Ergodic Problems of Classical Mechanics (The Mathematical physics monograph series)

ERGODIC PROBLEMS OF CLASSICAL MECHANICS THE MATHEMATICAL PHYSICS MONOGRAPH SERIES A. S. Wightman, EDITOR Princeton U...

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ERGODIC PROBLEMS OF

CLASSICAL MECHANICS

THE MATHEMATICAL PHYSICS MONOGRAPH SERIES A. S. Wightman, EDITOR

Princeton University

Ralph Abraham, Princeton University FOUNDATIONS OF MECHANICS

Vladimir I. Arnold, University of Moscow Andre Avez, University of Paris ERGODIC PROBLEMS OF CLASSICAL MECHANICS

Freeman .J. Dyson, The Institute for Advanced Study SYMMETRY GROUPS IN NUCLEAR AND PARTICLE PHYSICS

Robert Hermann, Argonne National Laboratory LIE GROUPS FOR PHYSICISTS

Rudolph C. Hwa, State University of New York at Stony Brook Vigdor L. Teplitz, Massachusetts Institute of Technology HOMOLOGY AND FEYNMAN INTEGRALS

.John R. Klauder and E. C. G. Sudarsban, Syracuse University FUNDAMENTALS OF QUANTUM OPTICS

Andre Lichnerowicz, College de France RELATIVISTIC HYDRODYNAMICS AND MAGNETOHYDRODYNAMICS

George W. Mackey, Harvard University THE MA THEMA TICAL FOUNDATIONS OF QUANTUM MECHANICS

Roger G. Newton, Indiana University THE COMPLEX j-PLANE

R. F. Streater, Imperial College of Science and Technology A. S. Wightman, Princeton University PCT, SPIN AND STATISTICS, AND ALL THAT

ERGODIC PROBLEMS OF

CLASSICAL MECHANICS

V. I. ARNOLD University of Moscow and

A. AVEZ University of Paris

o

ERGODIC PROBLEMS OF CLASSICAL MECHANICS

Library of Congress Catalog Card Number 68-19936 Manufactured in the United States of America

The manuscript was put into production on October 10,1967; this volume was published onJuly 5,1968

PREFACE

The fundamental problem of mechanics is computing, or studying qualitatively, the evolution of a dynamical system with prescribed initial data. Numerical methods allow one to compute the orbits for a finite time interval, but they fail as the time increases indefinitely. The three-body problem offers a typical example: Do there exist arbitrarily small perturbations of the initial data for which one of the bodieii moves to infinity? Mathematically speaking, the problem is the study of the orbits of a vector field on phase-space. Far from being solved, such a problem involves areas as various as probability and topology, number theory and differential geometry. Mr. Nicholas Bourbaki may forgive us for mixing so many structures. Maxwell, Boltzmann, Gibbs, and Poincare first proposed a statistical study of complex dynamical systems, which is now known as ergodic theory. [Ergodic theory was conceived for mechanics but applies to various other branches, such as number theory. For example, how are the first digits 1, 2, 4, 8, 1, 3, 6, ... of the powers 2" distributed? (See Appendix 12.)] But the mathematical definitions and the first important theorems are due to 1. von Neumann, G. D. Birkhoff, E. Hopf, and P. R. Halmos, and they appeared only in the thirties. During the past decade, a new step was taken, inspired by Shannon's information theory. The main result, due to Kolmogorov, Rohlin, Sinai, and Anosov, consists in a deep study of a strongly stochastic class of dynamical systems. This class is wide enough to include all the sufficicntly unstable classical systems. Among these systems figure the geodesic flows of space with negativc t"tfrvature, as studied by Hadamard, Morse, Hcdlund, E. Hopf, Gelfand, Fomin. On the other hand, Sinai proved that the Boltz-

vi

PREFACE

mann-Gibbs model, that is, a system of hard spheres with elastic collisions, belongs also to this class; this proves the "ergodic conjecture." This book is by no means a complete treatise on ergodic theory, and references are not exhaustive. The text presented here is based on lectures delivered during the spring and fall of 1965 by one of the authors, who also wrote Chapter 4. The second author is responsible for the proofs of Chapters 1,2, and 3. We thank Professors Y. Choquet-Bruhat, H. Cabannes and P. Germain, J. Kovalewsky, G. Reeb, L. Schwartz, R. Thorn, and M. Zerner, who welcomed the lecturer at their seminar. We also thank Professor S. Mandelbrojt, who suggested that we write this book. The final manuscript was read by Y. Sinai, who made a number of useful improvements for which we are sincerely grateful. The translator (A Avez) wishes to thank warmly Professors V. 1. Arnold, S. Deser, and A S. Wightman, who prevented him from many mistakes. V. I. ARNOLD A AVEz

CONTENTS

v

Preface

Chapter 1.

Chapter 2.

Dynamical Systems 1. Classical Systems 2. Abstract Dynamical Systems 3. Computations of Mean Values 4. Problems of Classification. Iwmorphism of Abstract Dynamical Systems 5. Problems of Generic Cases General References for Chapter 1

Ergodic Properties 6. Time Mean and Space Mean 7. Ergodicity 8. Mixing 9. Spectral InvaI:.iants 10. Lebesgue Spectrum 11. K -Systems 12. Entropy General References for Chapter 2

vii

1 1

7 9 11 12

13

15 15 16 19 22

28 32 35 51

CONTENTS

viii Chapter 3.

Chapter 4.

Appendixes.

Unstable Systems 13. C-Systems 14. Geodesic Flows on Compact Riemannian Manifolds with Negative Curvature 15. The Two Foliations of a C-System 16. Structural Stability of C-Systems 17. Ergodic Properties of C-Systems 18. Boltzmann-Gibbs Conjecture General References for Chapter 3

Stable Systems 19. The Swing and the Corresponding Canonical Mapping 20. Fixed Points and Periodic Motions 21. Invariant Tori and Quasi-Periodic Motions 22. Perturbation Theory 23. Topological Instability and the Whiskered Tori General References for Chapter 4

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.

The Jacobi Theorem Geodesic Flow of the Torus The Euler-Poinsot Motion Geodesic Flows of Lie Groups The Pendulum Measure Space Isomorphism of the Ilaker's Transformation and B (1/2, 112) Lack of Coincidence Everywhere of Space Mean and Time Mean The Theorem of Equipartition Modulo Some Applications of Ergodic Theory to Differential Geometry Ergodic Tran"slations of Tori The Time Mean of Sojourn The Mean Motion of the Perihelion Example of a Mixing Endomorphism Skew-Products Discrete Spectrum of Classical Systems Spectra of K-Systems

53 53 60 62 64 70 76 79

81 81 86 93 100 109 114

-tI5 117 119 120 121 123 125 127 129 131 132 134 138 143 145 147 153

CONTENTS

18. Conditional Entropy of a Partition a with Respect to a Partition f3 19. Entropy of an Automorphism 20. Examples of Riemannian Manifolds with 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34.

Negative Curvature Proof of the Lobatchewsky-Hadamard Theorem Proof of the Sinai Theorem Smale Construction of C-Diffeomorphisms Smale's Example Proof of the Lemmas of the Anosov Theorem Iritegrable Systems Symplectic Linear Mappings of Plane Stability of the Fixed Points Parametric Resonances The Averaging Method for Periodic Systems Surfaces of Section The Generating Functions of Canonical Mappings Global Canonical Mappings Proof of the Theorem on the Conservation of Invariant Tori under Small Perturbations of the Canonical Mapping

ix

158 163 168 178 191 194 196 201 210 215 219 221 227 230 235 243 249

Bibliography

271

Index

283

CHAPTER 1

DYNAMICAL SYSTEMS This chapter contains examples of dynamical systems and related problems.

§l. Classical Systems DEFINITION

1.1

Let M be a smooth manifold, /l. a measure on M defined by a continuous positive density, 1>t: M .... M a one-parameter group of measure-preserving diffeomorphisms. The collection (M, /l., 1>t) is called a classical dynamical system. The parameter t is a real number or an integer. If t ( R, the group

1>t

is usually defined in local coordinates by:

xi = fi(x 1 , ... , JIl), If t ( Z,

1>t

feomorphism

1>

i

= 1, ... ,

n

= dimM.

is the discrete group generated by a measure-preserving dif-

1>

=

1> 1-

Then the system is merely denoted by (M, /l.,

1»

and

is called the automorphism.

EXAMPLE

1.2. QUASI-PERIODIC MOTION

Let M be the torus I (x, y) mod

11-

The measure is dx dy, the group ¢ t

is a translation group:

x=l,

y=a

where a ( R, and dot denotes d/dt. Assume a p, q

(Z,

q

=

p/q rational:

>0

and p and q relatively prime. In the covering plane (x, y), the orbit with

1

2


initial data x(O) = x o' y(O) = Yo has the form: Y = Yo +

~(x-xo) q

•

As x = Xo + q, y takes the value Yo + p and the corresponding point on M coincides with the initial point (x o' Yo)' Thus, the torus is covered by

closed orbits. If a is irrational, each orbit is everywhere dense (Jacobi, 1835; see Appendix 1). More generally, let Tn = !(x 1, ... , xn) mod 11 be the n-dimensional torus with the usual measure dx 1 .•. dx n, and ¢ t the

y

I I

I

I

-r-------I I

I I

-,--I

I

I

I -T--------

I I I

I I I

I I I

I

I

I

I

I I I

I I

I I

I I I I

I

o

x

Figure 1.3

one-parameter group of translations defined by: Xi=W i ;

;=l, ... ,n; ,wlR n

Every orbit of ¢t is everywhere dense if, and' only if, k ( Z" and w· k = 0 imply k = 0 •

3

DYNAMICM, SYSTEMS

EXAMPLE

1.4. GEODESIC FLOWS

Let V be a compact Riemannian manifold; M

=.

tary tangent bundle. Given a unit tangent vector ~

Tl V denotes its unif

Tl Vx to V at x,

there is one, and only one, geodesic y passing through x with initial velocity vector ~. We denote by y (~, s) the point of y obtained from x in time s when moving along y with velocity 1. The unit tangent vector to y at y(~, s) is

(1.5) Formula (1.5) defines a one-parameter group of diffeornorphisms of M Tl V. DEFINITION 1.6

The group G t is called the geodesic flow of V. It can be proved that G t preserves the measure Il induced on M by the Riemannian metric of V

(Liouville's theorem).

SOME MORE EXAMPLES 1.7

. Appendix 2 describes the geodesic flow of the usual torus immersed in the Euclidean space £3. For the ellipsoid see Kagan [1], and for Lie groups with a left-invariant metric see Appendixes 3 and 4. One more word, in mechanics geodesic flow is called "movement of a material point on a frictionless surface without external forces. "

Other mechanical systems

involve more general flows. EXAMPLE

1.8. HAMILTONIAN FLOWS

Let PI"'" Pn; ql"'" qn (in short: p, q) be a coordinate system in

R2n , and (1.9)

H(p, q) a smooth function.

The equations

dq

aH

dp

dt

ap

dt

define a one-parameter group of diffeomorphisms of R2n. This group is called a Hamiltonian flow on R2n.

4


LIOUVILLE'S THEOREM

1.10

The Hamiltonian flow preserves the measure dP l

...

dP n . dq1 ••• dqn

Proof:

The divergence of the vector field (1. 9) vanishes:

...P-. aq

~

(aH ) + (_ aH) - 0 ap, ap . aq -

THEORil!M OF CONSERVATION OF ENERGY

(Q. E. D.)

1.11

The function H is a first integral of (1.9). Proof:

dH dt

=

aH • aH + aH • (_ aH) = 0 aq ap ap aq

Lee us denote a subset H(p, q)

=

(Q. E. D.)

h ( R by M. For almost every h,

M is a manifold. This manifold is invariant under the flow. COROLLARY

1.12

There exists an invariant measure on the manifold M. Proof:

The invariant measure on M is defined by: dp. =

da

,

I grad HII

I

=

length,

where a is the volume element of M induced by the metric of R2n. If(1.9) has several first integrals, namely 11 ,12 "", Ik , then the equations (1.9) determine a classical dynamical system on each (2n ..., k)-dimensional manifold: II

=

EXAMPLE

hI'"'' Ik = hkJ where the h's are constants.

1.13. LINEAR OSCILLATIONS IN DIMENSION 2

The Hamiltonian is:

Equation (1. 9) has two first integrals: II =

p/ + q/,

The corresponding manifolds II

=

12 =

pi + qi

hI' 12 : h2 are two-dimensional tori.

5

DYNAMICAL SYSTEMS

The dynamical systems that are induced on these tori are isomorphic to those of Example (1.2). Appendix 5 provides further examples. REMARK 1.14. GLOBAL HAMILTONIAN FLOWS More generally one may consider a symplectic

1

2n-dimensional mani-

fold M2n instead of H2n , and a closed one-form w 1 (= dH) instead of H. Equation (1. 9) becomes

x= where /: T* Mx

for any ~

t

f

f

~

/w 1 '

x

f

M2n

TM x is defined by

TM x' Let us now give some examples of the discrete case:

Z.

EXAMPLE 1.15. TRANSLATIONS OF THE TORUS Let M be the torus

!(x, y) mod 1\

with the usual measure dx dy. The

automorphism ¢ is ¢(x,y) = (x+wl' y+( 2 )(modl),

w1

f

H.

Each orbit of ¢ is everywhere dense if, and only if, k· w ( Z, for k imply k

=

f

i';

0 (see Appendix 1).

EXAMPLE 1.16. AUTOMORPHISMS OF THE TORUS Again M =

! (x, yY mod 1\

and dll = dx dy. The automorphism ¢ is de-

fined by:

¢

(x, y) = (X+ y, x+ 2y) (mod 1) .

The mapping ¢ induces a linear mapping in the covering plane (x, y)

- (1 1)

¢ As Det ¢

1

=

1 2

.

1, ¢ is measure-preserving. A set A is transformed under

A symplectic manifold M 2n is a smooth manifold, together with a global

closed two-form

0

of rank n. Example;

o

=

dpl\dq

on R 2n .

6


¢. and then

¢2

as pictured in Figure (1.17). The linear mapping ¢ has

two real proper values A1 and A2 :

0

< A2 < 1 < Al •

r

)I

Figure 1.17

7

DYNAMICAL SYSTEMS

¢nA

Then, for n large enough,

looks like a very long and very narrow rib-

bon of the plane. On M, this ribbon lies approximately in the neighborhood of an orbit of the system:

x= According to Jacobi's

1,

the~rem

y=,\-l:

(Example 1.2), and because .\1-1 is irra-

tional, 1>n A converges to a dense helix of the torus as n ... + "".

§2. DEFINITION

Abstract O)'1lamicaJ Systems

2.1 2

An abstract dynamical system (M, Il' 1>/) is a measure-space (M, Il) equipped with a one-parameter group 1>/ of automorphisms (mod 0) of

(M, Il), 1>/ depending measurably of t.

Thus, for any measurable sets A and B, 1l(1)/A nB) is a measurable function of t, and 1l(1)/A)

= Il(A)

for any t. In the future (M, Il) will al-

ways be a nonatomic Lebesgue space, that is (M, fL) will be isomorphic modulo 0 to [0, 1] with its usual Lebesgue measure. In particular Il (M) 1.

If

1>/

is the discrete group generated by an automorphism

merely denote the system by (M, Il'

1».

1>

=

1>1'

we

In the following we shall omit the

notation "mod 0." All of the preceding examples are abstract systems: a compact Riemannian manifold M ~ith its canonical measure Il (Il(M)

= 1)

is isomorphic to [0, 1]. EXAMPLE 2.2. BERNOUILLI SCHEMES

The space M.

Let Zn

=

10, 1, ... ,n-11 be the first n nonnegative integers.

M is the Cartesian product M

=

Z nZ of a countable family of Zn 's. Thus,

the elements m of M are the bilateral infinite sequences of elements of Zn:

m (M,

2 See Al'pendix 6 for these concepts.

8


The a-algebra of the measurable sets. It is the algebra generated by set!!,

of the form

i ( Zn

i (Z,

The measure p.. Define a normalized measure p.. on

Zn by setting:

p.(O) = PO'"'' p.(n-l) = Pn - 1 , L P j = 1.

We set p. (A II) = p.I for every i,

i.

The measure of a is the product-mea-

sure, denoted again by p.: if A!I (iI'"'' i k all different) are kII '''., A!k Ik distinct generators, the measure of their intersection is the product oftheir measures, that is, p.lm I a j

iI''''' a jk

I

=

p .... p . .

ikl

II

lk

(M, p.)' is clearly a Lebesgue space. The automorphism cp. It is the shift m = (.",8 j ,,,.)

where a;= a j _

1

..

m'= (".,a;,,,.),

for every i,' cp is a bijection.

To prove cp is measlire-

preservi'ng, it is sufficient to take into account the generators:

Hence: p.[cp(Aj»)

p.[A!H)

Pj

= p.(Aj).

Notation. The above abstract dynamical system is called a Bernouilli

scheme and denoted by B(p o'"'' Pn- 1 ). Remark. Tossing a coin involves the scheme B ('12,'12) • .This fact was first pointed out by

J.

Bernouilli. The elements of M

=

Z2Z ar,e indefinite

bilateral sequences of tosses: 0 means "head," 1 means "tail." The set A~(resp. A~) represents the set of the sequence in which "head" I ~ (resp. "tail") appears at the

ith

toss. Thus, it is quite natural to set:

.

.

p.(AI> =prob (AI> =

EXAMPLE

2.3.

1

2' .

THE BAKER'S TRANSFORMATION

Let M be the torus I(x, y) mod

11

with its usual measure dxdy.

DYNAMICAL SYSTEMS

9

'J

1~

____________. -____

o

------~~

x

1

=

< f 11 > . < 1 I g >

for every f, g ( L 2 (M, Il)· Proof:

If f and g are some characteristic functions, then (9.9) reduces tothe very definition of mixing (8.2). The general case is derived easily by observing that the space of finite linear combinations of characteristic functions is dense in L 2 (M, Il). In spectral terms, (M, Il' ¢t) is mixing if it is ergodic and the spectrum of Ut (except for A = 0) is absolutely continuous with respect to the Lebesgue measure. The converse is false. We say that

Ut has properly continuous spectrum if its only proper functions are constants. It can be proved (see Halmos [1]) that a dynamical system has properly continuous spectrum if, and only if, it is weakly mixing (see 8.9). We turn to the case in which the spectrum of Ut is discrete. EXAMPLE 9.10

Let M be the circle \z I z ( C, Izl translation ¢ (z)

= (). z,

()

=

e 2TTiW ,

tion zP, p ( Z:

11, Il its usual measure, ¢ the.

= W

(

R •. Let us consider the func-

sJ>

UzP

=

(Uz)P

Hence, the zP's are proper functions of U with corresponding proper values ()P. The set \zP I p.(

ZI.

which is called the discrete spectrum of U.

forms a complete orthonormal system of L 2 (M. Il); whence the definition: DEFINITION 9.11

A dynamical system

tM. p.. ¢) has properly discrete spectrum if there

is a basis of L 2 (M. p.). each function of which is a proper function of the induced operator U.

Let us turn back to (9.10). According to Theorem (9.7), the system is ergodic if, and only if, 1 is a simple proper value,. that is, if, and only if, pw

I

Z when p i 0, which means w is irrational. In otherwords, our sys-

26


tem is ergodic if, and only if, the orbits are dense on M (see Example 7.8 and Appendix 1). Observe that ergodicity implies that all the proper values (}P are distinct and simple. The system is not mixing; take f = g = zP

in (9.8); we get:

< VnzP IzP > = If p f, 0,

(Jpn •

lim (JPn does not exist and (9.9) is not fulfilled. These results n=oo

extend immediately to.the n-dimensional torus and suggest the following theorem. THEOREM

9.12

Let (M, Jl, ¢) be an ergodic dynamical system, V the induced operator. Then: (a) the absolute value of every proper function of V is constant a. e.;

(b) every proper value is simple; (c) the set of all the proper values of V is a subgroup of the circle group

Izlz

(C,

Izl

=

1\;

(d) if (M, Jl, ¢) is mixing, the only one proper value is 1.

Proof: Since V is unitary, every proper value A has absolute value 1. It follows that if f is a corresponding proper function f(¢x) = Al(x) a.e. implies

If(¢x) I =

If(x)1 a.e.

I

Hence, f(x)1 is invariant under ¢, and ergodicity implies that If

I is con-

stant a.e. (Corollary 7.6). In particular, f f, 0 a.e. Let h be another proper funetion with proper value A. Since f f, 0 a.e., hlf makes sense. We get:

V(!!) Vh f Vf =

h

I'

and hlf is an invariant function, so that h is a constant multiple of f. This proves (b). If A and Jl are proper values of V, with corresponding proper functions f and g, we get

27

ERGODIC PROPERTIES

VCt)

=

g~

= ::

=

A~-1 ·Ct)

.

Hence II g is a proper function of V with proper value A~-1

•

This proves

(c). Finally, if the system is mixing, take I = g equal to a proper function with proper value in (9.9), we get: lim < vn/l I>

n=oo

',

that is lim

An

constant.

n=oo

Hence A = 1, and (d) is proved. These properties of the discrete spectrum have been in some sense extended to the continuous part of the spectrum by Sinai [2], [3] (see, however, the recent paper of Katok and Stepin [1] for an example of a system whose 'maximal spectral measure does not dominate its convolution). Tht: group of the proper values is obviously an invariant of the dynamical system. If the spectrum is discrete, this group forms a complete system of invariants. More precisely, we have:

DISCRETE SPECTRUM THEOREM

9.13

(VON NEUMANN, HALMOS)

(a) Two ergodic dynamical systems with discrete spectrum are isomorphic

ii, and only ii, the proper values 01 their induced operator coincide. (b) Every countable subgroup 01 the circle group is the spectrum 01 an er-

godic dynamical system with discrete spectrum. The proof will be found in Halmos [1]. It is based upon the construction of some compact abelian group (character group of the spectrum of given ergodic dynamical systems with properly discrete spectrum). Then, one proves the isomorphism of our given dynamical system with a translation of this abelian group. To emphasize this result, we point out that the isomorphism problem is solved as far as the discrete spectrum case and abstract frame are con-

28


cerned. In contrast, no characterization is known for the spectrum of a classical system. For instance, does there exist a classical system whose discrete spectrum is a prescribed subgroup of the circle group? Appendix

16 contains some information related to this question. §10. Lebesgue Spectrum Let us begin with an example. EXAMPLE

10.1

We again consider Example (1.16): M is the torus l(x, y) (mod 1)\ with its usual measure;

4)

is the automorphism:

.¢(x, y)

=

(x + y, x + 2y) (mod 1);

U is the induced operator. It is well known thaUhe set

D

=

\e

p,q

(x, y) =

e 2TTi (pX+ qy),

p, q (

Z\

is an orthonormal basis of L 2 (M, p.). The set D can be identified with the lattice Z2

= \(p,

q)! C R2. Since Uep,q

= ept q, p+2 q '

U induces an au-

tomorphism u on D:

Let us show that (0,0) is the unique finite orbit of u. Assume that (p, q) ( Z2 has a finite orbit. This orbit is a bounded subset of R2, invariant under the linear operator of R2,

1>

has two proper values A1, A2, 0 < A2 < 1 < A1• Hence, 1> is "dilating" in the proper direction corresponding to A1 , and "contracting" in the proper direction corresponding to A2 • This implies that the only invariant

(under ¢) bounded subset of R2 is (0,0). (Q. E. 0.) We conclude that Z2 -\0,0\ splits into a set I of orbits of u, and each orbit is in an obvious one-to-one correspondence with Z.

29

ERGODIC PROPERTIES

Let us go back to D

=

Ie p, q I p,

q ( Z\. D -leo , 0\ splits into orbits

of V: C l , C 2 , ... , Ci,oo.; i (1. If fi,o is some element of C i , we may write

C., where f.l,n

=

=

Ii.l,n In ( Z\ ,

V n f.1, o. To summarize, if H., is the space spanned by the

vectors of C i , then L 2 (M, /1) is the orthogonal sum of the H/s and ofthe one-dimensional space of the constant functions. Each Hi is invariant under V and has an orthonormal basis

Ii./,n In ( Z\

such that:

Situations such as this occur often enough to deserve a definition. DEFINITION

10.2

Let (M, /1, tor. (M, /1,

eM

eM

be an abstract dynamical system, V the induced opera-

has Lebesgue spectrum LI if there exists an orthonormal

basis of L 2 (M, r11) formed by the function 1 and functions f.1,J. (i (1, j ( Z) such that: Vf 1,] . .

f1,]+ . . l' for every i, j.

=

The cardinality of 1 can be easily proved to be uniquely determined and is called the multiplicity of the Lebesgue spectrum. If 1 is (countably) infinite, we shall speak .of (countably) infinite Lebesgue spectrum. If 1 has only one element, the Lebesgue spectrum is called simple. An analogous definition holds in the continuous case. Let Vt be the one-parameter group of induced operators of a dynamical flow (M, /1, q, t)'

The flow is

said to have Lebesgue spectrum LI if every Vt (t .;, 0) has Lebesgue spectrum L I. REMARK

10.3

This terminology is derived from the following fact: Let Vt

=

J

00

-00

e 2TTit A dE (A)

.'

be the spectral resolution of Vt . It can be proved that (M, /1, q,t) has

30


Lebesgue spectrum if. and only if. the measure

< E(,\)f II> is absolutely

continuous with respect to the Lebesgue measure. for every I ( L 2 (M. /L) orthogonal to 1. THEOREM 10.4

A dynamical system with

Lebes~ue

spectrum is

mixin~.

Prool: From Theorem (9.8) we need to prove that: lim n"OO

< Un I I ~ > = < I 11 >. < 1 I ~ >

for every I. ~ ( L 2 (M. p.). This is equivalent to: lim n+ oo

for every I.

~

< Un I I ~ > = 0

orthogonal to 1. It is sufficient to prove this when I and

~

are basis vectors. for the general case follows by continuity and linearity

If 1= I j • i • ~ = Ik • r • then

which is null for n large enough. COROLLARY 10.5

The automorphism if> (x. y) = (x+ y. x+ 2y)(mod 1) of the torus M =

I (x. y)(mod

1) I

(Example 1.16) has Lebesgue spectrum (Example 10.1). Then. it is mixing and ergodic (Corollary 8.4). EXAMPLE 10.6

The Bernouillischemes have countable

Lebes~ue

spectrum.

In par-

ticular they are spectrally equivalent. Prool: We prove it for B(~. ~); the same statement holds for B(Pl'· ..• Pn) UI to minor modifications. Let us recall (see Example 2.2) that M = Z2 Z the space of the infinite bilateral sequences: m = ...• m_ l • mo' m l .···; mj (

\0.11.

31

ERGODIC PROPERTIES

The function 1 and the function j -1

Yn(x) =

if x = 0 if x = 1

1+ 1

form an· orthonormal basis of the space L 2(Z2. 1l ) associated to the n-th factor of M. From the product structure of M. we get an orthonormal basis of L 2 (M. Il) which consists of the function 1 and all the finite pro~ ucts

y

nl

..... y

fIJc

of the y 's with distinct indices n 1• ..•• n k n

.

U be the induced operator of the shift

.

Now. let ..

¢. Call two elements of the above

basis equivalent if some integer power of U carries one onto the other. The function 1 constitutes its own equivalence class; the other basis functions split into countably many equivalence classes. Each such equivalent class is in a one-to-one correspondence with Z: the action of U on the class is to replace the element corresponding to n

l

Z by the element

corresponding to n+ 1. To ~ummarize. there exists an orthonormal basis of L 2 (M. Il) consisting of the function 1 and of functions I.I,]. (i = 1. 2 •... ; j l Z) such that

UI 1,] . . = l.I, J.+1 for every i. j. The number i is the number of the equivalence class. the number j is the number of its element which corresponds, as described above. to j

l

Z. Thus. B(Yl. Yl) has countable Lebesgue spectrum.

Let (M 1.1l1' ¢1) and (M 2 .1l2' ¢2) be two Bemouilli schemes. There exist. from the above. orthonormal bases

I 1, I;~j I in

1

11, I;,j I

.. 10

L 2 (M 1.1l1) and

L 2(M 2 , 1l2) such that:

I~I, }·+1

•

for every i, j. The isometry of L 2(M l' Ill) onto L 2(M 2 , 1l2) defined by

1 ... 1, carries the spectral type of the first scheme into that of the other.

32


§ll. K-Systems In this section we define a class of abstract dynamical systems with strongly stochastic properties. DEFINITION

11.1

5

An abstract dynamical system (M, IL' ¢) is caIled a K-system 6 if there exists a subalgebra ff of the algebra of the measurable sets satisfying: (j' C ¢ff.

(a)

n 00

(b)

0,

¢nff =

n=-oo

where

0 is

the algebra of the sets of measure 0 or 1, 00

V

(c)

i,

¢n (j' =

n=-oo

where ¢, by abuse of language, is the automorphism of

f

induced by ¢.

The above conditions become, in the continuous case: (a ') 00

n ¢/i

(b ')

=

0

=

1.

t=-oo 00

(c ')

V ¢/f. t~-OQ

From the very definition, the isomorphic image of a K-system is a K-system. EXAMPLE

11.2

BERNOUILLI SCHEMES (SEE

2.2)

The BernouiIli schemes are K-automorphisms. Proof: Let B(PI"'" Pn) be a Bernouilli scheme. The algebra 1 is generatec 5 See~ Appen~dix 17 for notations as C, /\, V,.... The standard notations (Rohli, are.

I =~1ll, 0 = ;ll.

6 A. N. Kolmogorov [2) introduced this class under the name of quasi-regular sys terns.

33

ERGODIC PROPERTIES

by the:

Let ct be the algebra generated by the A Ij,s, i

< O. We know that:

¢(Aj) = Aj+l where ¢ is the shift. Hence ¢ct is the algebra generated by the k

~

At's,

1, and

ct c ¢ct, proving the property (a). On the other hand, every generator ¢q(A/)

=

Aj+q , i ~ 0, for q

=

A! of

1 is a

r-i. Hence we get the property (b): 00

V ¢nct Let us now prove the property (c). Let

93

=

i.

be the subalgebra of 1, each

element of which belongs to some subalgebra generated by a finite number of AI- To every A = fl(A) " fl(B)

93

f

for any B

there corresponds an N f

¢ -nct, n

fl(A) "fl(B) holds for every B

still holds for any A

f

i

f

Z such that fl(A

~ N (exercise). Hence fl (A

=

(n;; ¢-ncf.., Since ~ 1. this relation n,; ¢ -net. Especially: =

and B (

fl (B) = fl (B

n B)

that is fl(B) = 0 or 1, for every B. (

= [fl (8)]2 ,

n;; ¢-net.

We conclude:

hence: (Q. E. D.)

COROLLARY

n B)

n B)

11.3

The Baker's Transformation is a K-system.

34


Prool: This system is isomorphic to B (V2, V2) (see Appendix 7).

1104

EXAMPLE

Chapter 3 will be devoted to a wide class of classical K-systems. This class contains the automorphisms of the tori, the geodesic flows on compact Riemannian manifolds with negative curvature, and the Boltzmann-Gibbs model of particles colliding elastically.

11.5

THEOREM

A K-system has a denumerably multiple Lebesgue spectrum. In particular, it is mixing and ergodic (Theorem lOA). This theorem is due to Kolmogorov [2] for K-automorphisms and to Sinai [6] for K-flows. We sketch a proof for K-automorphisms, the complete proof will be found in Appendix 17. H8UH

U-1 H8H

...•......

.' •••••

o. o.

0

0.00

000

••

0

000.00

:UH

H

U-1H

....................

x Uh 1

x hl

x U-1h 1

..........................

Hl)

U-1hj .................. ;" ....

Hd

x h2

000

••••

0

••

00

oX

Uhj

x hj

II

Figure 11.6

Let (f be the subalgebra of Definition (11.1). We d~note the subspace of L 2 (M, f.L) generated by the characteristic functions of the elements of (f by H. If U is the induced operator, the properties (11.1) of (f are translated as follows:

n° ~

~

unH C," C UH C H C U- 1H C," C

U n=-oo

n=-oo

= L 2 (M, f.L),

where Ho is the one-dimensional space of the constants.

UnH

35

ERGODIC PROPERTIES

Let us select an orthononnal basis

Ih.1 I

on the'orthocomplement H 9VH

of VH in H. H. is the space spanned by the sequence ... , V-lh., h., I

I

]

Vh.,.... The H's are invariant under V, and their orthogonal sum is I

I

L 2 (M, p.) 9 Ho' Hence, if we set e .. 1,J

=

Vjh., I

i(Z+,

j(Z,.

the e 1,1 . .'s and the function 1 constitute an orthononnal basis of L2(M,~) such that:

for every i, j. We conclude that V has Lebesgue spectrum. The proof

~il1

be complete after it has been shown that the dimension of H 9 VH is infinite (see Appendix 17): V has a spectrum of infinite multiplicity.

§12. Entropy This section is devoted to the definition and the study of a non spectral invariant of dynamical systems introduced by A. N. Kolmogorov [4]. Throughout, z (t) denotes the function on [0, 1] defined by: z(t) = {-tLOgt if O

acts. Hence A = 1, and from (12.38):

O.

A PROBLEM 12.41

Whether the entropy h (1)) of a classical system depends continuously on

1>

is an open question.

REMARK 12.42

Kouchnirenko's theorem is connected to recent results of M. Artin and B. Mazur [1]: Let M be a smooth compact manifold, then for a dense set of C 1 -diffeomorphisms the number N (n) of isolated fixed points of 1>n, n

=

1,2, ... , is exponentially bounded from above:

c

='

C (1)),

A

= 1..(1))

•

REMARK 12.43

Recently, Kouchnirenko 12 introduced some new nontrivial invariants . of abstract dynamical systems: A-entropies. Let A be a monotone sequence of integers

12 See his report at the lnt. Math. Congr., Moscow, 1966.

ERGODIC PROPERTIES

49

Then the A-entropy of an automorphism ¢ with respect to a partition a is defined as:

-

As in Definition (12.23), A-entropy is:

One obtains the usual entropy if A

=

10, 1,2, ... 1. The A-entropies can

distinguish some systems with usual entropy O. Let us give an example. Let A .3) is 0

=

1) ;

II¢"(II ::;

A2

'11(11

if (( Ym , (0 < A2 < 1)

This is a characteristic example of the C-systems that we define next.

y

Figure 13.2

55

UNSTABLE SYSTEMS

DEFINITION 13.3

Let ¢ be a C 2 -differentiable diffeomorphism of a compact, connected, smooth manifold M. We denote the differential of ¢ by ¢*. (M, ¢) is called a C-diffeomorphism if there exist two

fi~/ds

of tangent

planes Xm and Ym 'such that: (1) TM m falls into the direct sum of X m and Y : m TM m =Xm mY, dimX m =kIO, dimYm m (2)

=

110.

For every positiVe integer and for some Riemannian metric g:

1I(¢,n)\~'11 2 a' i,n Iltll,

II(¢-n)*tll

11(¢n)*tll

II(¢-n)* til ~ a' e.\n Iltll,

:S b· e-.\nlltll,

:S b· e-.\nlltll, if if

t ( X m,

t ( Ym

The constants a, b, .\ are positive and independent for n and b

depend 2

0

·

but a and

on the metric g. Xm is called the dilating space, and Ym is the

contracting space. Example (13.1) is a C-diffeomorphism: a = b =

1,

e

.\.

\

= "1'

e

-.\

\

= "2

This definition extends to the continuous case (t (R): Let ¢ t be a oneparameter group of C 2 -differentiable diffeomorphisms of a compact, connected, smooth manifold M.

d~

(M, ¢t) is called a C-flow if:

(0)

the velocity vector

¢tmlt=o does not vanish;

(1)

TM m splits into a direct sum: TMm=XmmYmmz~,

where Zm is the one-dimensional space spanned by the velocity vector at m, and dim X m 2

=

k .;, 0,

dim Ym

=

I ,;, 0 i

And hence for every metric: Let gl and g2 be two Riemannian metrics on M.

Due to the

compactness

of M, there exist two positive constants a,

ail til 2:S I till :S f3lltll2

t(

f3

such that:

for every TM . . Thus, if the inequalities 2 hold for ~ I with constants a and b, they still hold for ~2 with corresponding constants

(al (3) a

and metric g.

(f3/ a) b.

This proves the independence of the definition from any

56


(2) for any positive real number t and for some Riemannian metric g:

11(1)/tll ~ aoeAtlltll, 11(1)_/t!1 ~ boe-Atlltll, if t (X m ; 11(1) /tll ~ boe-Atllt!!, 11(1)_/tll ~ aoeAtlltll, if t( Ym . The constants

a, b, A are positive and independent

for t and

t,

but

a

and b depend on the metric g. Condition (D) means that the system has no equilibrium position. Condition (2) describes the behavior of the orbits. A C-diffeomorphism or a C-flow will be called, for short, a C-system REMARK

13.4

It is easy to show that: (1) the subspaces Xm and Ym are uniquely determined (they are, respec-

tively, the "most dilating" and the "most contracting" subs paces of TM m );

(2) dim X m = k and dim. Ym = I do not depend on m (k is a continuous function of m, with integer values, on the connected space

M)~

(3) Xm and. Ym· depend continuously on m. Finally, observe that a C-system is not a classical system (see Definition 1.1) since we do not postulate the existence of an invariant measure. Now let us show how to construct certain C-flows from C-diffeom:>rphisms EXAMPLE

13.5 (SMALE)

The Space M. Let T2

Iu I 0

= I(x, y) mod 1 I be the two-dimensional torus and [0, 1] =

~ u ~ 110 We construct the cylinder T2 x [0, 1], and after we identi-

fy T 2· x

101

and T2 x 11 I according to:

«x, y), 1) ;: where

1>

(1) (x, y), 0)

=

«X+ y,

x+ 2y), 0) (mod 1) ,

is the diffeomorphism (13.1):

1> : ( ; ) ->

nD(;)

(mod 1) .

57

UNSTABLE SYSTEMS

c

u

il

Figure 13.6

We obtain a compact manifold M. Let (x, y, u) be a point of M. The mapping p: M ~ 51

= ! u (mod

Hence, M is a fibre

bundle 3

1) l, p (x, y, u)

=u

has rank 1 everyvihere.

with basis 51 and fibre T2

The flow ¢/. We define a flow ¢/ by its infinitesimal generator:

x = 0,

(13.7)

y

= 0,

u=

1 .

An Auxiliary Riemannian Metric Let Al and A2 , (0

y'

positive asymptotes negative asymptotes Figure 14.4

Let y (u, t)

= y (t) = y

etrized by arc length t.

be a geodesic emanating from u ( T1 V and paramLet x be a point of V. There exists a geodesic

Y1 . issuing from x and passing through- y (t1)' As t1 .... + 00, Y1 converges to a limit which is a geodesic y '(u', t) emanating from u' ( TVx ' For a suitable choice of the origin of y, it can be proved that:

(14.5)

distance (y (t), y '(t»

5. b· e->'"

t ,

t ::::

0,

where the constants b and A are positive and independent for y, y', t.

Geodesics such as y' are known as the positive asymptotes to y. They can be proved to be orthogonal trajectories to (n -1) -dimensional submanifolds (n = dim V): the so-called positive horospheres S+. Let us denote by S+ (u) the horosphere emanating from the origin of u ( T1 V and which is orthogonal to the positive asymptotes of y (u, t). This horosphere can be interpreted as an (n -1) -dimensional submanifold of T1 V: S+(u) is the union of its. orthogonal unitary vectors oriented as u. The tangent plane at u of S+(u) C T1 V is an (n -1) -plane Yu of T( T1 V).

62


Exchanging the role of t and - t, we define negative asymptotes and negative horospheres S- in the same way. The tangent plane at u of S-(u) C Tl V is an (n -1) -plane Xu of T (T1 Vu). From the very definition,

we have:

where' Zu is the one-dimensional space generated by the velocity-vector of the geodesic flow. That is the condition (1) of C-flows (Definition 13.3). Condition (2) comes from (14.5). Observe that the fields Xu and

¥u are completely integrable. Their

integral manifolds are the horospheres S+ and S-. Both of these foliations are invariant under the geodesic flow, for the horospheres are orthogonal trajectories of (n-l) -parameter families of geodesics of V . . We turn to prove that general C-systems admit two invariant foliations.

§15. The Two Foliations of a C-System Let(M, ¢) be a C-diffeomorphfsm; Xm (resp. Ym ) denotes the k-di-

mensional dilating space at m ( M (resp. the IGdimensional contracting space). A Riemannian. metric on M is definitively selected. Hence, Xm and Ym are:Euclidean su~spaces of TMm' SINA(THEOREM 8

15.1

Let (M, ¢) be a C-diffeomorphism, then: (1) There exist two foliations

X and

~ that are invariant under¢

and that are respectively tangent to the dilating field Xm and the contracting field Ym. Hence, these fields are always integrable. (2) Every diffeomorphism ¢': M

-+

M, C 2 -cIose enough to ¢, is a

C-dilleomorphism. The dilating and contracting foliations

X, and

~' of ¢'

depend (;Ontinu6usly on ¢'. 8 This w~s proved essentially in the paper by.V. I. Arnold and Y. Sinai [6]; al~hough

their discussion was concerned with the particular case of small perturba-

tions of automorphisms of a two-dimensional torus, the proof extend to the general case.

63

UNSTABLE SYSTEMS

Appendix 22 completes the proof we sketch here. CONSTRUCTION

15.2

The. space K of the fields p of the tangent k-planes to M inherits a natural metric Ip1 - P2 1 which makes it into a complete metric space. Let

p be such a field, and p (m) the k-plane of TM m' The diffeomorphism cp induces a mapping cp**: K ~ K: cp**(m)

=

cp*p(cp- 1 m).

where cp * is the differential of cp which maps a k-plane of TM m onto a k-plane of TM cp(m) The dilating and contracting fields X and Y of cp are clearly fixed points of cp**. It can be proved (see Appendix 22) that the axioms of Csystems imply that cp ** (or a positive integer power (cp **)n) is contracting in a neighborhood of the dilating field X:

(15.3) for

IX -P 1 1 < 0, IX -P 2 1 < 0,

°

small enough. Of course, (15.3)

and

still holds for any diffeomorphism cp'. C 2 -c1ose enough to cp, since cp'* is C 1-close to cp *. We deduce from the contracting mapping theorem that a mapping verifying (15.3) admits a fixed point. The fixed point of the mapping cp is X, but for cp' it is another field p'. Clearly:

p'

(cp ,**)n X ,

lim

=

n=~

cp'*p'(m)

=

p'(cp'm) ,

and the field p' is dilating for cp'. A similar study of (M, cp -1) leads to the contracting field of cp' that is close to Y. Thus cp' is a C.diffeomorphism. THE INVARIANT FOLIATIONS

15:4

First assume that (M, cp) possesses two invariant foliations

!

and

tangent, respectively, to X and Y. Then, the same property holds for (M, cp '). In fact, the invariant field p' of cp' is obtained as:

p'

=

lim (cp ,**)n X n=oo

y,

64


But (¢ ,**)n X is the field of the i 0) of each dilating sheet

f3

£

X', close to m,

clearly have some points neighboring ¢n m (see Lemma A, Appendix 25

69

UNSTABLE SYSTEMS

for the precise meaning). It can be proved (Lemma B, Appendix 25) that there exists, among these sheets, a unique sheet (j(m) such that its images

cp '" (j(m) are still close t~ cp"(m) for all n

0 separate the others, and the number of fixed points,

is even. Let us make correspond to any point x the vector whose extremity is

Te n x (see Figure 20.7). It is readily seen that the index (Appendix 27) of this vector field at a fixed point is: Ind

=

sign of ( dA • d~ \ dl d¢)

Thus, half of the fixed points have index + 1 and the others have index -1This means that half of these points are elliptic and half of hyperbolic type (an elliptie point has index + 1, an hyperbolic point has index -1). Elliptic and hyperbolic points are illustrated in Figure 20.7. Now, consider an elliptic~ fixed point: Ten x

=

x. The orbit of x is

x, Te x, ... , Ten-lx, therE!'fore:

All points of the orbit of x are fixed points of Ten and are elliptic since they l1ave the same proper value. Hence, the set of the elliptic fixed points splits into orbits consisting of bits, then there are

£.0

~

points. Let 1c be the number of such or-

elliptic points. This gives us 2kn fixed points

(elliptic and hyperbolic), as promised in §20.5.

ZONES OF INSTABILITY 20.9

We now turn to the neighborhoods of the above elliptic and hyperbolic fixed points. According to V. Arnold [7] (see also Appendix 28), each "generic" elliptic point is surrounded by closed curves that are invariant under Ten. These curves form" islands" (see Figure 20.10). Each island repeats in miniature the whole structure, with its curves C'. islands be-

91

STABLE SYSTEMS

tween these curves, and so on. Between these islands and curves

~

, re-

main zones around the hyperbolic points. In fact,4 the separatrices of hyperbolic fixed points of the Ten,s intersecting each other create an intricate network, as depicted in Figure 20.10. On discovering this, Poincart: wrote ([2], V.3, Chap. 33, p. 389): "One is struck by the complexity of thIs figure that I am not even attempting to draw. N othmg can give us a better idea of the complexity of the three-body problem and of all problems of dynamICS where there is no holomorphlc integral and Bohlin's series diverge."

Figure 20.10

The ergodic properties of the motion in zones of instability are unknown. There probably exist systems with singular spectrum and K-systems among the ergodic components.

4 See Poincare [21. V. Melnikov [1].

92


REMARK

20.11

The above argument does not prove there exist infinitely many elliptic islands for a given €« 1. Poincare's last geometrical theoremS proves that there exist infinitely many fixed points of ~n (n

->

+ 00) with index

0.1

o -0.1

-0.4 -0.3 -0.2-0.1

0 0.1

0.2 0.3 0.4 0.5 0.6 Y

Figure 20.12 Yr-~-.-r-.--~~~-r-.--r-~-r~-.r-~

.5 .4

"-,3

..

"

...: "

-.5-.4-.3-.2-.1

,/

/

0 .1 .2 .3 .4 .5 .6 .7 .8 .9 Y Figure 20.13.

5 See Poincare [3], G. D.'Birkhoff [1].

93

STABLE SYSTEMS

+ 1 inside the annulus located between the invariant curves

~

(Theorem

19.10). Perhaps some of these points are not elliptic but hyperbolic with reflection. Numerical computations 6 seem to support this conclusion. y 0.8 0.6

0.2

o -0.2 -0.04 -0.6 -0.8 -0.8 -0.6 -0.04-0.2

0

0.2 0.04 Q6 0.8 X

Flgure 20.14

We have borrowed Figures 20.12-14 from the work of M. Henon and

C. Heiles [1]. They depict the orbits of a mapping of type ~ computed with an electronic computer. All the points, exterior to the curves, belong to one and the same orbit!

§21. Invariant Tori and Quasi-Periodic Motions

The example we considered in §19 and §20 is a particular case of a situation which occurs for each system close enough to an "integrable" system. 6

See Gelfand, Graev. Sueva. Mlchai1ova. Morosov M. Henon. C. Helles [1].

b);

[3];

Ochozimskl. Sarychev •...

94


(A) INTEGRABLE SYSYEMS 21.1

If one takes a look at the "integrable" problems of Classical Mechanics,7 one finds that, for all of these problems, bounded orbits are either periodic or quasi-periodic. In other words, the phase-space is stratified

i~to

invariant tori supporting quasi-periodic motions.

E'XAMPLE 21.2 Assume the phase-space Q is the product of a bounded domain Bn of

Rn by the torus 1"'. Let P

=

(P l '

, .. , Pn )

be coordinates on Bn and q

=

(ql' "" qn) (mod 277) coordinates on Tn. The Hamiltonian equations, with

Hamiltonian function H = Ho(p), read: (21.3) The motion is quasi-periodic on the invariant tori P = Ct, with frequencies w (p). Frequen«ies depend on the torus: if

f 0, "

then, on each neighborhood o(the torus p = Ct , there are invariant tpri un which frequencies are independent and orbits everywhere dense (see Appendix 1). There exist other tori on which frequencies are commensurate; they are exceptional, that is they form a set of measure zero. Coordinates (p, q) of B n x 1'" are called "action-angle" coordinates. For all integrable systems, it can be shown (Appendix 26) that a certain (2n - 1)-dimensional hypersurface divides the phase-space into invariant domains each of which is stratified into invariant n-dimensional manifolds. If the domain is bounded, these manifolds are tori supporting quasiperiodic motions. The action-angle coordinates can be introduced into such a domain, thus, the system can be described by (21.3).

7 For instance, the motion of a free point along a geodesic on the surface of a triaxial ellipsoid or a torus (see (1.7) and Appendix 2), a heavy solid body (Euler, Lagrange, and Kovalewskaia cases),

95

STABLE SYSTEMS

(B) SYSTEMS CLOSE TO INTEGRABLE SYSTEMS 21.4 Now, we assume that the Hamiltonian function is perturbed:

the "perturbation" HI being" small enough." The Hamiltonian equations are then:

(21.5)

oH I oq

---,

p

q

For most initial data, A. N. Kolmogorov [6] proved that the motion remains quasi-periodic (see Theorem 21. 7). Consequently, (21.5) is not ergodic on the "energy surface" H = Ct and, among the ergodic components, there are components with discrete spectrum, the complement of which has small Lebesgue measure as H I is small. Assume that the function H (p) is anal ytic in a c0cnplex domain [0] of the phase-space: O(~p,

I~ pi

'R q, (0,

< p,

I~ ql

< p) .

Assume also that the unperturbed system is nondegenerate: Det

(21.6)

(U I~ 0

=

lo2H

Det __0_

op

op2

I

Select an incommensurate 8 frequency-vector w

oJ- O.

=

w*. The equations of

the invariant torus To(w*) of the unperturbed system (21.3) are p = p*, where wo(P*) = w*. Thus, the system (21.3) has frequencies w* on

To(w*). THEOR.EM

21.7

If HI is small enough, then for almost 9 all w*, there exists an invari-

ant torus T(w*) of the perturbed system (21.5) and T(w*) is close to To (w*). To be precise: That is, (w, k) ,;, 0 for all integers k. All, except for a set of Lebesgue measure zero.

96


For all K> 0 there exist E> 0 and a mapping p = p(Q), q = q(Q) of an abstract torus T =

I Q (mod

21T)} into

n such that,

according to (21.5)

Q = w*, and:

Ip(Q)-p*1 2, the n-dimensional invariant tori do not di-

vide the (2n -1) -dimensional manifold H

=

constant and those orbits that

do not belong to the tori T (w*) can travel very far along H

=

h (see §23).

(C) ApPLICATIONS AND GENERALIZATIONS 21.8 Theorem (21. 7) applies to the motion of a free point along a geodesic on a convex surface close to an ellipsoid or '8 surface of revolution. This theorem allows one to prove the stability in the plane restricted circular

three-body problem. 10

One can also deduce the stability of the fast rota-

tions of a heavy asymmetric solid body. 11 But this theorem does not apply if the unperturbed motion has fewer frequencies than the perturbed motion (degenerate case) for, in this case,

10 A. N. Kolmogorov [7]. 11 V. I. Arnold [5].

97

STABLE SYSTEMS

condition (21.6) does not hold: Det

- O.

The cases of "limiting degeneracy" of the oscillation theory (points of equilibrium, periodic motions) also require a particular study. In that direction we mention some results that generalize Theorem (21.7).

V. I. Arnold [7] proved the stability of positions of equilibrium and of periodic motions of systems with two degrees of freedom in the general elliptic case. As a corollary, A. M. Leontovich [1] deduced the stability of

the Lagrange periodic solutions for the reduced problem of the three-body (plane and circular).

V. Arnold [8], [9], [10], studied the generation ~f new frequencies from the perturbation of degenerate systems. As a corollary, one obtains the

perpetual adiabatic in variance of the action for a slow periodic variation of the parameters of a nonlinear oscillatory system with one degree of freedom, and also that a "magnetic trap" with an axial-symmetric magnetic

field can perpetually retain charged particles. Finally, quasi-periodic motions in the n-body problem have been found. If the masses of n planets..are small enough in comparison with the mass of the central body, the motion is quasi-periodic for the majority of initial conditions for which the eccentricities and inclinations of the Kepler ellipses are small. Further, the major semiaxes perpetually remain close to their original values, and the eccentricities and inclinations remain small (see V. Arnold [4]). On the other hand,

J. Moser [1], [5] generalized Theorem (21.7). Moser

abandons the requirement of analyticity of the Hamiltonian and substitutes

instead the requirement that several hundred derivatives exist.

For in-

stance, for systems with two degrees of freedom, it is sufficient (hat 333 derivatives exist! (D) INVARIANT TORI OF CANONICAL MAPPINGS 21.9 Theorem (21.7) can be reformulated by using the construction of the

98


"surfaces of section" of Poincare-Birkhoff. Assume that, in Equations (21.3), the first component w l of w is nonvanishing.

Consider a sub-

manifold ~2n-2 of the phase-space n2n whose equations are: ql

=

0,

H = h = constant. The orbit x (t) of (21.5) through a point x on ~2n-2 will, as t increases from zero, return to ~2n-2 and will cut ~2n-2 in a uniquely determined point Ax (Figure 19.11). If tile perturbation H 1 is small enough and w 1 (p)

0, the mapping A: ~2n-2

f.

-->

~2n-2 is well

defined in a neighborhood of the (n -1) -dimensional torus: p = Ct,

ql = O.

Since

then P2' ... , Pn; q2' ... , qn (mod 211) are "action-angle" coordinates in this neighborhood. The mapping A is canonical (see Appendix 31). Now, consider the unperturbed system (H 1

=

0). According to (21.3),

the map A may be written as follows: (k = 2, ... , n).

(21.10)

In other words, each torus p

=

Ct is invariant and rotates through w(p)

under the mapping A.

If the perturbation H 1 is small, then the corresponding canonical map1,2n-2

-->

~2n-2 is close to (21. l-O). The (n -1) -dimensional

invariant tori of

d'

are, obviously, similar to the n-dimensional invariant

ping

d':

tori of (21.5) and there is a theorem, similar to Theorem (21.7), for mappings (see Theorem 21.11). Let

n

again be the phase-space p, q:

Assume that B: p, q

-->

p '(p, q), q '(p, q) is a global canonical mapping,

that is:

¢Pdq=~Pdq, Y

for any closed curve y of

n

By

(see Appendix 33). Then, assume that the

99

STABLE SYSTEMS

functions p '(p, q), q'(p, q) - q are analytic in the complex neighborhood

[a] of a:

Let A: p, q

-+

p, q + w (p) be the canonical mapping defined by an analyt-

ic function w(p) in [a], and To(w*) the torus p = p*, w(p*) = w* that is invariant under A.. THEOREM

21.11

If B is close enough to the identity, then, for almost

12

all w *, there

exists a torus T (w*) that is invariant under BA and close to To(w*). To be precise, to any K> 0 corresponds an S> 0 and a mapping 0:

T

-+

A, p = p(Q), q = q(Q), of an abstract torus T = !Q (mod 27T)1 into

A, such that: O(Q+w*)

A

=

8·A·0(Q),

B

01-1 Q - Q+w*, and: Ip(Q)

-p*1 < K,

Iq(Q)-QI + 277)

Of course, for t '" 1, the eVQluti,on \Ht) -1(0)\ '" E

«

:=

F([, ¢),

E« 1.

1. Notable ef-

fects, of order 1, of the evolution appear only after a long enough time:

t '" 1/ E . Perturbation Theory proceeds to study the perturbed system as follows. Let

fi(n

be the mean:

One considers the "averaged system," or "system of evolution":

j = E·fi(]).

(22.4) For E

«

1, one expects that:

(22.5)

IHi) -j(t)1 « 1 for 0 < t < 1

E

where [(t), ¢(t) is a solution of (22.3) and j (t) is the solution of (22.4) with initial data: j (0) =

Ho).

13 Thls method goes back to Lagrange, Laplace, and Gauss, who used It in Celestia I Mechanics.

102


Now the problem arises as follows: what relations exist, for 0 between the perturbed motion [(t) and the "motion of evolution"

< t < 1.,

I (t)

E

?

Does the inequality (22.5) 'hold? For the simplest periodic motions (k = 1) it is readily proved -(see Appendix 30 and Bogolubov and Mitropolski [1]) that if

< C·E, for 0
1 and consider the system: ¢1 = 11 ,

¢2 = 12 ,

11 = E,

i2

= Eacos(¢1-¢2)·

Of course, the system of evolution is:

11 = E,

12 = 0

(corresponding to small arrows on Figure 22.7). Consider the following initial data: 11 =12 =1 1 "'1 2 = 1, 12 .J2

¢1

0,

1 ¢2 = arcos a 1 (t)

-----

~

--....,. ~ ~ ~

J

~

--....,.

--....,.

~

~

---+

~

---+ ~

(t)

-+ ~

11 , J 1 Figure 22.7

STABLE SYSTEMS

103

Then:

1. Thus,

I[(l/s) - ](l/s)1 In other words, after the interval of time

=

lis,

1. the averaged motion loses

any relation with the real motion which remains locked in by the resonance wI = w 2 •

(C) MATHEMATICAL FOUNDATIONS OF THE AVERAGING METHOD 22.8

There exist, at least, four distinct approaches to the problem of the mathematical foundations of averaging method. All four lead to rather modest results. (1) The neighborhoods of particular solutions (for example, positions of equilibrium

F

=

0) of the averaged system can be fairly well studied.

For instance, there exist attracting tori of (22.3) which correspond to the

< t < (0) obviously holds in the neighborhood of such a torus. N. N. Bogolubov [2], J. Moser attracting points of system (22.4). Stability (for 0

[2], [5], and 1._ Kupka [1] proved that attracting tori still exist for perturbed systems. This approach does not apply to Hamiltonian systems because attracting points do not exist according to the Liouville theorem (see 1.10).' (2) One can study the relations between I (t) and] (t) for most (in the sense of measure theory) initial data, neglecting points that correspond to resonance. For instance, Anosov [3] and Kasuga [1] proved theorems of the following type: Let R ( s, p) be the subset of

n

of the initial data such that

I[(t) -](t)1

for certain 0

>P

< t < lis. Then, lim measure R (S, p) e

-+

=

0 for all p> O.

0

This approach allows one to obtain similar results for systems much more general than (22.3); whence its weakness: estimates of the measure

104


of R (f;., p) are not realistic and one has no information concerning the motion in R(E,p). (3) One can study passages through states of resonance, (.4) One restricts oneself to Hamiltonian systems to obtain more information. (D) PASSAGE THROUGH STATES OF RESONANCE 22.9

Let us begin with an example: ¢1 = II + 12,

1>2 = 12,

il

= E,

i2

= E cos(¢1 -¢2)·

The averaged equations are (see Figure 22.10):

j2=0.

J 1 =E,

-- -- ---- -

-~ ,~

~

-

-

----.

-

--

-----

Figure

--

•

=

¢2(O)

J

-

22.10

Consider the initial data that correspond to resonance wI ¢1(O)

I (t)

= 11 (0)

=

12 (0)-1

=

= w2:

O.

The system is easily integrated: lI(t)-j(t}1

=

1/2(t}-11

=

fiEjTcosx2.dX, o

T=..jE/2t.

l?

105

STABLE SYSTEMS

For t

=

liE, obviously,

I/( t) - J( t) I =

C· YE.

Thus, the passage through the resonance w 1 = w 2 disperses the bundle of orbits I(t), cp (t), which in the beginning differ only by phases

cp(O). The scattering of 12 after going through the resonance is of the order of

Vc (see

quencies (k

=

Figure 22.10). For a general system (22.3) with two fre-

2), one obtains 14 the following theorem:

If the quantity:

does not vanish in {l, then we have the estimate: for all 0

(22.11)

< t < !. E

Condition A ;, 0 means that the system cannot remain locked in at any resonance: (22.3) implies

In example (22.6) condition A

changes sign at 11 A

t.

=

t.

0 is violated:

12 if a> 1. This example shows that condition

0 cannot be replaced by an analogous condition for the averaged sys-

tem. The idea used in proving (22.11) is that the scattering produced by each resonance is of the order C YEand that, among the infinitely many resonances

w/ w2

=

min, only the greatest (m, n

produces notable effects. 14 V. I. Arnold [12].

< In !. ) E

106


Passage through resonance!': for systems with more than two frequencies (k > 2) has not been studied. (E) EVOLUTION OF HAMILTONIAN SYSTEMS 22.12

Next, apply the. averaging . method to Hamiltonian systems (21.5). If condition (21. 6) of nondegeneracy holds, then most of the unperturbed orbits are ergodic on tori p = constant. Thus, it is satisfactory to write this system in the form (22.3), with I

= p, ¢ = q, k

=

I

= n:

aH 1 al

, ¢

w(I) +' E - -

~, i

- E--

aH 1 a¢

where

w The averaged system is ] = 0, for

o. In other words, there is no evo}ution for nondegenerate Hamiltonian systerns: ] = constant. Theorem (21.7) of conservation of quasi-periodic motions rigorously establishes this conclusion. In fact, Theorem (21. 7) implies that:

IHt) -](t}1 < (for all initial data if n

K for all t ( Rand

2, and

a2 H o aP

E
0). The intersections have equations: Y = Y·, J

Thus

Y

J

=

e -Ilt; • Yo ... 0 •

n contains the set of all the surfaces

g tl V that are parallel to M +

and converge to M+. These surfaces already obstruct M+ at TJ; this proves Lemma (23.S). (C) THE TRANSITION CHAINS 23.6

Assume that the dynamical system has transition tori T1, T2, ... , Ts' These tori will be said to form a transition chain if the departing whisker

M7 of every preceding torus T; is transverse to the arriving whisker M;-::'l of the following torus Ti+ 1

at some point of their intersection (see

112


Figure 23.7):

Mi nM;

;, 0, M; nM3- ;, 0, ... , M;_l nM; ;, 0.

M; M;_l Figure 23.7

LEMMA 23.8

Let T1 , T2 ,

..• ,

Ts be a transition chain. Then, an arbitrary neighbor-

hood V of an arbitrary point ~

f

M;- is connected with an orbit (t) to

an arbitrary neighborhood V of an arbitrary point TJ

(0)

f

V,

(t)

f

f

M; :

V for a certain t.

Proof: Consider the future U ~ torus, then

Uobstructs Mi

Ut > 0 V(t) at ~1 ~

the open set U. Let ~; be a point of

of V. Since Tl is a transiti~n

Mi nM;.

Thus, M; intersects

M; nu, then there exists a neigh-

borhood V 1 of ~; that belongs to U. The future of V 1 belongs to U and it is sufficient to perform the same argument s times to prove that U obstructs

M;

at TJ.

(Q. E. D.)

(D) AN UNSTABLE SYSTEM 23.9

Let U ~ R2 x T3 be the five-dimensional space 20 11' 12 ; B on

Ts' LEMMA

23.12

Each manifold Tw defined by the equations 11 =

M.

This

curve y is invariant under ¢ and y is dense on M, for .:\2 - 1 is irrational Qacobi's theorem, Appendix 1). Let m

=

(x, y) be a point of y. Of course,

we have:

and 0

0 correspond two trigonometrical polynomials ~- and ~+ such that: Pe -(z)

< fez) < Pe + (z) for every z ( M

and j(Pe+(z)-Pe-(Z))d ll

x) are: bk =

1

e- 277ik (x -0), f(x)d/l = e 277ik ' 0 . a k .

M

The invariance of I is equivalent to b k = a k for any k, that is a k = 0 or k·

0

(

Z. 132

133

APPENDIX 11

If the

w;'s

are integrally independent, the second case occurs only for

k = O. Thus a o is the only Fourier coefficient possibly nonzero, f is constant, and (M, /L, ¢) is ergodic (see 7.2).

If a k f- 0 exists such that k· w ( Z, then f (x)

=

e 21Tik · x is a non-

constant invariant function and (M, /L, ¢) is not ergodic. Remark. In the continuous case (M, /L, ¢t)' where

¢ t:

e 21Tix

->

e 21Ti (X + tw) ,

we have a similar result: (M, /L, ¢t) is ergodic, if, and only if, k ( Zn and k·w = 0 imply k = 0 (or if, and only if, the orbits are everywhere dense; see Jacobi's theorem).

APPENDIX 12

THE TIME MEAN OF SOJOURN (See Chapter 2, Section 7) TH~OREM

A12.1

, An abstract dynamical systeJll (M, /l, ¢ /) is ergodic if, and only if, the sojourn time reT) in an arbitrary measurable set A of an orbit

I¢ /x I 0 $. t s; TI is asymptotically proportional to the measure of A:

(A12.2)

.

dT)

T->oo

T

11m - -

=

It (A),

for all measurable A and almost every initial point x ( M. Proof:

* Assume (M, /l, ¢/) is ergodic and A is measurable. We have f(x) for every f ( Ll (M, It) and for almost every x ( M (see 7.1). Take f

=

T XA

(characteristic function of the set A), we obtain:

for almost every x. The converse is derived at once: (A12.2) implies ergodicity. It is sufficient to observe that the characteristic functions

XA

generate Ll (M, /l).

Theorem (A12 .1) clearly holds in the discrete case (M, Il. ¢ ). 134

135

APPENDIX 12

EXAMPLES AI2.3: TRANSLATIONS OF TORI Let M be the n-dimensional torus

I e 27Tix I x

£

Rn I, Il the usual mea-

sure, and ¢ the translation:

If k ( Zn artd k· (U

(

Z imply k

=

0, then (M, Il, ¢) is ergodic (see Appen-

dix 11). Thus, relation (AI2.2) holds for almost every initial point. This can be rephrased as follows: denote by r(N, A) the number of elements of the sequence

that belong to A, then: r(N, A) (A) · 11m - - - = Il

(AI2.4)

N~oo

N

for almost every initial point e 27Tix . If A is Jordan-measurable, that is, if ~A is Riemannian-integrable, then (AI2.4) holds for every initial poiRt. To prove it, it is sufficient to use the theorem of Appendix 9 and to take f

= ~A'

Extension to·the continuous case holds good. This result i!l

known as the theorem of equipartition modulo 11 and is due to P. Boh1 [1], W_Sierpinskii, and H. Wey1 [1], [2], [3] ~ It is one of the first ergodic theorems. Historically, it originated from an attempt to solve the Lagrange problem of the mean motion of the perihelion (see Example 3.1 and Appendix 13). Here follow some applications. 2 ApPLICATION AI2.5: DISTRIBUTION OF THE FIRST DIGITS OF 2" (see Example 3.2) The first digit of 2n is equal to k if, and only if: k • lOT $. 2n

le 21Tix I x

e 21Ti (X+ a).

(

RI.

the usual measure /l, and the translation ¢:

(M, /l, ¢) is ergodic, for a is irrational (see Exam-

ple 7.8). Thus, the sequence 1(na) I n ( N I is equidistributed. In particular, take A

=

[Log 1ok, Log10(k+ 1)] in relation (A12.4), we have: lim

r(N, A)

-N-

N->oo

=

/l(A)

1

= Log 10(1-+; k-).

But r(N, A) is nothing but the number of elements of the sequence 1, 2, ... ,

2N - 1 , the first digit of which is k. Thus, if we go back to the notation of ExamRle (3.2), we have:

Consequently, the proportion of 7's is greater than the proportion of 8's in the sequel!,ce of thEt~fi,,~t digits of

12n In

=

1,2, ... 1. This is not what one

expects from an inspection of the first terms: 1,2,4,8, 1,3, 6, 1, 2,5, ... This is due to the fact that a = 0, 30103 '" REMARK

is very close to 3/10.

A12.6

Since the sojourn time in a domain A of a point of an ergodic system is proportional to the measure of A, it is natural to ask about the dispersion. Let us mention some results due to Sinai [1]. Let ¢t be the geodesic flow of the unitary tangent bundle Tl Y of a surface .y of constant negative curvature. If A is a domain of Tl Y with piecewise differentiable boundary, then the mean sojourn time of a geodesic ¢tX in this domain has a Gaussian distribution and verifies the central limit theorem:

137

APPENDIX 12

lim !l T-+oo

where

~ x ITT(X) T

TT(x) =

measure

- !l(A)

< ;; yT

I t I ¢tX

f

A, 0

f

=

$nl 217

:s. t :s. Tl

f

Ca e- u2/2 • du

-00

and C is a constant.

APPENDIX 13

THE MEAN MOTION OF THE PERIHELION (See Example 3.1 and Appendix 12) The problem of mean motion arises from the theory of the secular perturbations of the planetary orbits (Lagrange [1]). One asks for the existence and estimate of: (A13.1)

n

n

1 Arg iwkt , ~ ak . e t-++oo k= 1

= lim

where w k ' t ( Rand a k f, 0, ak ( C. In other words, if one considers a plane linkage AoAl ... An consisting of the links A k _ 1 Ak of fixed lengths lakl, moving with constant rotation-velocity w k ' we are interested in the mean rotation-velocity of the vector AO An (see Figure A13.1). THEOREM A13.2 (See H. Weyl, [1]- [5].) Assume that the wk's

(A13.3)

W·

are

integrally independent, that is:

k = 0 and k ( Zn imply k = O.

Then the mean motion 0 exists and is expressed as:

(A13.4) The Pk's depend on the lakl only. Ifl p(ak ; al' ... , ~k' ... , an) is the probability that an (n-l) -linkage with prescribed sides al' ... , ~k' ... , an spans a distance inferior to ak (see formula A13.12), then: 1

A

means cancellation.

138

139

APPENDIX 13

y

o =AO Figure A13.1.

x

.

Case n '" 3: initial position of the linkage .

(A13.S) In particular, for n '" 3, if there exists a triangle the sides of which &re

lall, la2 1, la3 1 and the

angles of which are Al , A 2 , A 3 , then (Bohl's

formula): (A13.6)

n '"

_A_l_w..;.1_+_A...;:2:....w.....;2'--.+_A_3:....w_3;:.... 17

The case in which no triangle can be constructed was investigated by Lagrange [1]. In the general case A. Wintner [1] found the expression:

in terms of the Bessel functions ]0 and ]1· Relation ~ Pk an "addition" theorem for these functions.

=

1 provides

140


Proof of Theorem A13.2 THE CORRESPONDING DYNAMICAL SYSTEM

A13.7

Let us consider the dynamical system (M, p., ¢>t)' where M

=

Tn

= Izlz = (zl, ... ,zn)\'

is the n-dimensional torus, p. is the usual measure, and ¢>t is the translation group:

The phase space of the n-linkage is M, and ¢>t depicts the movement. Let us define a function a on M by: n

(A13.B)

a(z) = Arg

I

lakl

Zk'

O:s a < .211 •

k=l This function is discontinuous over the slit l:

= I z I a(z) = 01,

and is not

defined on the so-called singular manifold S = Izll:~lak~zk = 01 which consists of all possible states of a closed n-linkage with the prescribed sides lakl. Nevertheless, the function:

(A13.9) is analytic outside of S. The limit (k13.1), if it exists, is nothing but the time mean ~ of {3:

(A13.10) where

THE SPACE MEAN

A13.11

The system (M, p., ¢>t) is ergodic, for the

(Uk's

are integrally indepen-

dent (Appendix 11). If the function {3 were Riemannian-integrable, then,

141

APPENDIX 13

according to the theorem of Equipartition modulo 1 (Appendix 9), the time mean

f1*

~

n

n ~ i3

the Birkhoff theorem implies that

f1

is Lebes-gue-integrable. Thus,

for almost every initial phase.

This suggests the study of the space mean shows that

f1

f1.

would be equal to the space mean

We only know (see A. Wintner [1]) that

i3.

Relation (A13.9)

depends linearly on the (U/s. Therefore,

f1

depends lin-

early on the (Uk's:

To compute Pl (for instance) we set: (Ul

~

217,

~

(U2

•••

~

(Un

~

0 •

We have:

1 -

217

f1 (217,0, ... ,0)

,

where

Relation (AI3.8) allows one to carry out the integration over (Jl:

if Ila2Ie217i(J2 + ••• + Ian Ie 217i (Jn I < lall if Ila:ile217i(J2 + ••• + Iani e 217i (Jn I > lall Thus we obtain:

where

This proves relation (AI3.S). Relation ~ Pk ~ 1 is derived easily by setting (Ul

~

•••

~

(Un

~

2" •

142


A13.13

EXISTENCE OF THE TIME MEAN

Hence, formulas (A13.4) and (A13.s) are proved for almost every initial phase Arg a k . To prove them for all initial phases we use a special device inaugurated by Bohl [1] for n n

=

3 and improved by Weyl [4], [5] for

> 3. We define a function on the torus M by: N (z)

=

algebraic number of the points of intersection of the curve

I¢/z, -271 < t < 01 with the slit ~. We count + 1 a point of intersection z. for which (3 (z.) > 0 and -1 if J

(3(z.) J

J

< O. (See Figure AI3.1s.) It can be proved that

N(z) is bounded.

N=O Fii:Ure

~13.1S

Thus, according to (AI3.10), the following relation holds uniformly over

m: (~13.14)

Since the function N is piecewise nian-integrable, the time mean

N*

continuo~s

and, in particular, Rieman-

exists everywhere and is equal to N

(Appendix 9). From (AI3.14) one deduces that {3* where and is constant.

=

*=

N

N exists every-

(Q. E. D.)

APPENDIX 14

EXAMPLE OF A MIXING ENDOMORPHISM Let us consider the transformation: 1

cf>: (x, y) ... (2x, 2y) (mod 1) of the torus M

= l(x, y) mod 11

carrying the usual measure dx dy. cp-' A

M

I'

~

~ ~

~

@ ~

~

@

~

~

~

@

G

~

~

~

Figure A14.l

~--------------------~

Which is called "multiplication of loaves" since Figure (A14.1) shows the solution of a well-known historical problem.

143

144


To be more explicit, we write: ( (2x, 2y)

¢ (x, y)

~ (2x,2y-l)

t

(2x-1,2y)

if 0 $ x,

ym(a)

= m({3)

< >m(a)

C ')J{(f3}

ai )

=

iEI

V

m(a)

iEI

(See A. N. Kolmogorov [3] and V. Rohlin [3]).

§ 8. Entropy of a Given {3

Let a

=

IAi I i

= 1, ... , d and {3 =

IB j 1 i

= 1, ... , 81 be- two finite mea-

surable partitions. We can assume, without losing generality, that any element Ai or B j has positive measure (see A 18.2).

160


DEFINITION

A 18.6

Let z (t) be the function over [0, 1] defined by:

~

=

z(t)

- t log t if 0 < t

o

:s 1 •

ift=O.

f3

The conditional entropy of a with respect to

is:

h(alf3} = ~ Il(B j ) ~ z(Il(A/B j )), i where

Il(A./B.) 1

]

=

Il(A. nB.) 1

]

Il(B.) ]

is the conditional measure of Ai relative to Bj" We tum next to the proof of Theorem (12.5) which we reformulate. THEOREM

Let a

12.5

=

f3 = IBjl,

1Ai\'

=-ICkl

y

be finite measurable partitions.

Then: h (al f3) ~ 0 with equality if, and only if a

(12.6)

:s f3;

h(aVf3.(y) = h(aly) + h(f3/aVy);

(12.7)

:s f3

(12.8)

a

(12.9)

f3:s

(12.10)

y

> Maly)

:s h (f3/y) ;

> h(a/y)

~ h(alf3};

h(aVf3/y)

:s h(aly) + h(f3/y)

•

Proof: Proof of (12.6) is left to the reader as an easy exercIse. The elements of aV f3 and aVy are, respectively, of the form: Ai B j and Ai

n Ck ·

Therefore

h (aV f3/y)

But we have

n

161

APPENDIX 18

jl(A,nBjnC k )

=

jl(Aj~Ck)

jl(AjnBjnC k )

jl(C k )

jl(A,nC k )

jl(C k )

= jl(A/CkhdB/ A( n'C k )

and we deduce relation (12.7): h(aV{3/y)

= -

!

jl(AjnBjnC k ) Log jl(A/C k )

i,j,k

= -

!

jl(A j

n C k ) Log

jl(A/C k ) -

i,k

h(aly) + h({3/aVy)

Let us prove relation (12.8): If a :::. {3, then aV {3

(3 and relations

=

(12.6) and (12.7) imply: h({3/y)

=

h(a/y) + h({3/aVy) ~ h(aly) •

Let us prove relation (12.9): Since lk jl(CkIB j ) = 1 and jl(C/B j ) ~

0, the concavity of z (t) implies:

Since {3 :::. y, each B j is the disjoint union of some Ck's; therefore we have:

where the sum extends over those C k,'s be longing to B j

.

We deduce:

162


!

z(p.(A/Ck))·p.(CkIB j ) $. z[p.(A/B;)].

k

Multiplying both members by p. (B j) and summing over i and j yields (12.9). Finally, (12.10) is a consequence of (12.7) and (12.9): a V y ::: y implies h «(3/ aVy) $. h «(3/y)

and h(aV(3/y)

=

h(a/y) + h«(3/aVy) $. h(aly) + h«(3/y).

The preceding definitions and properties extend to denumerable measurable partitions (see Rohlin and Sinai [5]).

APPENDIX 19 ENTROPY OF AN AUTOMORPHISM (See Theorem 12.26) The purpose of this appendix is to prove the following theorem dl.e to Kolmogorov. THEOREM

1

A 19.1

If ¢ possesses a generator a, then h (¢) = h (a, ¢):

The proof breaks into several lemmas. Denote by F the set d all finite measurable partitions. Given a, {3

f

F we write

Ia, {31

= h (a

I (3) +

h ({31 a). LEMMA

19.2

Ia, {31 is a distance on F. Proof: It is clear that

Ia, {31

~

O. From formula (12.6) of Chapter 2 we de-

duce:

Ia, {31

=

0

> h (a I (3)

It is also evident that

= h ({3

la, (31

I a)

=

='

0

>a $. {3 and {3 $. a

>a = {3.

1{3, al. According to (12.11), (12.12), and

(12.9) we have: h(a/y) = h(aVy) -h(y) $. h(aV{3Vy) -h({3Vy) + h(j3vy) -h(y) = h(aI{3vy)+h({3/y) ~ h(a/{3)+h({3!y)

1 The proof follows Rohlin [4].

163

164


and symmetrically: h(y/ a)

5 h({3/a) + h(y/{3) •

Addition yields:

la, yl ~ la, {31 + 1{3, yl • LEMMA A

19.3

Given ¢' h (a, ¢) is a continuous function on F in' its argument a. More precisely:

Proof:

Given a, {3 ( F, we set:

an = aV¢a •.• V¢n-l a ; {3n = (3V··· V¢n-l{3 • From (12.11) of Chapter 2 follows: h({3n/an) -h(an /{3n) = [h(an V{3n) -h(an)]-[h(an V{3n)-h({3n)] = h ({3n) -h (an) •

Since h (

I) ~

0, we deduce:

On the other hand, from (12.10) of Chapter 2 follows: .h(an /(3 )n = h(aV···V ¢n- 1 al{3n ) < h(a/{3 n ) + ••• + h(¢n-l a/{3 n ). .

Similarly, from (12.9) and because {3, ... , ¢n-l {3

Symmetrically: Addition yields:

513

fT ,

'we have:

165

APPENDIX 19 _

Dividing both sides of this inequality by n and passing to the limit as n ->

00,

we obtain Lemma (A 19.3).

LEMMA A 19.4

If aI' a 2 , ... is a sequence of finite partitions such that

00

V n~l

m(an )

~

~

1 .

then the set B of partitions {3 ( F, such that {3

~ an

for at least one

value of n, is everywhere dense in F. Proof: We-rreed to prove that for every finite partition a and every 0 > 0 there exist an n and a {3 ( B such that:

{3

s.

an'

la, {31 < 0 .

Let AI" .. , Am be the elements of a.

is dense in

( m(an)

1,

Since

for every 0' > 0 there eXist an n and subsets AI',···, A~_l

such that: i~1 .... ,m-1.

Let us denote by {3 the partition of M into sets B 1 •

It is clear that {3

s.

an. On the other hand:

la. (31 ~

r hi (3) + h ({31 a)

...•

Bm defined by:

166


k

IL(A)

!.

Il(B k nA j )

j

Il(B k )

- ! k

=

Il(A j n B k )

Il(A) !

- !

j

Log [ Il (A I. n B k) Il (A j) Log

]

[1l(B k nA.h I

Il(B k )

Il (B k)

J

- 2 ! IL(AjnB k ) Log IL(AjnB k ) + ! IL(A) Log IL(A j ) i

i,k

These formulas show that and vanishes when

Al

=

la, ~I

depends continuously on

A l , ... , A~_l

=

Ai, ... , A~_l

A m _ 1 . Therefore, if 8' is small

enough, then la,~1 < 8. Proof of Kolmogorov theorem:

Assume that ¢ possesses a generator a. We set, for i\

l

F and q =

0,1, ... :

We have:

From Lemma (A 19.4) it follows that the set B' of partitions that ~

~

~ l

F such

$. an for at least one value of n is everywhere .dense in F. Let

be an element of· B '. Clearly:

Therefore, from (12.12) of Chapter 2 follows: h (~m) $. h (iin+m_ l

)

167

APPENDIX 19

n + m -1 m

Now, observe that:

q

q

h(>.V ••• V ¢2 q -2>..)

2q-1

2q-1 Thus, passing to

~he

... 2h (A, ¢), as q ...

00

•

q

limit as m ... + 00, we obtain h ({3, ¢)

s

h (a, ¢) .

Recall that B' is everywhere dense in F and that h ({3, ¢) is continuous in {3 (Lemma A 19.3), then: h (a, ¢) that is:

~

sup h ({3, ¢) B'

sup h({3, ¢) F

~ h(¢) ,

·\PPENDIX 20

EXA\IPLES OF RIEMANNIAN \IANIFOLDS

\"\ITII NEGATIVE CURVATURE (See 14.1, Chapter 3) Consider the proper affine group G of the real line

I tit

( R I. An ele-

ment g' of G has the form:

g:

t

-+

yt +

x, y (R,

x,

y > 0,

and can be denoted by (x, y). Given g'

we obtain:

= (x', y'),

g'(g(t))

y'(yt+x)+x'

=

=

y'yt+y'x+x'.

Therefore, if we denote the group operation by .L this may be written: (x', y').L (x, y) = (y'x + x', y'y) •

The neutral element is e = (0, 1) an~ the inverse of (x, y) Both .L and g

-+

IS

(_xy-l, y-l).

g-l are smooth operations. Thus, G is a L"ie group that

is diffeomo.rphic to. the upper half-plane I (x, y) I y >

01.

to. a Riemannian manifo.ld. THEOREM A 20.1.

THE RIEMANNIAN METRIC OF G

The leFt-invariant metric of G which reduces to

at the neutral element e

=

(0, 1) is: dx 2 + di

y2

168

No.W we turn G in-

169

APPENDIX 20

Proof: To any element X = (x, y) of G corresponds the left translation LX· L x(U)

=

x 1. U,

where U

=

(u, v)

l

G .

We have:

!:) ,

( -u-x y-' y the tangent mapping of which is:

(A 20.2) Define a metric over the Lie algebra TG e by setting:

This defines a left-invariant metric at each point X:

Therefore, if X = (x, y), (A 20.2) implies:

(~1)2 + (~2)2

In other words, the mejric is: (A20.3)

DEFINITION A 20.,+

The upper half-plane G endowed with the metric (A 20.3) is called the Lobatchewsky-Poincare plane. It can be useful to represent a point (x, y) of G by the complex number z = x + iy. THEOREM A 20.S.

THE ISOMETRIES OF G

The symmetry (x, y)

-+

(-x, y) and the homographies:

170


(A20.6)

az + b cz + d ;

z ... z

a, b, c, d (

R. ad-bc

1

preserve the metric (A 20.3).

Proof: Proof is purely computational and easy if one observes that: ds 2

=

-4dzdz h - - - - , were z (z_z)2

=

.

X-ly.

THEOREM A20.7. ANGLES

The angles of metric (A20.3) coincide with Euclidean angles. Consequently, words such as "orthogonal," and so on can be used unambiguously.

Proof: i's proportional to dx 2 + di .

THEOREM A 20.8. GEODESICS

The geodesics of (A 20.3) are the straight lines: x

=

constant, y > 0

and the upper half-circles centered on ox. In particular, there exists one, and only one, geodesic passing through two given distinct points. Proof: Let ab be a segment of x

0, y > O. For any arc y joining a and

b we have:

f

ds 2

•

ab

This proves that x = 0, y > 0 is a geodesic. An image of this geodesic under any isometry (A20.6) is still a geodesic. We obtain so all the upper half-circles centered on ox and the halfstraight lines x

=

constant, y > O. In fact we obtained all the geodesies

for, given a vector u ( Tl G, there exists a half-circle centered on ox (ora parallel to oy) which is tangent to u.

171

APPENDIX 20 THEOREM

A20.9.

CURVATURE

The Gaussian curvature of (A 20.3) is equal to -1. Proof: The Gaussian curvature K is constant,

tor the metric is invariant

under a transitive group of isometries. The Gauss-Bonnet formula applied to a geodesic triangle L\

=0

A+ B+ C ':"

ABC gives:

1T

+

ff

K· da

=

1T

+ K· area L\ •

t" ~

~

The particular case of Figure (A 20.10) gives A

B

C

=0

O. As the

element of area is da = (dxdy)/y2, we obtain: area L\ We conclude that K

=

1T •

-1.

y

----O~=-B---------/------~~==~C~--~x r Fi~re

A20.10

172

ERGODIC PROBLEMS OF CLASSICAL MECHANICS ;:

THEOREM A20.11. ASYMPTOTIC GEODESICS

Let y (u, t) = y (t) be a geodesic parametrized by arc length t, and g

G. The geodesic passing through g and

f

as t1

->

}J!J.- has a

limit position

+ 00 (resp. -00). This limit position is the geodesic passing

through g and the intersection y(+oo) (resp. y(-oo)) of y with ox. Geodesics emanating from y(+oo) (resp. y(-oo)) are called the positive (resp. negative) asymptotes to y. Proof: Let y(t 1) be a point of y. The geodesic passing through g and y(t 1) is a circle centered on ox, possibly reduced to a straight line (A 20.8). From the very definition of the metric (1\20.3), y(tl) runs to-ox as tI

-+

+ 00 (resp. - 00), that is y (t 1) converges to the, intersection y (+ 00)

(resp. y(-oo)) of y with ox (see Figure A20.12). Thus our geodesic has a limit position, namely the upper half-circle centered on ox and passing through g and y (+ 00) (resp. y (-00)). Consequently, this limit position is a geodesic. y

'Y(-m)

Figure A 20.12

x

173

APPENDIX 20

DEFINITION A 20.13. HOROCYCLES

1

The orthogonal trajectories of the positive (resp. negative) asymptotes to yare called the positive (resp. negative) horocycles of y. THEOREM A20.14.

The positive (resp. negative) horc.::ycles of yare the Euclidean circle of G which are tangent to y

=0

at y(+oo) (resp. y(-oo)). In particular,

the straight lines y = C > 0 are horocycles. They are positive horocycl/ of the axis oy (y

-+

00).

Proof:

The 'positive (resp. negative) asymptotes to y form the upper part of the pencil of circles that are orthogonal to y

=

0 at y(+oo) (resp. y(-oo)).

Theorem (A 20.14) follows at once from the elementary properties of conjuii!,ate pencils of circles. The points y(+oo) and y(-oo), which do not belong to G, have to be removed. THEOREM A20.1S. RIEMANNIAN CIRCLES

The Riemannian circles of (A 20.1) centered at m form the upper part of the pencil of circles whose radical axis is ox and whose Poncelet points consist in m and the symmetric m' of m with respect to ox. Proof:

The Riemannian circles centered at m are the orthogonal trajectories of the geodesics emanating from m. This family of geodesics is

nothi~g

but the upper part of the pencil of circles passing through m and m'. (Q. E. D.) In particular, the power of any point d of ox with respect to one of these Riemannian circles centered at m is:

(see Figure A 20.17). 1 Notion due to Lobatchewsky (in Greek, "horos" = horizon).

174


THEOREM

A 20.16

Horocycles are Riemannian circles the radii of which are infinite and the centers of which are at infinity (on y

=

0). '/

Proof: Consider the Riemannian circle passing through a fixed point n of a geodesic y and centered at m ( y (see Figure A 20.17). If m moves to infinity along y, that is, if m converges to ox, then mm' ... O. Therefore, the power of any point of ox with respect to our circle tends to zero. Thus our circle has a limit position which is the circle tangent to ox at y (+ 00) and which passes through n. Theorem (A 20.16) shows that this limit position is an horocycle. Conversely, any horocycle is obtained from the (Q. E. D.)

above construction.

'Y(+OO )

m' Fi\t:ure A20.17

175

APPENDIX 20 THEOREM

A 20.18

Let y(u, t) and y'(u', t) be two geodesics which are positively (to fix the idea) asymptotic one to the other. We denote their arc length counted from their origins nand n' by t. Then, after a suitable selection of nand n " we have: d (y (t), y '(t)) $. nn 'e- t ,

t 2. 0,

where d means the Rierrftmnian distance, and nn' is the arc-length of the horocycIe. Proof: Origins nand n' are selected on the same horocycle 1 (Figure A 20.19). Denote by m and m' the intersections of y and y' with another horocycle 2. Arcs

nm

and

0

'm' are equal, for 1 and 2 are parallel curves: n 'm' =

nm =

Let us compute the arc

mm'

t.

that belongs to 2. Horocycle 2 has the equa-

tion: x

=

r sin u, y

= r + r cos

u.

Thus, with obvious notations: m' -l-+-d -Cu-o-s-u

{

Symmetrically, on horocycle 1: nn

,

=

un'

tg -

2

U

- tg ~

2

A straightforward computation with y and y' leads to: t =

t

~

om =

n'm'

Log 1tg

I

Log tg

~n I -

Log I tg u;

Uri -

Log tg

I

I'

U; 'I .

176


y

r

x

Figure A 20.19

Consequently:

tg et

u _n_

2 u

tg .2!.. 2

tg

u

~

2--u

,

tg .2!.. 2

mm

I

=

u

tg ~ - tg 2

u

tg

,

~

- tg

2

I-t

nn • e

Theorem (A 20.18) follows from d(m, m ')

:s.

mm' .

u

-!!.

2

u .2!!_

2

nn

mm

177

APPENDIX 20 GENERALIZATION A 20.20

The manifold V is the upper space xn > 0 of Rn endowed with the metric: (dx )2 + •.. + (dx )2 1

n

(x )2 n

V is the Lobatchewsky space of constant curvature - 1. The horocycles

are (n -1) -dimensional manifolds, namely the planes xn

constant and

the Euclidean spheres of V which are tangent to the plane xn

=

o.

APPENDIX 21

PROOF OF TilE LOBATCHEWSKY-IIADAMARD THEOREM (See 14.3, Chapter 3) § A. Manifolds of Neglltive Curvature Fa:>,

of negative

,t us recall some classical properties of Riemannian manifolds '-M

'me.

THEOREM A 21.1

Let V be a complete, simply connected Riemannian manifold of negative curvature. Then: (1) There exists one, and only one, geodesic passing through two

dis'tinct given points; (2) V is diffeomorphic to the Euclidean space;

(3) let ABC be a geodesic triangle whose angles are A, B, C 'and whose sides are a, b, c. Then:

Proof will be found in S. Helgason [1], A direct consequence is the following corollary; COROLLARY A 21.2

Under the above assumptions, Riemannian spheres of V are convex, that is, a geodesic has at most two common points with a sphere.

178

179

APPENDIX 21

§ B. Asymptotes to a Given Geodesic As usual, y (x,

II,

t)

= y (t) = y

denotes a geodesic emanating from x,

with initial velocity-vector u and arc length t. The point of y corresponding to t is denoted also by y (t). The Riemannian distance of two points a and b is denoted by

la, bl.

Denote a complete, simply connected Rie-

mannian manifold of negative curvature by V. THEOREM A

21.3

Let v' be a point of V. The geodesic joini!lg v' to a point y (t) ( y converges to a limit as t

->

+00 (resp. t

->

-(0).

This limit is a geodesic.

froof: (See Figure A 21.4.)

v' Y(v',u',t)

----------------~------------~----~--~Y

Fi~re

A21.4

The points v' and y (t 1 ) define one, and only one, geodesic y (v', u l' t ). We set

s1 =

Iv: y(t 1)1.

Take t2 > t1 and apply relation (3) of Theorem

(A21.1) to the geodesic triangle v.', y(t 1), y(t 2 ); With obvious notations

we have:

On the other hand, the triangular inequality applied to v, v' y (t 1) gives:

t1 whence:

lv, v'l

~ s1 ~ t1 +

lv, v'l '

180


Similarly:

We deduce: 1, that is to say:

Thus, according to Cauchy, u 1 converges to a limit u' as t1

->

+

00.,

The geodesic y (v', u ',t) is the limit position of y (v', u l' t), for the exponential mapping Expv' is continuous.

y(v', u', t) is called a posi-

tive asymptote to y. Negative asymptotes are defined in the same way (t 1

->

-(0).

REMARK

A 21.5

It is readily proved that the positive asymptote to y emanating from a

given point of the positive asymptote y (v', u', t) is nothing but y (geometrically). Therefore, we may speak of a positive asymptote to y without referring to a definite point v'. Furthermore, the set of the positive asymptotes to y is a (dim V -1)- parameter family of geodesics. § C. The Horospheres 1 of V

The Riemannian manifold V is again complete, simply connected, and of negative curvature. Let y (v,

11,

t )., =' y (t) be a geodesic and v' an ar-

bitrary point of V. LEMMA

A21.6

converges to a finite limit L (v': y, v) as t differentiable function of v' and v. 1

See A. Grant

[1].

->

+ 00, and this limit is a C 1 .

181

APPENDIX 21

Proof: Take t2 > t l' The triangular inequality applied to v, Y (t l)' Y(t 2 ) gives: ¢(t2)

lv', y(t2)1 -Iv, y(t2)1 ::: lv', y(tl)1 + Iy(tl)' y(t2)1 -Iv, y(t2)1 =

Iv,' yUl)1 -Iv, y(tl)1

=

¢(tl) .

Therefore, ¢ (t) decreases monotonicall::::. On the other hand, ¢ (t) is bounded, for the triangular inequality applied to v, v', y (t) gives:

This proves the existence of: ¢(t) = L(v'; y, v) .

lim t ..... +

00

The second assertion follows from the inequality:

that is

Obviously: L(v'; y, v) - L(v'; y, Vl) = vVl

(A 21. 7)

where

vVl

is the algebraic measure of

vVl

'

on the oriented geodesic y .

DEFINITION A 21.8

The locus of the points x for which L (x; y,O) = 0 is called the

pos~

itive horosphere through 0 of y and will be denoted by H+(y, 0). According to Lemma (A21.7), H+(y,O) is a Cl-differentiable submanifold. of dimension (dim V-I). Let vt be an arbitrary point of y. Relation (A 21. 7) shows that H +(y, 0) has equation:

L(x; y,vt )

=

OVl

Now we obtain the horospheres as spheres with center at infinity and radius

182


infinite. The Riemannian sphere, the center of which is a and passing through b, will be denoted by L (a, b). LEMMA

A21.9

L (y(t),O) converges to H+(y,O)

as t -+ +00.

Proof:

Let x be a point of H+(y, 0), we have: ¢(t) '" Ix,y(t)I-IO,y(t)1

On the other hand, ¢ (t)

~

-+

0 as t

-+

+00.

O. Therefore, L (y (t), 0) intersects the geo-

desic segment xy (t) at a point b(t) (see Figure A 21.10).

-----+--------~--------+-------~y

o

Figure A21.10

We have: lx, b(t)1 = Ix,y(t)I-ly(t), b(t)1

Ix,y(t)I-IO,y(t)1

-+

0 as t

-+

+00.

This means that every point of H+(y, 0) is a limit point of the spheres L(y(t),O) as t

-+

+00. Conversely, we prove that SUGh a limit point be-

longs to H+(y,O). Let b(t) be a point of L (y(t), 0) and x

=

lim b(t). t -+ +00

The triangular inequality gives:

Ilx, y(t)1 -10, y(t)11 < I lx, y(t)1 -Ib(t), y(t)11 + Ilb(t), y(t)1 -10, y(t) II Ix, b( t) I

-+

0 as

t

-+

+ 00 •

183

APPENDIX 21

Therefore: L (x; y, 0)

~

0, that is, x

f

H+(y,O).

COROLLARY A21.11

Horospheres are convex, and strictly convex if the curvature of V

IS

boundoo from above by a negative constant. Proof: H+(y,O) is the limit of the balls passing through 0 and the center of

which goes to infinity along y, and these balls are convex (see A 21.2). LEMMA

A21.12

Let H+(y, 0) and H+(y, 0') be two horospheres of y. If a (H+(y, 0) and a' ( H+(y, 0'), then la, a'i :: 10,0'1. Proof:

Assume la, a' I

< 10, 0' I. From (A 21. 9) we conclude that to each

corresponds a point a (t)

and a point a'(t)

f

~ (y (t),

f

~ir(t),

0) such that:

lim a(t) ~ a, t ... + 00 0') such that: lim

t ... + 00

a'(t)

=

a'.

Thus, for t large enough, we have: la(t),a'(t)1

< 10,0'1.

To fix the ideas assume that the point 0' lies between the points 0 and y (t). We obtain the following contradiction: laCt), y(t)1 ::; laCt), a'(t)1 + la'(t), y(t)1 ~ 10,y(t)1

LEMMA

=

< 10,0'1 +

la(t),y(t)I·

la'(t), y(t)1

(Q. E. D.)

A21.13

Two positive horospheres H+(y, 0) and H+(y, 0 ') cut off an arc of length 10,0' I on every positive asymptote to y.

184


Proof:

Y(a',Uo,t) -----+--------~------------~-----------Y ·0

FIgure A 21.14

Let y(a', u, t) be a positive asymptote to y that intersects H+(y, 0') at a '. The points y (t) and a' define a geodesic on which we select a point a(t) such that la(t), a'i = -L(a'; y,O) = 10,0'1 and a' lies between a (t) and y (t) (see Figure A 21.14). Since the exponential mappinl Exp a' is continuous, we obtain: . a(t) = a

lim

f

y(a',u',t) and la,a'i

-L(a'; y,O).

We deduce: I la,y(t)I-IO,y(t)1 I :; Ila,a(t)1 + la(t),y(t)I-IO,y(t)11 =

I Ia, a (t) I + Ia', y (t ) I -I

°"

y (t) II

->

0 as t ... + 00 •

Thus, a ( H+(y, 0). (Q. E. D.) THEOREM A

21.15

The positive asymptotes to yare the orthogonal trajectories of the positive horospheres of y. Proof: DireCt consequence of (A 21.12) and (A 21.13). Finally, observe that negative horospheres H-(y, O) can be defined as above from the negative asymptotes (t

->

-00).

185

APPENDIX 21

§ D. The Horospheres of Tl V

The unitary tangent bundle of V is denoted by Tl V and p: Tl V

-+

V

is the canonical projection. Let u be a point of Tl V; u defines a geodesic y (pu, u, t)

= y (u, t)

y (t) .the lift of which, in Tl V, is denoted again by y (t). From § B we know there exist two horospheres H +(y, pu)

=

H +(u) and H -(y, pu) =

H -(u) passing through pu. The set of the unitary vectors orthogonal to H+(u) (resp. H-(u)) along H+(u) (resp. H-(u)) and oriented like u is a

(dim V-I)-dimensional submanifold }(+(u) (resp. }(-(u)) of Tl V. The }{'s are called the horospheres of Tl V. THEOREM

A 21.16

(1) The y(u, t)'s and the }{+(u)'s, }{-(u)'s are the sheets of three

foliations of Tl v. (2) At each point u ( Tl V these foliations are transverse, that is:

T(T1 V)u where

X:

(resp.

X;;,

=

X;

E9

X;; Zu

Zu) is the tangent space of }(+(u) (resp. }(-~u),

y (u, t) ) at u.

(3) These foliations are invariant under the geodesic flow ¢t:

Proof: (1) Follows from the very construction of the sheets. (2) Follows from the strict convexity of H + (resp. H -, see A 21.11). (3) Follows from Theorem (A 21.15). The invariance of the foliations reduces the study of the differential

¢; to the study of its restriction to }(+(u) (resp. }(-(u) ) and y(u). we assume definitively that V is the universal covering

Iv

Now,

of a compact

. Riemannian manifold W of negative curvature. In particular, 'the curvature of V is bounded from above by a negative constant _ k 2 .

186


LEMMA

A21.17

Let rs(t) be a one-parameter family (s> 0) of numerical, C"-differentiable functions. Assume that: 2 r. s > - k ·r s

(k = constant

> 0)

for every s,t 2: 0, and rs(O) > 0, rs(s) = O. Then:

; (;1

~

a'ektll(ll,

The positive constants

IlcP~t(11 ~a.ektll(11 if(lX:,

11¢>~lll S

a and

b

b'e-ktll(11

are independent of

if ( l X~ t and

~,

and

!I

denotes the length of a vector of Tl V equipped with its natural Riemannian metric. Proof: We p.rove the first inequality, the others can be proved in the same way. Let y (0, u, t)

=

y (t)

=

y be a geodesic of V, and let x be a point of

H+(y,O), close enough to O. There is a well-defined geodesic y s(x, us' t) = y s(t) passing through x and y (s)

the Riemannian distance of y(t) and

l

y. Our first purpos"e is to compute

ys(t), regarded as ~lements of

Tl V.

Let r s(t) be the Riemannian distance of their projections y(t) and y s(t) on V. To compute r s(t) '·we consider a Jacobi field 2 1/1 (t) -along y, 'that is orthogonal to y and vanishes for t

=

s. By definition:

where R ( , ) is the curvature tensor and V is the covariant derivative along y. By definition the sectional curvature in the two-plane (y,l/I) is:

< R (y, I/I)y, 1/1 > :11/1 112 We know that p(y,l/I)

On the other hand,

2 See

J.

Milnor

[1].

S

_k 2 , consequently:

188


V

=

YlV 2 \11/I112

=

Yl d2211 1/1112 , dt

I\VI/I 112 ? (~ 111/1 11)2 Therefore, the length ls(t). of I/I(t) verifies:

that is

rs

> k 2 • I s , and I s (0) > 0, -

ls(s)

=

0 .

Lemma (A 21.17) and the classical possibility to select the Jacobi field

1/1 such that: r s(t) = I s(t) + 0(1)

if x is close enough to y imply: r s(t) < r s(O) •

(A21.19)

cosh[k(s-t)] , for 0 ::;, t ::;, s . cosh (ks)

Now it is readily seen that the angle of y and y s at y (s) converges zero as s

->

+ 00. Thus,

i)s)

->

0 as s

->

~o

+ 00, and Lemma (A 21.17) im-

plies again:

I; s(t)1

k· r (0) sinh [k (s - t)] s

, for 0 ::;, t ::;, s .

sinh (ks)

+00, ys(t) converges to a point y'(t) of the positive asymptote

y'(x, u', t) to y, and

ys(t)

converges to V(t). If r(t) denotes the dis-

tance of y (t) to y '(t), then the inequalities (A 21.19) and (A 21.20) imply (s

->

+(0):

IHOI < kr(O) e- kt ,

for t > 0 .

189

APPENDIX 21

Thus, the Riemannian distance of y(t), V(t)

Tl V verifies:

f

We easily deduce th.e first inequality of Theorem (A 21.18).

.

.

Due to this theorem, the sheets J{+(u) (resp. J{ -(u) ) are called the ~

"contracting" (resp. "dilating") sheets of Tl V.

§ E. Proof of the Lobatchewsky-Hadamard theorem 3 THEOREM

A 21.21

Let W be a compact, connected Riemannian manifold of negative curvature, then the geodesic flow on Tl W is a C-flow. Proof: Let V =

IV

be the universal covering of W equipped with the inverse

image of the Riemannian metric of W under the canonical projection":

IV . . W. V

satisfies the assumptions of preceding sections. Thus the geo-

desic flow on Tl V verifies the conditions of C-flows: condition (0) is trivially fulfilled; condition (1) follows from Theorem (A 21.16); condition (2) follows from Theorem (A 21.18). We finish the proof by proving that" is compatible with the three foliations of V

=

IV

and Tl IV. The first ho-

motopy group "l (W) is isomorphic to a group of automorphisms of

IV,

for

W is connected. The group "l(W) acts also as a group of automorphisms

of Tl IV: if u', J{ ±.(u

U" f

Tl

IV

are congruent mod" 1 (W), then J{ ±.(u ') and

") are themselves congruent mod" 1 (W).

REMARK

A 21.22

The horospheres of a compact, n-dimensional manifold Ware diffeomorphic to Rn-l. In fact, let us consider the horosphere J{ +.

It is a

See J. Hadamard [l]. Proofs of SectlOns Band C are mainly due to H. Busemann: Metric Methods in Finsler Spaces and Geometry. Ann. Math. Study. No.8.

3

Princeton University Press.

190


paracompact manifold. Let S be a compact subset of J{+. Then 1>tS is covered by a disk D of

1> t J{ +

(take t large enough). The counterimage

1>; 1D i~

a disk which covers S in J{ +. Therefore, J{ + is diffeomorphic

to Rn -

according to the following lemma of Brown (Proc. Amer. Math.

1,

Soc., 12(1961), 812-814) and Stallings (Proc. Cambridge Philos. Soc., 58 (1962), 481-488): Let M be a paracompact manifold such that every compact subset is contained in an open set diffeomorphic to Euclidean space. Then M itself is diffeomorphic to Euclidean space. This result

~oes

not hold for noncompact manifolds.

Consider the

space I(x, y) I y > 0, x-(mod 1)1 endowed with the metric:

The Gaussian curvature is equal to - 1 and the universal covering space is the Lobatchewsky plane (see Appendix 20). The curve y horocycle homeomorphic to S1.

FIgure A 21.23

=

1 is an

APPENnIX 22 PROOF OF THE SINAI THEOREM (See Section 15, Chapter 3) Let (M, ¢) be a C-diffeomorphism and X m {resp. Ym ) the k-dimensional dilating space at m Crespo the I-dimensional contracting space). A Rie-

mannian metric is definitively selected on M. Thus, Xm and Ym are Euclidean subspaces of TMm' THE METRIC SPACE OF THE FIELDS OF TANGENT k-PLANES

A22.1 The tangent space TMm is the direct sum Xm

III

Ym . Therefore, the

equation of a k-plane V m C TM m' transverse to fm' is:

where x ( Xm , y (fm , and P(Um): X m ... fm is a linear mapping. We define a metric in accordance with the norm of the linear mappings P(V): if V m and U:n are two k-planes of TM m ' then we set:

IVm -V'[ m

=

IIP(U m )-P(U')II m

=

sup X(Xm

.lx[

=

dim M).

1,2) is the direct sum of two subspaces Xj and Yj:

Xj

III

lj,

dim Xj

=

k,

dim Y.I

=

I:

R2 be a linear mapping such that: X2 ,

AX I

{

Y2 ,

Ixl

for x ( Xl

IIAyl < alxl

for y ( YI

IIAxl

(A 22.3)

AYI =

~ fL

where fL and a are constants. Let us denote by

Ci'

the operator induced by A, which makes corre-

spond to the k-planes of RI the k-planes of R 2 . If V and V' are transverse to YI , then:

Proof: By definition:

lCi'v - Ci'v'l

=

sup IP(Ci'U)x - P(Ci'v')x1

Ixl < I

X(X 2

=

sup

Ixl

IA[P(U)A-Ix] - A'[P(U')A-Ix]1

(x,

195

y, z, X, Y, Z) = (Ax, p.y, vz,

AX, ii Y, i7 Z)

,

where:

A

1> 1>

=

2 + V3,

v = (2-V3)2,

P. = Av = 2-y3,

is an automorphism of G, because p. = Av. Therefore, ¢;f

f, and

defines a diffeomorphism ¢ of M by:

(M, ¢) is a C-Diffeomorphismo

An element of the Lie algebra TG e of G is of the form

C:iO;g) The metric

of TG e defines a right mvariant metric on G and, consequently, a Riemannian metric on M = Golf. The Lie algebra TG e splits into the sum X + Y, where the elements of X (respo Y) are of the form:

(respo)

Next, by right translations, the splittmg TG ~ of every point

IS

imposed on the tangent space

g of G:

TG~ = X~ + ~ Thus, the tangent space TMm at m ( M splits into: ™m

= Xm+ Ym

It is ea,sily checked that the linear tangent mapping" d9 and contracting on Ymo

IS

dilating on X m

APPENDIX 24

SMALE '5 EXAMPLE (See Section 16, Chapter 3) Smale [2] proved the following theorem, which gives a negative answer to the "problem of structural stability": are the structurally stable diffeomorphisms dense in the C 1-topology? THEOREM

A24.1

There exists a diffeomorphism phism

1/1',

C 1-c1ose to

1/1,

1/1:

T3 ... T3 such that no diffeomor-

is structurally stable.

We tum to the construction of

1/1.

§ A. The Auxiliary Diffeomorphism ¢> Let T2 be the torus I(x, y) mod 1\. We define a diffeomorphism ¢>1 of

T2 x Iz 1-1 ~ z ~ 1\ onto itself by setting:

{

¢>. 1·

(p . . q P (P

(mod 1)

z ... ~z

Let By, be the ball (Figure A 24.2) of T2 x R with center (0, 0, 2) and ra-

dius

1/2: x2

We defioe a diffeomo",hism

+;+ (z_2)2

~; ¢>1:

{Of

:v: i~~

~ ~ .

T2 x Iz 1

y ... ~y

z ... 2z-2 . 196

°~ z ~

3\ by setting

APPENDIX 24

197

Now, the torus T3 is T2 x S1, where S1 is [-3,3] with endpoints identified.

z

x: dilating direction Y: contracting direction

o

t

t Figure A 24.2

The following lemma is easily proved.

x

198


LEMMA

A24.3

There exists a diffeomorphism ¢: T3 ... T3 such that: (1) its restriction to T2 x Iz 1-1 S; z S; 1\ is ¢l'(2) its restriction to By, is

¢i ,-

(3) ¢ leaves {(O, 0, z) 10 < z S; 2\ invariant with no fixed point. PROPERTIES OF

¢. A 24.4

T2 x 10 \ is obviously an invariant torus under ¢. The restriction of ¢ (or ¢l) to T 2 x 10\ is nothing but the diffeomorphism of Example (13.1):

(~)

(A 24.5)

. . 0 0(;)

(mod 1).

Let us recall some properties of this diffeomorphism: There exist two foliations

X and

~ on T2 x 10\. They correspond, respectively, to the dilat-

ing and the contracting eigenspaces Xm and Ym of (A 24.5). Every sheet of

X (or

~) is everywhere dense in T2 x 10 \. The periodic points 1 of ¢

are dense in T2 x 10\. This fact can be proved by observing that every rational point (pi q, p 'I q) ( T2 is periodic. Now we pass to the diffeomorphism ¢: T3 ... that the periodic points of

cp,

in T2 x I z

I-

those of (A 24.5), as do also the foliations

1 S; z

r. s:.

It is easy to see

1 \, coincide with

and ~ in T2 x 10\. The foli-

X

ation ~ generates an invariant contracting foliation of T2 x 1z S; 1\, whose sheets are the "planes" of the form Y x Izl-1

I-

1 S; z

s:. z S; 11,

where

Y is some sheet of ~. § B. The Diffeomorphism ifJ

The diffeomorphism ifJ is obtained by perturbing. cp_ Let Go be the ball of T3 with radius d and center (0, 0, %): x 2 + ; + (z _ %)2 We set G =

d2 .

cp-l Go' ¢ tx, y, z) = (x', y: z '),

1 That is, the points'; ( integer N.

s:.

T2x

10\

such that

¢N';

and we observe that d can

= .; for some nonvanishing

199

APPENDIX 24

be chosen small enough for ¢G

nG =0 .

We define. our desired diffeomorphism tjJ(x, y, z)

=

~

¢ (x, y, z)

r/J by setting: = (x', y', z ')

outside G

(x' + 1/1I>(x, y, z), y', z ') on G,

where II> is a nonnegative COO function with compact support in G and nondegenerate maximum value + 1 at ¢-1(0, 0, %), and finally 1/

>

°

is

small enough so that tjJ is a diffeomorphism. PROPERTIES OF tjJ. A24.6

" z)\ Now the curve tjJI(O,.O,

°s.

z

This bump lies in the region T2 x I z ¢

=

S. 21 has a bump B (see Figure A24.2). \-1 $. z S. 11 where tjJ coincides with

¢l' This region is foliated into contracting planes (see A 24. 4). Let

~xla) + (ylb) =

1 be the equation of such a contracting plane in the chart

(x, y, z). Among these planes intersecting the bump B, we select the plane ~, for which a is maximum (see Figure A 24.2). Either j= contains a peri-

odic point of tjJ, or it does not. In the first case, the bump is called periodic and in the second case, nonperiodic. LEMMA A24.7 tjJ is not structurally stable.

This follows from two remarks: (1) An arbitrarily small change of 1/ in the definition of tjJ gives

rise to a diffeomorphism tjJ" arbitrarily C1-close to tjJ and similar to tjJ. The density of the periodic points (see A 24.4) implies that we can suppose the bump of tjJ is periodic and the bump of tjJ" is nonperiodic, and vice versa. (2) If tjJ and tjJ" are in the opposite cases there is no homeomorphism h: T3

->

T3 close to the identity" such that tjJ".

h establishes a one-to-one correspondence planes, and periodic points of tjJ and tjJ ".

b~tween

h=

h· tjJ. In fact,

bumps, contracting

200


LEMMA

A24.8

Every diffeomorphism

!/J',

Ci·close to

!/J,

possesses an invariant tom.

similar to T2 x 101, a bump, a contracting sheet similar to

1,

and so on:

Complete. proof of this lemma is announced by Smale [2]. Now Theorem (A 24.1) follows readily from Lemma (A 24.7) and (A 24.8).

APPENDIX 25

PROOF-OF THE LEMMAS OF THE ANOSOV THEOREM (See Section 16, Chapter 3) Lemma A

Let (M, ¢) be a C-diffeomorphism. We select definitively a Riemannian metric on M. Since M is compact, there exists a number d > 0 such that, whatever be the ball B (p ; d) C TM with radius d and center p ( M, . p the restriction Exppl B(p; d)

of the exponential mapping at p is a diffeomorphism. Let l¢nm I n ( Z! be an orbit of ¢' A chart of a neighborhood of this orbit is (B, 1jJ-1), where B is the sum of the balls B

n

= B(¢nm, d)

C TM,J..n o..p

m.

and the restriction IjJ I-B is Exp ,J..n' . Let us denote by X the dilating n "fJ m n k-space X (¢n m) of TM¢nm' and by Yn the contracting I-space y(¢nm), Tke invariant dilating and contracting foliations rand new foliations on B:

Finally, ¢ induces a mapping:

201

Y

of M induce

202


such that the restriction ¢1 I

Bn

maps B

n

into B

n+

l' with obvious restric.

tions concerning the range of ¢1' Assume d small enough, then the sheets of the foliations

X1

and '!:J 1 can be regarded as sheets of the Euclidean

space XnED Yn of origin 0 = ¢nm, In which their equations are, respectively: y

= y(O)

+ in(x, y(O)) and

x

= x(O)

+ gn(Y' x(O))

where x ( X n , Y ( Yn and where in' gn' and their first derivatives can bl: made arbitrarily small by a suitable choice of d. a n of '!:J n which passes through the center 0 of

Consider the sheet

The mapping:

e: whose restriction

eIY

n:

I Yn I n Yn

->

( ZI an

->

I an In (

is defined for

ZI

,

y ( Yn n B n

by

is a diffeomorphism. Therefore, y ( Yn can be regarded as coordinates on

Figure A2S.1

203

APPENDIX 25

The diffeomorphism ¢1 maps an into a n+ 1 (see Figure A 25.1). the coordinates y, this defines a mapping ¢2:

In

and

ASSERTION A 25.2 It follows from the very definition of C-systems that the restriction

is contracting:

(A 25.3) where

e is a constant.

REMARK A 25.4 To be precise, (A25.3) holds for a certain iteration ¢~ of ¢2: we must "kill" the constant b in the definition of C-systems. For simplicity, we assume that (A 25.3) already holds for v = 1. Now let ¢' be a diffeo· morphism C 2.close to ¢' Then ¢' is a C-diffeomorphism (Sinai theorem, Section 15) and the foliations :t~ = 1j1- 1:t.' 'lJ; =rljl-1'lJ' and the mapping ¢; induced by ¢': ¢;: B ... B.

¢; = 1j1-1¢'1jI,

¢;

IBn:

Bn ... B n+ 1 ,

are defined as above. If ¢' is C 2·close enough to ¢. then the sheets of

:t~ are close to those of :t1 and transverse to the sheet an' Therefore, there exists a projection IT:

which makes correspond to each point a

l

Bn' the intersection ITa of an with

the sheet of:t; passing through a (see Fig~re A 25.5). Now let us consider the mapping (Figure A 25.5):

¢;

=

e- 1 IT¢;e,

204


cp'

1

Figure A 25.5

ASSERTION A25.6

If ¢' is C 2-close enough to ¢, If

¢;. is C-c1ose to ¢2: to any

> 0 corresponds a positive 0 su~: Lnat 11¢;'y-¢:yll

Yn1

X n can be regarded as coordinates on Yn1.

The diffeomorphism ¢1 maps Yn1 into Y;+1 ~ In the coordinates x, this defines a mapping: ¢3 = ~-11:D:

lXnln

Obviously ¢3 (0) = 0,

l

Z!

->

lXnln

l

ZI,

206


ASSERTION A 25.11 It follows from the very definition of C-systems that ¢31x : X n

n

~

Xn+ J

is dilating: (A25.12)

REMARK

A 25.13

In fact, (A 25.12) holds for a certain iteration of ¢3. For simplicity we assume that (A 25.12) already holds for ¢3. Now let ¢' be a diffeo-

f3 n =

morphism C 2 -close to ¢. Consider the sheet foliation

¢'In

f3

C Bn

X; = rjJ-J'X' which passes through e¢';(O) (n::. 0).

of the

(See Figure

A25.14.) According to Lemma A, this sheet is close to the center 0 of Bn.

Let y = hn(x), (x ( X n ), be the equation of

enough to ¢' then we can choose x ( Xn

n Bn

f3 n .

If ¢' is close

as local coordinates of

f3 n :

the mapping E

which is defined by x ~ (x, hn(x)) for x ( Xn

Ixn :

1

..

FIgure A 25.14

f3 n '

n Bn is a diffeomorphism. Yn+l

q>'

Xn ~

207

APPENDIX 25

From the very construction of the (3n's, we see that ¢;

maps (3n into

(3n+1' Therefore, this defines a diffeomorphism:

ASSERTION A 25.15

If ¢ and ¢' are C 2 -close enough, then ¢3 and ¢; are C 1-close: To any ce > 0 corresponds a positive 0 such that 11¢-¢'ll c 2 < 0 implies:

(A25.16)

n 8 n , n ~ O. This is a direct consequence of the construction of the Yn , (3n (see Sinai theorem, Section 15), and of the

for any x, xl' x2 ( Xn

fact that the (3n's are C 1-close to the Yn's.

LEMMA B. A25.17 II ¢' is C 2 -close enough to ¢' then there exists a well-defined sheet

o(

'Y'

such that ¢,n

o is close to

¢nm for any n ~ O. To be precise,

there exists one and only one point Xo ( Xo such that 11¢;n xoll

R, T IR n : R n

->

~

O.

Rn+ 1 be diffeomorphisms such that:

(1) K(O) = 0,

IIK(x) -K(y)11 > 0 Ilx - y II,

(2) IlL II $. E,

IIL(x) - L(y)11
1, 0 - E > 1,

for any x, y (R. Then, there exists one and only one point x ( Ro such that' the sequence Tnx is bounded, and (A 25.19)

IiTnx II $. _ E _ 0-1

for any n ~ O.

208


Proof:

The mapping T-11 R n : R n

->

R n- 1 (n > 1) is obviously a diffeomor-

phism. On the other hand: II (Kx - Ky) + (Lx - Ly) II

ilTx- Tyll

2:. IIKx-Kyll-IILx-Lyll 2:. (8- E)llx-yll; therefore: (A 25.20) Let bn(c) be the ball Ilx II ::; c of Rn' Since IITxl1 = IIKx + Lxii 2:. IIKxll-IILxll > 811xll- E , we have: (A25.21) As:;ume c large enough, that is: 8c - E 2:. c.

(A25.22)

Then (A25.21) implies Tbn(c) ) bn+1(c), therefore T-1bn+1(c) C bn(c) ,

consequently T-1b1(c) ) T- 2 b 2(c) ) •.. ) T-nbn(c) ) ....

But, according to (A 25 .20), we have: diam~er

T-nb n (c) ::; 2c(8'- E)-n

->

0 as n

->

+

00

•

Therefore nn> 0 T-nbn(C) reduces to a unique point x (bo(c)'

This

finishes the proof if one observes that c = E/(8 -1) verifies (A 25.22) . . Proof of Lemma B. A2S.23

According to (A 25.11) and (A 25.15), the mapping ¢; verifies the conditions of the preceding lemma. It is sufficient to set K

=

¢3' L

=

¢~ - ¢3' and change E into ce in condition (2). If we take ce = E(8 -1)

in (A25.19), we obtain 11¢;nxoll < E. This proves Lemma B.

209

APPENDIX 25

To summarize, we found a contracting sheet 0 (

'l:J' which remains

close to the orbit ¢nm for n ::: 0 (in the sense of Lemma 8). If ¢' is close enough to ¢'

°

is close to ¢nm, even for n

< O. To prove it, it

is sufficient to apply Lemma A t~ ¢ -1. The foliation

'!:I'

is the dilating

foliation of ¢,-1 and the sheet 0 is close to m. Consequently, according to Remark (A2S.10), the sheets ¢mo, (n

< 0) stay in the neighborhood

of the orbit ¢nm (in the sense of Lemma A):

Therefore, Lemmas A and 8 imply the following assertion: ASSERTION A2S.24 If ¢' is C 2-close enough to ¢' then there exists a sheet ;5 C Bo of

the foliation '!:I~ such that the sheets

¢t 8 C Bn'

(-00

< n < 00), stay

inside an E-neighborhood of the center of Bn. Using the same argument for ¢,-1, we find a sheet

f3 8

erties. Since the sheets

C Bo of the foliation ~; with similar prop-

and

fj are transverse in B o' there exists one 8 n fj in an E-neighborhood of the

and only one point of intersection z =

center of Bo. The desired homeomorphism k of the Anosov theorem is defined by setting k (m)

=

y,z. One easily checks that all the preceding con-

structions depend continuously on m. This proves that k is an homeomor: phism. The relation ¢'k to the identity.

= k¢

is obvious, as is the fact that k is E-close

APPENDIX 26

INTEGRABLE SYSTEMS (See Section 19, Chapter 4)

J. Liouville

.

(A26.1)

p

proved that if, in the system with n degrees of freepom:

aH aq

= --,

q "" ap aH ,

p = (Pl" .. , p ), q = (ql'···' q ), n

n

n first integrals in involution 1

(A26.2) are known, then the system is integrable by quadratures. Many examples of integrable problems of classical mechanics are known. In all these examples the integrals (A 26.2) can be found. It was pointed out long ago that, in these examples, the manifolds specified by the equations F,

f; "" constant turn out to be tori, and motion along

=

them is quasi-periodic (compare with Example 1.2). We shall prove that such a situation is unavoidable in any problem admitting single-valued integrals (A 26.2). The proof is based on simple topological arguments. THEOREM

A 26.3

Assume that the equations F;

=

f;

=

constant, i

=

1, ... , n, define an

n-dimensional compact, connected manifold M = Mf such that: (1) at each point of M the gradients grad F; (i

=

1, ... , n) are linearly

independent; 1 Two funchons F(p, q) and G(p, q) are in involution if their Poisson bracket vanishes idenhcally: (F, G) = aF aG ap aq

_ aF aG aq ap

210

== 0 •

211

APPENDIX 26

(2) a Jacobian Det lal/af I, which is defined below (A 26.7) does not

vanish identicaIIy. Then: (1) M is an n-dimensional torus and the neighborhood of M is the direct product Tn

Rn;

X

(2) this neighborhood admits action-angle coordinates (I, ¢», (I ( B n C Rn , ¢> (mod 2") ( Tn), such that the mapping I, ¢> ... p, q is canonical 2 and Fi

=

F/I).

Thus, Equations (A 26.1) may be written: I

=

1>

0,

=

w (I), where w (I)

=

aH , aI

and the motion on M is quasiperiodic since H = Fl = H(I) and Equations (A 26.1), in action-angle coordinates, are Hamiltonian equations 2 with corresponding Hamiltonian function H(J).

Proof: NOTATIONS A

26.4

,

We use the following notations. Let x

= (p, q)

be a point of the phase

space R2n; we shall denote by grad F the vector gradient F

Xl

, ... , F

x2n

of

a function F(x). The Hamiltonian equations (A 26.1) then take the form: (A26.5)

x=

1=(O-E) E-O

I grad H,

where E is the unit matrix of order n. We introduce in R2n the skewscalar product of two vectors x, y:

[x, y] = (Ix, y) = -[y, x], where ( , ) is the usual scalar product. As can be easily verified, [x, y] expresses the sum of the areas of the projections of the parallelogram with sides x, y onto the coordinate planes Piqi (i = 1, ... , n). Linear transformations S, which preserve the skew-scalar product [Sx, Sy}

--------------------See Appendix 32.

=

[x,

Jd

for all x, y,

212


are called symplectic. For instance. the transformation with matrix I is symplectic. The skew-scalar product of the gradients [grad F, grad G] is called the Poisson bracket (F, G) of the functions F, G. Obviously. F is a first integral of the system (A 26.5) if and only if its Poisson bracket (F, H) with the Hamiltonian vanishes identically. If the Poisson bracket

of two functions vanishes identically. the functions are said to be in invo-

lution. THE CONSTRUCTION. A 26.6

Consider the n vector fields: ~,~ I grad ~, (i ~ 1 •...• n). On account of the nondegeneracy of I and the linear independence of the grad F;'s. the vectors

~;

are linearly independent at each point of M.

Let us consider the system (A 26.5) with Hamiltonian F;. Since (F;, F.) ~ O. all the functions F. are first integrals. and every orbit lies wholJ

J

lyon M. Therefore the velocity field Finally. the fields

fJ

and

f,

~; ~ I

grad F, is tangent to M.

commute, for their Lie bracket is nothin~

but 3 the velocity field of the system (A 26.5) with Hamiltonian (F., F.) ~ O. , J Thus, M is a connected. compact orbit of the group Rn acting smooth· ly and transitively; therefore we proved that M ~ Tn. Besides. M being specified by the equations F;

~

f; ~ constant. the fields grad F; define a

structure of direct product in the neighborhood of M. Now. let us choose the torus Mf: F

~

f in the neighborhood of M and

consider the n integrals (A26.7) over the basic cycles y /f) of the torus Mf" Since the l; ->

p, q, which defines action-angle coordi-

nates: p

(A26.9)

LEMMA

as

as oq

af

A26.10

The one-form pdq of M([) is closed. Proof:

It is sufficient to prove that the integral of pdq along infinitely small

parallelograms lying in M([) vanishes. If D is a parallelogram with sides ~,.,." then ~ pdq (i.e., the sum of the areas of the projections of D onto

the coordinate planes P; q;, i = 1, ... , n) is the skew-product [~,.,.,] of ~ and .,.,. Suppose now that ~ and .,., touch M([) at a certain point. In accordance with (A 26.6) any vector tangential to M ([) is a linear combination of the n vectors f grad F;. But these vectors are skew-orthogonal since, in accordance with (A 26.2), [grad F , grad F.]

0 ,

J

1

and thus, since f is symplectic,

[! grad F;, f grad Fj J

=

O.

Therefore [~,.,.,] ~O, as required. The integral (A 26.8) can therefore be regarded as a many-valued function S and Equations (A 26.9) define, locally, a canonical transformation f, ACTION-ANGLE VARIABLES A

1>

->

p, q.

26.11

In fact formulas (A 26.9) define a global canonical mapping in which p and q have period 211 with respect to

1>.

To prove it, we observe that, for

214


every I, the differential of S([, q) is a global one-form on M([). fore d¢, as defined by (A 26.9), is also a global

There-

one~form.

Let us compute the periods of the one-forms d¢, over the basic cycles of the torus Mf' According to (A 26.7) we have:

f

Yi

d¢i

=

f d(aS) al. Yj'

=

~ d/.

f

'Yj

dS

=

~(2771.) = 2770 .. d / . } · '} ,

Therefore the variables ¢, are angular coordinates on the torus M(l) and our theorem is proved.

APPENDIX 27

SYMPLECTIC LINEAR MAPPINGS OF PLANE (See Section 20, Chapter 4) Let A be a symplectic linear mapping of the plane (p, q). The mapping A preserves the area-element dp 1\ dq, therefore Det A

=

1. Consequent-

ly, the product of the proper values Al and A2 of A is equal to 1. Besides, Al and A2 are roots of the real polynomial Det (A -AE). Therefore, either Al and A2 are both real, or they are complex conjugate: A2 = Xl •. In the first case, we have:

(A27.1)

p

q

Figure A 27.3

215

216


In the second case, we have: (A 27.2)

and the roots belong to the unit circle (see Figure A 27.3). The third and last possible proper value coniiAuration is:

q

p

hyperbolic rotation

p

hyperbolic rotation with reflection

Figure A27.5

217

APPENDIX 27 EXAMPLE

A27.4

The hyperbolic rotation: p, q

2p, Yz q ,

-+

or the hyperbolic rotation with reElection: p, q In both cases the orbit Tnx of x

2p, - Yz q (see Figure A 27.5).

-+ -

(p, q) belongs to the hyperbola pq

=

=

constant. Of course, the fixed point 0 is unstable. From classical theorems of linear algebra it follows that every mapping A of the first type (Ai of, A2 ; Ai' A2 ( R) is an hyperbolic rotafion, possibly with reflection. This means that, up to a suitable change of variables, A may be written

under the form: P, Q

EXAMPLE

->

1 Q• AP, X

A27.6

A r9tation through an angle a belongs to the second class (Ai A2 = ei~:

p, q

-+

p cos a- q sin a, p sin a + q cos a .

q

x p

Figure A27.7

=

e- ia,

218


This rotation transforms into an "elliptic rotation" (see Figure A27.7) under a linear change of ,,:ariables. In this case the orbit r'x of x = (p, q) belongs to an ellipse centered at O. The fixed point 0 is obviously stable. Classical theorems of linear algebra show that every mapping A of the second type (1'\11 = 1'\21 = 1, '\1';' '\2) is an elliptic rotation. In the first case (A 27.1), the fixed point 0 is called an hyperbolic point and one says that A is hyperbolic at O. In the second case (A 27.2)

the fixed point 0 is called an eIIiptic point and one says that A is eIIiptic at O. Finally, the third case (,\ 2 = 1) is called the parabolic case. REMARK

A27.8

Every canonical mapping A ~ close enough to an eIIiptic mapping A, is eIIiptic. In fact, the roots '\1 and '\2 depend continuously on A and are

restricted to lie either on the real axis or on the unit circle (see Figure A 27.3). Therefore, these roots cannot leave the unit circle, except at points ,\ = :!::. 1 which correspond to the parabolic case. Finally, we define the topological index of a vector field at a fixed point. Let us consider a vector field ';(x) of the plane p, q, with an isolated fixed point ';(0) This defines a mapping of the unit circle x2

= p2

+

l

=

O.

= 1 onto itself:

If E is small enough, then the topological degree of this mapping does not depend on E and is called the index of .; at 0, or the index of O. Now, consider the vector field ';(x) = Ax - x.

If the mapping 1\ is

nonparabolic, then 0 is an isolated fixed point of ';(x). THEOREM

A27.9

An eIIiptic point, or an hyperbolic point with reflection, has index + 1. An hyperbolic point has index - 1.

Proof consists in a mere inspection of Figures (A27.S) and CA27.7).

APPENDIX 28

STABILITY OF THE FIXED POINTS (See Section 20, Chapter 4) Consider an analytical canonical mapping A of the plane p, q, with

(0,0). Assume that 0 is elliptic, that is, that the differential of A at zero has proper values A1 = e- 1a , A2 ~ e ia . It has been fixed point 0

=

known since G. D. Birkhoff's time 1 that, if al2rr is irrational, to every s > 0 corresponds a canonical mapping B = B(s) of a neighborhood of 0

B: p, q

->

P, Q,

B (0)

=

0,

which reduces A to a "normal form":

that is as follows. Let I, ¢ be the canonical polar coordinates: 21 = p2 + Q2, 2/'= p'2 + Q'2,

¢

=

arctg(P/Q)

¢'

=

arctg(P'/Q')

then: I'-I

(A28.1)

¢'-¢

=

The coefficients a, a 1 ,

= O(ls+1)

a+

a1 i+ ai 2 + .•. + a/ s +

...

do not depend on the mapping R (~) by which A

is reduced to the form A '. If a

~ 2rrm/n

0([S+1).

and if there is a nonvanishing

coefficient a 1 , a 2 , ... , Birkhoff says that A is of "generic elliptic type." 1 Dynamical Systems, Chapter 3.

719

220


THEOREM A 28.2

(See Arnold [7]).

The fixed point of a

~eneric

elliptic mapping is stable.

The proof consists in applying the construction of Theorem (21.11) of Chapter 4 (see Appendix 34) to the mapping (A 28.1): for 1« I, 0(18+1) is regarded as a perturbation of the mapping l'

=

I,

Similar theorems are obtained concerning the stability of equilibrium positions and elliptic periodic solutions of Hamiltonian systems with two degrees of freedom (see Arnold [7]).

J. Moser [1]

obtained the strongest re-

sult in that way: MOSER'S THEOREM A28.3

The fixed point of an elliptic canonical mapping A of the plane is stable provided that: (1) a f, 217

j!- ,

217 :

(2) a 1 f, 0 ;

(3) A is C333 -differentiable. (As it is pointed out in a recent paper of Moser [6], this number of derivatives can be fairly well reduced.) A complete proof will be found in

J. Moser [1].

REMARK A 28.4

If a

=

Civita [1].

217m/3, th~n the fixed point can be unstable, as shown by Levi-

APPENDIX 29

PARAMETRIC RESONANCES (See Section 20, Chapter 4) The analysis of the stability of the fixed point (0,0) of a linear mapping of the plane is due to Poincare and Lyapounov. Only in recent times (1950), were these results extended by M. G. Krein [11. [2] to systems with many degrees of freedom. Krein's investigations have been enlarged by Jacoubovich [11. Gelfand and Lidskii [41. and so on. J. Moser [3] published a report of Krein's theorem. Let A be a linear symplectic mapping 1 of the canonical space R2n. We say that A is stable if the sequence An is bvunded. We say that A is parametrically stable if every symplectic mapping, close to A, is

st~ble.

We proved in Appendix 27 (and used it in Section 20, Chapter 4) that every elliptic mapping of R2 is parametrically stable. M. G. Krein displayed all the parametrically stable mappings of R2n. LEMMA A 29.1 (Poincare-Lyapounov) Suppose A is a symplectic mapping and ,\, is a proper value of A. Then 1/,\"

1

X, and I/X

are proper values of A.

A preserves the skew-scalar product

product and 1 =

(~=~), E

[e-,." 1 = (1

e-, .,,), where (

, ) is the inner

= unit matrix of order n. Therefore, we have:

[Ae-. A."l

=

[e-. ."l 221

and

A iA

= 1.

222

ERGODIC PROBLEMS OF CLASSICA.L MECHANICS

Prool: It is sufficient to prove that the characteristic polynomial of' A is real

and reciprocal. In fact, we have: p(..\)

= Det(A-..\E) =

Det(-lA,-1 1 + ..\p)

= Ded-A '-1 + ..\E) = Det(-A- 1 + ..\E) = Ded-E + ..\A) = ..\2n. Det (A_..\-1 E) = ..\2n. p(..\-I)

•

From thi~ lemma the following corollary is readily deduced; COROLLARY A29.2

The proper values of A divide into couples and "quadruples." Couples are lormed by ..\

~nd

..\-1, ..\ belonging to the real axis or the unit circle:

1..\1 = 1. Quadruples are formed by ..\,X,..\-I, and X-I (see Figure A 29.3) .

• 1.3

I

I I

I

\ I

\I

~.l

\I'

,.

'~'I

;\3

\ \

Flgure A 29.3

223

APPENDIX 29

COROLLAR Y

A 29.4

If the proper values are simple and lie on the unit circle IAI = 1, then A is parametricalIy stable, because if all the proper values are simple and lie on IAI

= 1,

then:

(1) A is stable (for obvious reasons of normal form); (2) all the proper values of a symplectic mapping A', close enough

to A, lie on IAI = 1. In fact, assume the contrary, then A' would have' two proper values A and X-I close to a unique isolated proper value of A. (see Figure A 29.3). Let us now assume definitively that :i 1 are not proper values of A. Krein classified the proper values belonging to the unit circle IAI = 1: they split into positive and negative proper values. First assume that all the proper values are simple; we prove the following lemma: LEMMA A29.5

Let ~1 and ~2 be the proper vectors with corresponding proper values

Al and A2 · Then, either AIA2

=

1, or [~1' ~2] = O.

Proof: Since A ~1

Al ~1 and A~2 = A2~2' we have:

[A~I' A~2]

=

Al A2 [~1' ~2]

=

[~1' ~2]

(Q. E. D.) COROLLAR y A

Let

(J

29.6

be a plane, invariant under A and correspollding to conjugate

proper values AI' A2, IAll = IA21 = 1. Then: (1) (J is skew-orthogonal to every proper vector ~3 corresponding to another proper value A3 ; (2) the skew-product [~,

1)]

of noncolinear vectors ~ and

1)

of

(J

IS

nonvanishing. Assertion (1) is a direct consequence of AIA3 ~. 1, A/\3 ~ 1: in view

224


of Lemma (A 29.5) we have [~l' ~3] We have also [~1' ~1]

= 0,

=

[~2' ~3] = O. Suppose [~1' ~2] = O.

and assertion (1) implies [~1' ~3]

= 0 for

every

~3' Therefore [~1' 71] = 0 for every 71, which is impossible. Consequent-

ly [~1' ~2] f, 0 and assertion (2) holds good. DEFINITION

A29.7

A proper value A, such that IAI

=

1, A2 f, 1, is called a positive (resp.

negative) proper value of A if: [A~,~] > 0 (resp.

< 0) for every

~ of the

real invariant plane a corresponding to the proper values A and X • This definition is correct. Indeed, the vectors

A~

and

~

of a are non-

co:·near because A2 f. O. Therefore, in view of Corollary (A 29.6), [A~,~]

f. 0 on

Consequently [A~,~] has constant sign for every ~ (a.

a.

REMARK A

\

29.8

The sign of a proper value lias a simple geometrical meaning. The plane a

admits a canonical orientation, for [~, 71] f. 0 if ~ is nonparallel to 71'

Therefore, oile may speak of positive (or negative) rotations. The restriction of A to a is an elliptic rotation through an angle a, 0

< lal < 7T. The

proper value A is positive (resp. negative) if A rotates a through a posi-

tive (resp. negative) angle. Krein's main result is: collision of two proper values with identical

signs on the unit circle IAI

=

1 does not provoke instability. In contrast,

two proper values with opposite signs can leave the unit circle after they have collided, so forming a "quadruple" with their conjugates (see Figure

A 29.3). To be p~ecise, let A(t) be a symplectic mapping -.yliich depends continuouslyon a parameter t, and the proper values of which are different from :. 1 if It I < A are simple collide for t THEOREM

an~

=

T.

Suppose that, for t < 0, all the proper values Ak of

lie on the unit circle, while certain of these proper values

O.

A 29.9

If all the proper values that collide have identical sign, then they re-

225

APPENDIX 29

main on the unit circle after the collision and the mappinl1 A remains sta-

ble for t < E, E > O. We shall prove the theorem in the simplest case in which all the proper values A, 1A > 0 collide. The general case can be reduced to this c'ase by selecting a canonical subspace R21(t) corresponding to th'e I colliding proper values and their conjugates. 1'0 fix the ideas, suppose that the proper values, Ak are positive:

[Ae-. e-] > 0 for e- ( Uk ' where Uk is the plane generated by e-k , ~k (Ae-k = .Ake-k).

Proof of the Theorem. A29.10 Consider the quadratic form [A e-. e-]; its polar bilinear form is nondegenerate. We have, indeed:

[Ae-. 1/] + [A1/. e-] Suppose [(A-A- 1 )e-. 1/]

(A 2 - E) Ae-

=

=

[Ae-. 1/] - [A -1 e-. 1/]

=

=

[(A - A-I) g. 1/] •

0 for every 1/, then (A-A- 1 )e-

=

0 and

O. Thus, 1 would be a propervalue of A2. which contr"dicts

the condition of Theorem (A 29.9) (A

del1enerate for It

I'k lie

on the unit circle

1.\1

=

1,

>.;

t

1. Besides, the

quadratic form [A~.~] is definite on every invariant subspace correspg.nd. ing ." the multiple proper values t.. k

,

Xk

.

APPENDIX 30

THE AVERAGING METHOD FOR PERIODIC SYSTEMS (See Section 22, Chapter 4) Let

n=

8 1 )( SI be the phase space, where 8 1 =

an open bounded subset of RI and SI sider ¢-periodic smooth functions

F:

n ...

RI,

f:

CI)

II =

([1' ... ,11)\ is

I¢ (mod 21T)} is a circle. We con-

=

(J), F (I, ¢), HI, ¢):

n ...

RI,

CI):

8 1 ... RI ,

and finally, E« I denotes a small parameter. THEOREM

A30.I

n:

We consider the systems defined in (A 30.2)

, ~ =

CI) ( [ )

1I

E· F ([, ¢)

=

+ Et([, ¢)

and

j

(A30.3)

= E·

F(j), where F(j)

=

---.L 21T

If

CI) ( [ )

,;,

0 in

n,

f

21TF (j, ¢) d¢ .

0

then the solutions I (t) and ] (t) of (A 30.2) and (A 30.3),

with equal initial data [(0) 1[(t)-](t)1

= ]

(0), satisfy 1 :

< C·E for every t, O:s: t ::: liE,

where C is a constant which does not depend on E,

1

We suppose that J(I) ( 8' for every t, 0::: t

227

:S

l/f .

228


Proof:

Let us improve (A 30.2) by using a new variable: (A 30.4)

P ~ P(I, ¢) ~ 1+ Eg(J, ¢),

From (A 30.2) and (A 30.4) follows: (A 30.5)

In order to cancel the terms of order E, we set: (A30.6)

g(I,¢) ~

I

¢ -

F(P)-F(P,¢) d¢; cu (P)

o

this expression is well-defined since cu(p) .;, 0,

f

2TT

(ii - F)d¢

=

0,

o

and therefore g(¢ + 2TT) ~ g(¢). Now, our system (A30.S) may be written: (A 30.7)

Let P(t) be the solution of (A 30.7) with initial data P(O) ~ J(O) (A30.8)

= /(O):

p(t) " P([(t), ¢(t».

Obviously, (A 30.7) implies: (A 30.9)

IP(t)-J(t)1

< Cl'

E

for every t, 0 $. t $. liE.

Finally, from (A 30.4), (A30.6) and (A 30.8) follows: (A30.10)

IP(t}-J(t}!

< C2 •

E

for every t.

Inequalities (A30.9) and (A 30.10) conclude the proof. They prove also that the motion decomposes into the averaged motion and fast small oscillations (see Figure A30.U).

229

APPENDIX 30

I (t )

"-="_- J(t)

t=o

Figure A30.1l

APPENDIX 31

SURFACES OF SECTION (See Section 21. 9, Chapter 4) Let H(p, q) be the Hamiltonian function of a system with n degrees of freedom (therefore the phase space is 2n-dimensional). Let L: H = h, q1 =

°

be a (2n - 2) -dimensional submanifold of the "level of energy"

H = h. If, in a certain domain Lo of L, P = (P2' ... , Pn)' Q = (q2' ... , qn) forrt: a local chart and q1 10 0, L is called a surface of section (see Figure A31.1). Assume that an orbit of the Hamiltonian system, through a

------;...... y

,I

, I

I

----__\r-"

I

x

Figure A 31.1

point x of a point

L o' returns to Lo' Then, in view of q1 10,0, the orbit through

x: on Lo sufficiently close to x, will, as 230

t increases, return .to

APPENDIX 31

231

Lo and will cut Lo in a uniquely determined point Ax '. In this manner, we define a mapping A:

THEOREM 1

A 31.2

The mapping A is canonical, that is for every closed curve y of L l , we have:

f

(A 31.3)

PdQ =

Y

where PdQ

=

f

PdQ,

Ay

P2 dq2 + ... + Pn dqn .

Proof: Consider the orbits emanating from y in the (2n + 1) -dimensional space I(p, q, t)l. The curves y and Ay of the space I(p, q)l are the projections of

two closed curves y'and Ay'of !(p, q,Ol which are formed respectively, q

p Figure A 31.4

by the initial points (t

=

0) and the end points of tbe above orbits (see Fig-

ure A31.4). Therefore, we have by the 1

Poinc~re-Cartan

theorem:

A proof of this well-known theorem has apparently never been published.

232


L pdq- Hdt

(A 31.5)

~,

where pdq

=

f

pdq- Hdt ,

Ay'

PI dql + .•• + Pn dqn· But H

=

=

constant along y' and Ay'

then:

.! Hdt ~y'

o.

Besides. we have:

f

pdq

Ay'

f

pdq.

Ay

constant on 1. we also have:

In view of ql

Thus. finally.

f

f

,pdq-Hdt = jPd Q,

pdq-Hdt

=

Ay'

Y

y

1.

PdQ.

Ay

and (A 31.5) implies (A 31.3). This proves the theorem. EXAMPLE A

31.6

Consider the problem of the "convex billiard table" (Birkhoff). Let

r

be a closed convex curve of the plane E2. Suppose that a material point

o

N Figure A31.7

233

APPENDIX 31

M moves inside r and collides with r according to the law "the angle of

incidence is equal to the angle of reflection" (see Figure A 31. 7).

The

states of M, immediately before aAd immediately after a reflectim, are determined by the angle of incidence a, 0 :::; a :::; 217, and the point of incidence. The point of incidence A is defined by the algebraic length q2 of the arc OA of r (0 is an arbitrary origin). In other words, the set of the states of M, immediately before and immediately after a reflection, form a torus

r2

=

la(mod 217), q2 (mod L)! in the phase space (L is the length

r2

of r). We obtain naturally a mapping A of a subset of

info another

·one: the state which immediately follows a reflection is transformed into the state immediately preceding the next reflection. THEOREM A 31.8 (G. D. Birkhoff)

I

=

sin a· d q2 /\ da is invariant under the mapping A .

Proof:

Between two reflections, the motion of M is determined by Hamiltonian equations in the corresponding four-dimensional phase-space. In the neighborhood of the above torus

r2, let us select special coordinates. Our point

M is well-defined by coordinates (ql' q2)' where ql

=

MN is the distance

from M to rand q2 is the algebraic length of the arc ON. Coordinates ql and q2 (mod L) .are clearly Lagrangian coordinates in a neighborhood of r. Let Pl and P2 be the corresponding momenta (the mass of M is supposed to be 1). On r, P l and P2 coincide obviously with the components of the velocity-vector v: Pl

= Ivl·

sin a,

P2

= Ivl·

cos a.

The Hamiltonian function H is the kinetic energy: H

v2

=""2

In the four-dimensional space I(P l , P2 ' ql' q2)~' consider the surface ~ whose equation is: 1, M ( r) :

234


From one reflection to the next, the motion defines a mapping :\: L

->

L.

The coordinates P2' q2 are local coordinates of L (a 1= 0) which is a Surface of section. In view of Theorem (A 31.2), the mapping t\ is canonical and therefore preserves the two-form: dP 2 /\ dq2 = sin a' dq2 /\ da . (Q. E. D.) An elementary proof of this theorem, due to G. D. Birkhoff [1], requires ex~ tensive computations.

APPENDIX 32 FUNCTIO~S

TilE GENERATI:\G

OF

"APPI~GS

CANONICAL

(See Section 21, Chapter 4) The following results are due to Hamilton and Jacobi. §A. Finite Canonical Mappings

Let x ~ (p, q), (p ~ (Pi' ... , Pn)' q ~ (ql' ... , qn))' be a point of the .canonical space R2n. The differentiable mapping:

is

call~d

canonical if A preserves the Poincare integral-invariant:

f

(A 32.1)

pdq

y

1.

~

pdq,

Ay

fOf- any closed curve y. Let a be an arbitrary two-chain. Relation (A 32.1) lmplies that A preserves the sum of the areas of the projections of a 'into the coordinate planes Pj' qj: (A 32.2)

[(a)

~

ff

a

dp 1\ dq

~

ff

dp 1\ dq

~

[(Aa) .

Aa

In other words, the two forms dp 1\ dq and dP 1\ dQ coincide: (A 32.3)

dp 1\ dq = dP 1\ dQ, where P = P(p, q), Q = Q(p, q) .

If the domain of A is simply connected, then conditions (A 32.1) and (A 32.2) are equivalent. Relation (A 32.3) shows that:

235

236


=

pdq + QdP, where P

P(p, q),

Q

=

is a closed form of R2n (since dp 1\ dq + dQ 1\ dP

A(x)

=

Q(p, q), =

0).

Therefore:

f

x pdq + QdP, where P = P(p, q), Q = Q(p, q), Xo

defines, locally, a function on R2n. Suppose that ql'· .. , qn; PI'···' Pn , form a local chart in some neighborhood of the point x, that is: Det(

~:) ~

O.

Then, ~(x) can be regarded as a function of P, q, defined in the neighborhood of the point P, q: (A 32.4)

A(P, q)

=

f

(P, q)

pdq + QdP, where p

=

p(P, q),

Q

=

Q(P, q).

DEFINITION A 32.5

The function A(P, q) is called the generating function of the canonical mapping A. Of course, A is only defined locally and up to a constant. From (A32.4) follows: (A32.6)

LEMMA

Q,

aA aq

= p.

A32.7

Let A (P, q) be a function such that:

in the

nei~hborhood

of a point (P, q). Then, Equations (A 32.6) can be

solved locally with respect to P and Q:

P

= P(p,

q), Q

= Q(p, q),

and the fuactions P, Q determine a canonical

mappin~

A.

237

APPENDIX 32

In fact, pdq + QdP is a closed form on R2n; thus dp 1\ dq = dP 1\ dQ . (Q. E. D.) Unfortunately, the generating fur-:-tion A is not a geometric object: A not only depends on the mapping A, but also on the coordinates p, q of R2n.

According to (A 32.6), the generating function of the identity 1 is Pq. Thus, every canonical mapping, close enough to the identity, has a generating function close to Pq. §R. Infinitesimal Canonieal Mappings

Consider a family of canonical mappings SE' the generating functions Pq + ES(P, q; E) of which depend smoothly on a parameter E «1. The

mapping

Se is close to the identity, if E is small. According to (A 32. 6),

the Taylor expansions of P(p, q) and Q(p, q) with respect to E are: (A 32.8)

where S

=

S(p, q; E) ,

By definition, the infinitesimal canonical mapping Se is a class of equivalent families Se: two families Se and SE' are equivalent if ISE- Se' I

=

O(E 2 ), DEFINITION

A32.9

The function S(p, q) on the phase-space is called the generating function of the infinitesimal mapping SE (or Hamiltonian function). Of course, S is defined up to a constant. Now we prove that the function S is a geometric object: S neither depends on the canonical coordinates p, q, nor on the choice of a representant SE in the class of equivalence: it is a mapping S: R 2n ... Rl. In fact, let y be a curve of R2n 1 ThiS is a way to memorize (A 32.6).

238


joining x and y:

ay = y - x. We set

Ye = Se Y , and we denote by a(E) the strip form-

ed by the curves Ye" 0 < E' < E, and oriented in such a way that

aoe

=

x

Y - Ye + ••• (see Figure A32.10).

Figure A32.10

Let us set: (A 32.11)

![a(E)]

11

=

dp/\dq.

a(e)

According to (A 32.2), this integral does not depend on the canonical coordinates and, according to (A 32.1), does not depend on the curve Y, but only on x and y. LEMMA A32.12 The generating function S. of the infinitesimal canonical mapping Se

is given by: (A32.13)

~

S(y) - S(x) =

![a(E)]1

dE

' e=O

and does not depend on the choice of the canonical coordinates p, q. Proof:

.

Let us set Se x (A32.14)

X

=

ox

= (op,oq).

E(S(y) -S(x)) = E

According to (A 32.8), we have:

f~ yap

~

f

dp +

~

dq

aq

(oqdp-opdq) + 9(E2) •

Y

On the other hand, according to (A32.11), the integral of dp /\ dq along a(E) is;

(A32.1S)

239

APPENDIX 32

Formulas (A32.14) and (A32.1S) imply (A32.13).

(Q. E. D.)

One can express the invariance of the generating function S in another form. Let -\ be a finite canonical mapping and SE an infinitesimal canonical mapping. The canonical mapping T E = AS E A-I is clearly infinitesimal. LEMMA

A32.16

The generating functions Sand T of the infinitesimal mappings SE and TE are related by: T(Ax)

(A 32.17)

S (x) + constant.

Proof; Let YE and a (E) be the curve and the surface of Lemma (A 32.12). The curve y' = Ay joins the points Ax and Ay. Besides, the curves

TE ,y', 0

~ E' ~ E,

form a strip T(E), which is nothir.g but:

(A 32.18) From (A 32.13) follows: (A32.19)

S(y)-S(x) = ~ J[a(E)), dE

T(Ay)-T(Ax) = ~[[r(E)]. dE

But the mapping A is canonical. Thus, according to (A32.2) and (A 32.18), we have: [[o(E)] = [[r(E)].

Comparison with (A 32.19) yields (A 32.17).

(Q. E. D.)

COROLLARY A32.20

Let '\ and CE be infinitesimal canonical mappings with corresponding generating functions Band C, and let ,\ be a finite canonical mapping .• Then, the infinitesimal canonical mapping: (A 32.21)

has the following generating function:


240 (A 32.22)

B '(x) =

c (x)

+ B (x) - C (A -1 X) + constant.

In fact, (A 32.8) implies that the generating function of the product of two infinitesimal mappings is the sum of their generating functions, and also that the generating function of the inverse mapping C;-1 is - C. Relation

(A32.22) is easily derived from these remarks and from Lemma (A 32.16). §C. Lie Commutators and Poisson Brackets Given two infinitesimal canonical mappings Ae and Be' there exists one and only one infinitesimal canonical mapping Ce such that:

(A32.23)

AaBbA_aB_b

Cab + O(a 2 ) + O(b 2 ); a, b ... O.

=

The mapping Ce is called the Lie commutator of Ae and Be . LEMMA

A 32.24.

C

The generating function

of

Ce

is equal, up to a sign, to the Pois-

son bracket of the generating functions A and B of Ae and Be :

(A32.2S)

'lie

=

-

[V A, V Bl.

V

=

gradient.

We use the notation [x, y] = (lx, y) as in Appendixes 26 and 27.

Proof:

Again let Y be a curve joining x and y:

ay

=

y - x. We consider the

five-sided prism (see Figure A 32.26) formed by four strips: a1

BeY' -b < E < 0,

aa 1

Y - Y1 + ''',

a2

\y~. -a<E

n

is globally canonical if it is homotopic to the identity and satisfies

f

(A33.1)

pdq = Y

f

pdq

Ay

for anyone-cycle y (even nonhomologous to zero). In conformity to Appendix 32, the mapping A is locally determined by a generating function Pq + A (P, q), provided that

Det (~: )

~

0,

(A33.2) p

=P

+

aA ,

aq

Q

aA

= q+ - ,

ap

Thus, locally, the function A (P, q) verifies: (A33.3)

A (P, q) =

j

(P, q)

(Q- q)dP + (p-P)dq.

243

E~GOD/C PROBLEMS OF CLASSICAL MECHANICS

244

Let us set: A(x) = A(P(x), q(x)) ,

where x

LEMMA

=

(p(x), q(x)) (

n.

A33.4

The mapping (A 33.2) is globally canonicai if and only if the function A (x), defined by (A 33.3), is single-valued on

n.

Proof:

Let

y be a

closed curve of

(A33.S)

n.

Let us prove that:

f 1. Whether T and AT intersect, for n> 1, if condition (A33.9) is not fulfilled, is an open question.

If condition (A 33.9) can be relaxed from Lemma (A 33.8), we ob1ain many "recurrence theorems" of the following type: Assume that the initial values a"

b i of the axis of the Kepler ellipses,

in the plane many-body problem, are such that the ellipses do not intersect. Then, whatever

T

be, there exist initial phases 2 1., g. such that the axis

.

"

of the ellipses return to their initial values after a time REMARK

T.

A 33.14

If we drop condition (A 33.9), Lemma (A 33.8) cannot hold without assuming that i\ is a diffeomorphism, because (even for n

=

1) regular and

globally canonical mappings can be constructed such that T and AT do not intersect. 2 Phases 1/. ~/ are angles (mod 271); ~/ determines the position of the ellipses and 1/ determmes the position of the planets on these ellipses.

247

APPENDIX 33

§C. Fixed Points

Now, let A be a global canonical mapping of the following particular type: (A33.1S)

A:

p., q -+ p, q 0, for any element w of the set no(K) (defined

below). Let us denote by

n (K)

(A34.8)

lei(k,c.» - 11

for any w, Iw'-wol

=

< K~II, II

the set of the wo satisfying:

=

> KN-l/

n+2, and for any k, Ikl

o) - 11

n (K) c

> K

Ik I-II,

no(K) •

LEMMA A34.10 Almost every (in the Lebesgue measure sense) point Wo belongs to

n (K)

for some K> 0 (hence, Wo ( no(K)).

4 The technique of evaluating small denominators was extensively worked out by C. L. Siegel [2],

[3],

in connectIOn with similar problems.

APPENDIX 34

253

Proof:

Let

n

be a bounded domain of the space

< d for some

lk,d = Iwollei(k,Cil) - 11

Then, clearly: only on.

n.

mea~ (lk, d

n m :s.

Iwol.

Let:

w, Iw-wol

< dl .

C· d, where the constant C depends

Relation (A 34.8) holds outside of Uk Ik,Klkl- v ' But we have

since

~

!k I-v

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