minimizing measure, i.e. there exists c E H i (M, R ) s u d l that # E .A4 ~. In view of the above two Propositions, it is clear t h a t minimizing measures completely describe the ergodic theory of the invariant set AL of all minimizing trajectories. It m a y seem t h a t the dynamics on such an invariant set is restricted to a special behaviour. However, as pointed out in [Mfi], the dynamics on AL can be a.s c o m p l i c a t e d as t h a t of any vectorfield X on M . In fact, if we consider the autonomous Lagrangian (20.19)
L ( x , ~ ) = II( - X(x)]] 2 ,
then every every solution of the differential equation ~ = X ( x ) on M is a minimizing t r a j e c t o r y of the Euler-Lagrange flow associated to L, as the reader can easily verify.
Proposition 20.8. [Ma12,Props. 2-3-4] supp Jt4 ~ is a compact subset of T M • S 1. ~~rthermore, for any m e a s u r e # E .A4 L , supp(#) C supp.h4 ~ ==~ # E A4 c . T h e arguments employed in the proof of the above P r o p o s i t i o n s 20.6 and 20.7 del)ends essentially on cutting and 1)asting minimizing trajectories a n d using the
180
continuity of pot) and the lower semicontinuity of A ( # ) combined with the KryloffBogoliuboff procedure of constructing invariant measures. In the proof of Proposition 20.8, the main point is that, if ]1~11is unbounded as (x, ~, t) varies over T M x S ~, then it is possible to construct an incomplete (i.e. not C 1) trajectory of the EulerLagrange flow, in contradiction with our hypotheses. The next pr()perty we will state is the Lipschitz property of the support of M r which is the main result of [Ma12]: T h e o r e m 20.9. [Ma12, Th. 2] It'1( : T M x S 1 ~ M x S 1 is the cmmnical proiection, then, for any c E H i ( M , R), the restriction z r l s u p p . M c is in.iective and its inverse (zr [ s u p p 3 d ~ ) -1 : ~r(supp.M ~) --, s u p p . M ~ is a Lipschitz map, i.e. there exists a construct C > 0 such that, I'or emy x, y E ~r(suppAd~), we have dist (~r-~ (x), 7r-~ (y)) <_ C dist (x, y) .
The intuitive idea of the proof of Theorem 20.9 is the following. There is a well known "curve shortening" l e m m a in basic Riemannian geometry which goes as follows. Let a and fl be curves on a Riemannian manifold joining points P , P' aad Q, Q~ resp. Suppose that a and fl cross. Then there exist curves a, joining P and Qt, and b, joining Q and pi, such that (20.20)
length (a) + length (b) < length (c,) + length (fl) .
A similar "shortening lemnm" h()lds fin tile action flmctional associated to a Lagrangian L. In fact, the following holds [Ma12, w L e m m @ if K > 0, then there exist constants e, 6, rl, C > 0 such that, if a, ~ : [to - e, t0 + e] --* M are trajectories of the Euler-Lagrange flow, with a(to - e) = P, c~(t0 + e) = P ' and f ~ ( t o - e) = Q, fl(to + e) = Q', Hdc~(t0)II, I]d~(to)N < K , d i s t ( ~ ( t o ) , ~ ( t o ) ) < 6, and dist (dc~(t0), dfl(to)) > C dist (or(t0), r then there exists C 1 curves a, b : [to e, t0 + e] ~ M such that a(to - e) = P , a(to + e) = Q' and b(to e) = Q, b(to + e) = P ' , and -
(20.20')
-
A(c~) + A([~) - A ( a ) - d(b) >_ q (list (da(to), d[~(to)) 2 .
Thus, if r were not injeetive on supp M c, or its inverse were not Lipschitz, it would be possible to construct a probability measure # E AdL for which A c ( # ) < A~(Mr contradicting the definition of M ~. This result would be achieved by "cutting and pasting" trajectories using the "curve shortening" lemma and the Tonelli's Theorem. Then the Kryloff-Bogoliuboffargument would provide the required measure, because of the continuity of p(#) and the lower semicontinuity of A(#). The details of the arguments sketched above can be found in [Ma12, w167 T h e first applications of Theorem 20.9 are to the description of the case M = S 1, thereby completing the picture given by Proposition 20.4 and re-obtaining the basic results found in w P r o o f o f P r o p o s i t i o n 20.4
181
Suppose 13 : H I ( S X , R ) - R --~ R is not strictly convex. Then the g r a p h of 13 intersects a line l in R 2 in a segment I not reduced to a point. Let (h0,13(h0)) and ( h i , 13(hi )) be the endpoints of I. These points are e x t r e m a l points of the e p i g r a p h of 13, hence there exist action mininfizing ergodic measures #o and #1 whose r o t a t i o n n u m b e r is h0 resp. hi. Each of t h e m is contained in M r where c E H I ( S 1, R ) -= R is the slope of the line l. By T h e o r e m 20.9, the projection zr of supp M r on S 1 x S 1 is injective. But this contradicts the fact t h a t two Birkhoff generic orbits 70, 71 in zr(supp(#o)) resp. z r ( s u p p ( F 1 ) ) m u s t cross, since they have different rotation nulnbers. On the other hand, they a.re the projections of distinct (and therefore disjoint) trajectories of the Euler-Lagrange flow on T S 1 x S 1. []
Let h E H I ( S 1 , R ) , let l C H I ( S 1 , R ) x R be a s u p p o r t i n g h y p e r p l a n e of the e p i g r a p h of the flmction 13, which pa.sses t h r o u g h (h, fl(h)) and let c E H 1(S 1, R ) be the slope of l. Let Mh = T S 1 x {0} V1s u p p M r By the strict convexity of/3 ( P r o p o s i t i o n 20.4) Mh is well defined (i.e. it is independent of the choice of l), since in this case M r = .A/[h. T h e set Mh is a closed invariant set for the time-1 Poincar6 m a p fL : T S 1 --~ T S i associated to the Euler-Lagrange flow. These m a p s inchide "twist inappings" defined ill w as a p a r t i c u l a r case. C o r o l l a r y 2 0 . 1 0 . The projection 7rl of Mh (C T S 1) on S 1 is injective mid the inverse rr~ 1 : 7rl(Mh) --+ Mh C T S 1 is Lipsdlitz. Corollary 20.10 includes T h o r e m 14.1, now r e - o b t a i n e d as an i m m e d i a t e consequence of Theoreln 20.9. Let ~r : R 2 ~ S 1 x R = T S 1 be the s t a n d a r d projection and let ~/h = ~ - l ( M h ) C R 2. Let zrl : R 2 ~ R be the projection on the first factor a n d let ))~ denote a lift of fL to the universal cover R 2. Since, by Corollary 20.10, the p r o j e c t i o n 7rl : iV/t, --~ R is injective, f4r/, inherits an order s t r u c t u r e from t h a t on R. C o r o l l a r y 2 0 . 1 1 . ]L : Ml, -~ -~/h is order preserving. Consequently, it'h is irrational, Mh supports a unique invarimlt m e a s u r e /th, which is the mlique minimM measure of rotation l m l n h e r h. P r o o f . T h e order preserving p r o p e r t y follows i m m e d i a t e l y from the injectivity of the p r o j e c t i o n of s u p p M r on S 1 x S 1, which is the content of T h e o r e m 20.9 in the case M = 8 1 . T h e unique ergodicity of the closed inw~,ria.nt set Mh is a s t a n d a r d consequence of the order preserving property. T h e proof is the same as in the case of an order preserving homemorl)hisnl of the circle of irrational r o t a t i o n number. Fina.lly, since all m i n i m a l nleasures of r o t a t i o n n u m b e r It are s u p p o r t e d in Mh, it follows t h a t there is a unique measure #h haxing such properties. []
182
A different application of Theorem 20.9, to small perturbations of symplectic diffeomorphisms having an invariant torus, can be found in [Mal2, w There, it is exploited the remark that we can use the Lipschitz property asserted by Theorem 20.9 (and the a priori bound on the Lipschitz constant which can be obtained through it) to localize the invariant set supp Ad ~. The result can be stated as follows. Let f be a symplectic diffeomorphism of the symplectic manifold (N, w), i.e. f*oa = w, where N is a. 2n-dimensional manifold and w is a closed non-degenerate 2-form on N. A K.A.M invariant torus of f is an n-dimensional submanifold T of N such that f ( T ) = T and f i t is smoothly conjugate to ,an irrational translation on the n-torus T" by a vector p = (pl,-.., p,) E R " which satisfies a Diophantine condition, i.e. there exist eontants C, fl > 0 such that (20.21)
Ik0 + k~p, + ... + k , p , I > C(Ik~ I + . . . + I<,l) -~ ,
for all k = (kl,..., k,) E Z" \ {0}. P r o p o s i t i o n 20.12. There exists co E H I ( T ", R) = R " (which is the derivative at p of the function fl associated to the Lagrangian system obtaJned by interpolating f in a neighborhood ot" the invariant K.A.M. torus T), such that the tbllowing holds. It'c is close enough2 to co in H~(T ", R) = R " and g is sutt~ciently close to f in the C 1 topology on Hamiltonian perturbations of f, then there exists a g-invariaat set T~ associated to c (which is suppA4 ~ for the associated Lagrangian system). Furthermore, T~ is a Lipschitz graph over T and it converges in the Hausdorff topology to T as c tends to co in R n and g tends to f in the C 1 topology. Finally, i f g has a K.A.M. torus sut~ciently Cl-close to T, then that torus is one of the sets
We will now turn to the existence results of ergodic minimal measures for generic Lagrangian systems obtained by Marl4 [Mfi] following the approach sketched above. In Ma.fi4's results a central role is played by the following class of minimal measures: A measure # E C will be said a uniquely minimal measure of a Lagrangian L if M e ( L ) = {#}, for some c 6 H 1(M, R). As a consequence of Proposition 20.8 uniquely minimal measures are uniquely ergodic (i.e. if # is a uniquely minimal measure then u = # for every probability measure # such that supp ( , ) C supp (#)). In particular, t', is ergodic. T h e o r e m 20.13. [Mfi, Th.B] Let L be a Lagrea~gian satist}'ing the hypotheses of positive definiteness, superlinear gz'owth (Mong ~qbers) and completeness of the Euler-Lagrange flow. Then: a) For every c E H I ( M , R ) there exists a residual subset A(c) C C ~ 1 7 6 x S 1) such that r 6 A(c) implies cardAd~(L + 'r = 1. b) There exist residuM subsets A C C~176 M x S 1) and 7-/C H i ( M , R ) such that 6 , 4 and c 6 ~ imply card.AdC(L + r ) = 1. The core of Theorem 20.13 is a), while b) is essentially a corollary of a) via standard arguments on the residuality of points of continuity of upper semicontinuous functions. An interesting problem posed by Theorem 20.13 is whether one can replace "residuality" by "fifll measure". A result in a certain sense "dual" to Theorem 20.13 is the following:
183
T h e o r e m 2 0 . 1 3 ' . [Mfi, Th.C] Let L be a Lagrang'iaaa sa.tistjying the same hypotheses as in Theorem 20.13. Then:
a) For every h E H i ( M , R ) there exists a residual subset A(h) C C~176 x S ~) such that r E A(h) implies cardh4h(L + r = 1. b) There exist residual subset, s A C C ~ ( M x S ~) emd 7"l C H i ( M , R ) such that r E A and h E 1"t imply caxd,a, th(L + r = 1. Again, one may ask whether "residual" can be replaced by "full measure" in Theorem 20.13' or whether the set 7"/ in part b) of the theorem contains those h for which A,Q,(L + r contains an ergodic measure. The last question can be reformulated as follows: are the nlinimai ergodic measures determined by their homology? The proofs of Theorems 20.13 and 20.13' are contained in [Mfi], together with other results in the salne spirit. We would also like to mention recent results obtained by the first author [Ma13] in the direction of extending to more degrees of freedom the vm'iational construction of orbits sketched in the second part of w in the case of twist maps of tile annulus. The main step is a definition of the appropriate vatiationai principle and Peierls's batTiers for Lagrangian systems, which generalize (at least partially) the corresponding concepts for twist nlaps. These generalizations are then applied to the construction of wandering orbits. However, the results obtained so fat- [Ma13] fall far short of what one would expect in more degrees of freedom.
184
References
[Anl] S.Angenent: Monotone recmrence relations, their Birkhoff orbits, and their topological entropy, Ergodic Th. Dynam. Sy~s. (to appear) [An2] S.Angenent: A remark on the topological entropy and invariant circles of an area preserving twist map, in Twist mappings and their applications, R.McGehee and K.R. Meyer editors, New York, Springer-Verlag, 1992.
[Au] S.Aubry: The twist map, the extended Frenkel-Kontorova. model and the devil's staircase, Physics 7D (1983), 2~0-258. [Au-LeD] S.Aubry-P.Y.LeDaeron: The discrete Frenkel-Kontorova model and its extensions I: exact results for the ground states, Phy~ica 8D (1983), 381-~22. [B-M] J.Ball-V.Mizeh One dimensional variational problems whose minimizers do not satisfy the Euler-Lagrange equation. Arch. Ration. Mech. Anal. 90 (1988), ,Y25-388. [Ba] V.Bangert: Mather sets for twist maps and geodesics ported 1 (1988), 1-~5.
on
tori, Dynamics Re-
[Bil] G.D. Birkhoff: Surface transformations and their dynamical applications, Acta Math. 43 (1922), 1-119. Reprinted in Collected Mathematical papers, American Math. Sot., New York, 1950, Vol. II, 111-229. [Bi2] G.D. Birkhoff: On the periodic motion of dynamical systems, Acts Math. 50 (1927), 359-379. Rel)rinted in Collected Mathematical papers, American Math. Sot., New York, 1950, Vol. II, 333-353. [Bi3] G.D. Birkhoff: Sur quelques combes ferm~.es remarquables., Bull. Soc. Math. de France 60 (1932), 1-26. Reprinted in Collected Mathematical papers, American Math. Soc., New York, 1950, Vol. II, 418-443. [B1] S.Bullet: Invariant circles for the piece-wise linear standard map, Comm. Math. Phys. 107 (1986), 2~1-262. [De] A.Denjoy: Sur les courbes d6finies par les 6quations differentielles 5. la surfime du tore, J. Math. P',.re.~ Appl. 11 (1932), 333-375. [F] G.Forni: Construction of inva'riant measures and destruction of invariant curves for twist maps of the annulus, Ph. D. Thesis, Princeton University, October 1993. [Gr] J.M.Greene: A method for determining stochastic transition, J. Math. Phys. 20 (1979), 1183-1201.
[Hd] G.A. Hedhmd: Geodesics on a two-dimensional Rienlannian manifold with periodic coetIicients, Ann. of M,,th. 33 (1932), 719-739.
185
[He] M.R. Herman: Sur les courbes invariantes par les difffiomorphismes de l'anneau, Vol. I & I I , Asterisque 103-104 (1983) ~'~ 144 (1986). [La] V.F.Lazutkin: The existence of caustics for a billiard problem in a convex domain, Math. USSR Izvestija 7 (1973), 185-214. Translation of Izvestija, Mathematical series, Academy of Sciences of the USSR, 37, 1973.
[L-L] A.J. Lichtenberg-M.A. Liebermann: Regular and Ch.aotic Dynamics, SpringerVerlag, New-York 1983 (Second Edition 1992) [MK-P] R.S.MacKay-I.C.Percival: Converse KAM: Theory and Practice, Comm. Math. Phys. 98 (1985), 469-512. [Mfi] R. Marl& Properties and Problems of Minimizing Measures of Lagrangian Systems, prei)rint , 1993.
[Mal] J.N. Mathe,': Existence of quasi-periodic orbits h)r twist homeomorphism of the ammlus, Topology 21 (1982), 457-467. [Ma2] J.N. Mather: Glancing billiards, Ergod. Th,. Dynam. Sys. 2 (1982), 397-403. [Ma3] J.N. Mather: letter to R.S. MacKay, February 1984. [Ma4] J.N. Mather: Non-existence of invm'iant circles, Ergod. Th. Dynam. Sys. 4
(1984), 301-309.
[Ma5] J.N. Mather: Non-uniqueness of solutions of Percival's Euler-Lagrange equations, Commun. Math. Phys. 86(1983), 465-473. [Ma6] J.N. Mather: More Denjoy invariant sets for area preserving diffeomorphisms, Comment. Math. Helv. 60 (1985), 508-557. [Ma7] J.N. Mather: A criterion fi:n the non existence of inwl.ria.nt circles, Publ. Math. LII.E.S. 63 (1985), 153-204. [MaS] J.N. Mather: Modulus of continuity for Peierls's barrier , Periodic Solutions o] Itamiltonian systems and Related topics, ed. P.H. Rabinowitz et al. NATO ASI Series C 209. D. Reidel, Dordrecht (1987), 177-202. [Ma9] J.N. Mather: Destruction of inva.riant circles, Ergod. Tit. Dynam. Sys. 8
(198S), 199-214. [Mal0] J.N. Mather: Minimal measures, Comment. Math. Helv. 64 (1989), 375-
394. [Mall] J.N. Mather: Varia.tional construction of orbits for twist diffeomorphisms, J. Amer. Math. Soc. 4 (1991), no. 2, 203-257.
186
[Mal2] J.N. Ma.ther: Action minimizing invariant measures fi)r positive definite Lagrangian systems, Mah. Z. 207 (1991), 169-207. [Ma13] J.N. Mather: Variatiolml construction of orbits of twist diffeomorphisms II, to Bernard Malgrange on his 65th Birthday, preprint (to appear in the Proceedings of the Malgrange Fest). [Mol] J.Moser: Stable and Random motions in Dynamical Systems, Princeton Univ. Press, Princeton, 1973. [Mo2] J.Moser: Monotone twist mappings and the calculus of variations, Ergod. Th. Dynam. Sys. 6 (1986), 401-413. [P-deM] 3. Palis-W.de Melo: Geometric Theory of Dynamical Systems: An Introduction, Springer-Verlag, New York-Heidelberg-Berlin, 1982. [P-M] R.Perez-Marco: Solution complSte au problSme de Siegel de lindarization d'une application holomorphe au voisinage d'un point fixe (d'aprSs J.-C. Yoccoz), S~minaire Bourbaki, ~i~rn, e anv,~e, 753 (1991-92), 273-,']09. [Pel] I.C. Percival: A variational principle for invariant tori of fixed frequency, J. Ph,ys. A: Math.. and Gen. 12 (1979), No. 3, L.57. [Pe2] I.C. Percival: Variational principles for invariant tori and cantori, in Syrup. on Nonlinear Dynamics and Beam-Beam Interactions, (Edited by M. Month and J.C. Herrara), No. 57 (1980), 310-320. [Po] H.Poinca.r& Oeuvres, Vol. I., Gauthier-Villars, Paris, 1928-1956. [Re] R.T.Roekafella.r: Conve'~ Analysis, Princeton Math. Set., vol. 28, Princeton University Press, Princeton, 1970. [Rs] H.Rfissmann: On the h'equencies of quasi-periodic solutions of nearly integrable Hamiltonian systems, Preprint, Euler International Mathematical Institute, St. Petersburg, Dynamical Systems, 14-27 October 1991. [S-Z] D.Salamon-E.Zehnder: KAM theory in configuration space, Comm. Math. Helvetici 64 (1989), 83-132. [Sw] S.Schwartzman: Asymptotic cycles, Ann. Math. II Set., 66 (1957), 6270-28~. [Yo] J.-C.Yoccoz: Conjugaison Diff~rentiable des Diff6onmrphismes du Cercle dont le Nombre de Rotation V6rifie une Condition Diophantienne, Ann. Scient. Ec. Norm. Sup., 17 (198~), 333-359.
188 LIST OF C.I.M.E. SEMINARS
1954 -
Publisher
i. Analisi funzionale
C.I.M.E.
2. Quadratura delle superficie e questioni connesse 3. Equazioni differenziali non lineari
1955 -
4. Teorema di Riemann-Roch e questioni connesse 5. Teoria dei numeri 6. Topologia 7. Teorie non linearizzate in elasticitY,
idrodinamica,aerodinamica
8. Geometria proiettivo-differenziale
1956 -
9. Equazioni alle derivate parziali a caratteristiche reali i0. Propagazione delle onde elettromagnetiche ii. Teoria della funzioni di pid variabili complesse e delle funzioni automorfe
1957 - 12. Geometria aritmetica e algebrica
(2 vol.)
13. Integrali singolari e questioni connesse 14. Teoria della turbolenza
(2 vol.)
1958 - 15. Vedute e problemi attuali in relativit~ generale 16. Problemi di geometria differenziale
in grande
17. Ii principio di minimo e le sue applicazioni alle equazioni funzionali
1959 - 18. Induzione e statistica 19. Teoria algebrica dei meccanismi automatici
(2 vol.)
20. Gruppi, anelli di Lie e teoria della coomologia
1960 - 21. Sistemi dinamici e teoremi ergodici 22. Forme differenziali e loro integrali
1961 - 23. Geometria del calcolo delle variazioni 24. Teoria delle distribuzioni 25. Onde superficiali
1962 - 26. Topologia differenziale 27. Autovalori e autosoluzioni 28, Magnetofluidodinamica
(2 vol.)
189 1963 - 29. Equazioni differenziali astratte 30. Funzioni e variet~ complesse 31. Propriet~ di media e teoremi di confronto in Fisica Matematica
1964 - 32. Relativit~ generale 33. Dinamica dei gas rarefatti 34. Alcune questioni di analisi numerica 35. Equazioni differenziali non lineari
1965 - 36. Non-linear continuum theories 37. Some aspects of ring theory 38. Mathematical optimization in economics
1966 - 39. Calculus of variations
Ed. Cremonese, Firenze
40. Economia matematica
"
41. Classi caratteristiche e questioni connesse 42. Some aspects of diffusion theory
"
1967 - 43. Modern questions of celestial mechanics 44. Numerical analysis of partial differential equations 45. Geometry of homogeneous bounded domains
1968 - 46. Controllability and observability 47. Pseudo-differential
operators
48. Aspects of mathematical
logic
1969 - 49. Potential theory 50. Non-linear continuum theories in mechanics and physics and their applications 51. Questions of algebraic varieties
1970 - 52. Relativistic fluid dynamics 53. Theory of group representations
and Fourier analysis
54. Functional equations and inequalities 55. Problems in non-linear analysis
1971 - 56. Stereodynamics 57. Constructive aspects of functional analysis 58. Categories and commutative algebra
(2 vol.)
190 1972 - 59. Non-linear mechanics 60. Finite geometric structures and their applications 61. Geometric measure theory and minimal surfaces
1973 - 62. Complex analysis 63. New variational techniques in mathematical physics 64. Spectral analysis
1974 - 65. Stability problems 66. Singularities of analytic spaces 67. Eigenvalues of non linear problems
1975 - 68. Theoretical computer sciences 69. Model theory and applications 70. Differential operators and manifolds
1976
-
Ed Liguori, Napoli
71. Statistical Mechanics 72. Hyperbolicity 73. Differential
tt
topology
1977 - 74. Materials with memory 75. Pseudodifferential
operators with applications
76. Algebraic surfaces
1978 - 77, Stochastic differential equations 78. Dynamical systems
" Ed Liguori, Napoli and Birh~user Verlag
1979 - 79. Recursion theory and computational complexity 80. Mathematics of biology
1980 - 81. Wave propagation 82. Harmonic analysis and group representations 83. Matroid theory and its applications
1981 - 84. Kinetic Theories and the Boltzmann Equation
(LNM 1048) Springer-Verlag
85, Algebraic Threefolds
(LNM
947)
86, Nonlinear Filtering and Stochastic Control
(LNM
972)
(LNM
996)
1982 - 87. Invariant Theory 88. Thermodynamics 89. Fluid Dynamics
and Constitutive Equations
(LN Physics 228) (LNM I047)
"
"
191 1983 - 90. Complete Intersections
(LNM 1092) Springer-Verlag
91. Bifurcation Theory and Applications
(LNM 1057)
92. Numerical Methods in Fluid Dynamics
(LNM 1127)
1984 - 93. Harmonic Mappings and Minimal Immersions
(LNM 1159)
"
95. Buildings and the Geometry of Diagrams
(LNM 1181)
"
(LNM 1206)
"
97. Some Problems in Nonlinear Diffusion
(LNM 1224)
"
98. Theory of Moduli
(LNM 1337)
1985 - 96. Probability and Analysis
1986 -
(LNM 1161)
94. Schr6dinger Operators
(LNM 1225)
99. Inverse Problems I00. Mathematical
(LNM 1330)
Economics
(LNM 1403)
i01. Combinatorial Optimization
1987 - 102. Relativistic Fluid Dynamics 103. Topics in Calculus of Variations
1988 - 104. Logic and Computer Science 105. Global Geometry and Mathematical
Physics
1989 - 106. Methods of nonconvex analysis 107. Microlocal Analysis and Applications
1990 - 108. Geoemtric Topology: Recent Developments 109. H
Control Theory
ii0. Mathematical Modelling of Industrical
(LNM 1385)
"
(LNM 1365)
"
(LNM 1429)
"
(LNM 1451)
"
(LNM 1446)
"
(5NM 1495)
"
(LNM 1504) (LNM 1496) (LNM 1521)
"
Processes
1991 - iii. Topological Methods for Ordinary
(LNM 1537)
Differential Equations 112. Arithmetic Algebraic Geometry
(LNM 1553)
"
113. Transition to Chaos in Classical and
(LNM 1588)
"
(LNM 1563)
"
(LNM 1565)
"
(LNM 1551)
"
Quantum Mechanics
1992 - 114. Dirichlet Forms 115. D-Modules,
Representation Theory,
and Quantum Groups 116. Nonequilibrium Problems in Many-Particle Systems
192 1993 - 117. Integrable Systems and Quantum Groups 118. Algebraic Cycles and Hodge Theories 119. Phase Transitions and Hysteresis
1994 - 120. Recent Mathematical Methods in
to appear to appear (LNM 1584)
to appear
Nonlinear Wave Propagation 121. Dynamical Systems
to appear
122. Transcendental Methods in Algebraic
to appear
Geometry
Springer-Verlag
