Topics in Ergodic Theory.

231 Pages·1994·13.121 MB·English

by  Yakov G. Sinai

Topics in Ergodic Theory
G.
VA. SINAI
This book concerns areas of ergodic
theory that are now being intensively
developed. The topics include entropy
theory (with emphasis on dynamical
systems with multi-dimensional time),
elements of the renormalization group
method in the theory of dynamical sys
tems, splitting of separatrices, and
some problems related to the theory of
hyperbolic dynamical systems.
Ya. G. Sinai is Professor of Mathemat
ics at Princeton University.
Princeton Mathematical Series, 44

TOPICS IN ERGODIC THEORY
PRINCETON MATHEMATICAL SERIES
Editors: Luis A Caffarelli, John N. Mather,
and Elias M. Stein
I. The Classical Groups by Hermann Wey/
3. An Introduction to Differential Geometry by Luther Pfahler Eisenhart
4. Dimension Theory by W. Hurewicz and H. Wallman
6. The Laplace Transform by D. V. Widder
7. Integration by Edward J. McShane
8. Theory of Lie Groups: I by C. Chevalley
9. Mathematical Methods of Statistics by Harald Cramer
10. Several Complex Variables by S. Bochner and W. T. Martin
l l. Introduction to Topology by S. Lefschetz
12. Algebraic Geometry and Topology edited by R. H. Fox, D. C. Spencer, and A. W. Tucker
14. The Topology of Fibre Bundles by Norman Steenrod
15. Foundations of Algebraic Topology by Samuel Eilenberg and Norman Steenrod
16. Functionals of Finite Riemann Surfaces by Menahem Schiffer and Donald C. Spencer
17. Introduction to Mathematical Logic, Vol. I by Alonzo Church
19. Homological Algebra by H. Cartan and S. Eilenberg
20. The Convolution Transform by I. I. Hirschman and D. V. Widder
21. Geometric Integration Theory by H. Whitney
22. Qualitative Theory of Differential Equations by V. V. Nemytskii and V. V. Stepanov
23. Topological Analysis by Gordon T. Whyburn (revised 1964)
24. Analytic Functions by Ahlfors, Behnke, Bers, Grauert et al.
25. Continuous Geometry by John von Neumann
26. Riemann Surfaces by L . Ahlfors and L. Sario
27. Differential and Combinatorial Topology edited by S. S. Cairns
28. Convex Analysis by R. T. Rockafellar
29. Global Analysis edited by D. C. Spencer and S. lyanaga
30. Singular Integrals and Differentiability Properties of Functions by E. M. Stein
31. Problems in Analysis edited by R. C. Gunning
32. Introduction to Fourier Analysis on Euclidean Spaces by E. M. Stein and G. Weiss
33. Etale Cohomology by J. S. Milne
34. Pseudodifferential Operators by Michael E. Taylor
36. Representation Theory of Semisimple Groups: An Overview Based on Examples by Anthony
W. Knapp
37. Foundations of Algebraic Analysis by Masaki Kashiwara, Takahiro Kawai, and Tatsuo Kimura.
Translated by Goro Kato
38. Spin Geometry by H. Blaine Lawson, Jr., and Marie-Lauise Michelsohn
39. Topology of 4-Manifolds by Michael H. Freedman and Frank Quinn
40. Hypo-Analytic Structures: Local Theory by Franr:ois Treves
41. The Global Nonlinear Stability of the Minkowski Space by Demetrios Christodoulou and Sergiu
Klainerman
42. Essays on Fourier Analysis in Honor of Elias M. Stein edited by C. Fefferman, R. Fefferman,
and S. Wainger
43. Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals
by Elias M. Stein
44. Topics in Ergodic Theory by Ya. G. Sinai
TOPICS IN
ERGODIC THEORY
Ya. G. Sinai
PRINCETON UNIVERSITY PRESS
PRINCETON, NEW jERSEY
1994
Copyright © 1994 by Princeton University Press
Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540
In the United Kingdom: Princeton University Press, Chichester, West Sussex
All Rights Reserved
Library of Congress Cataloging-in-Publication Data
Sinai, IAkov Grigor 'evich, 1935-
Topics in ergodic theory/ Ya. G. Sinai.
p. cm. - (Princeton mathematical series: 44)
Includes bibliographical references and index.
ISBN 0-691-03277-7
I. Ergodic theory. 2. Topological dynamics. I. Title.
II. Series.
QA61 l.5.S56 1993
515.42-dc20 93-16644
This book has been composed in Times Roman
Princeton University Press books are printed on acid-free paper
and meet the guidelines for permanence and durability of
the Committee on Production Guidelines for Book Longevity of
the Council on Library Resources
Printed in the United States of America
I 3 5 7 9 10 8 6 4 2
CONTENTS
Preface vii
I. General Ergodic Theory
Lecture I. Measurable Transformations. Invariant 3
Measures. Ergodic Theorems
Lecture 2. Lebesgue Spaces and Measurable 16
Partitions. Ergodicity and Decomposition into Ergodic
Components. Spectrum of Interval Exchange
Transformations
Lecture 3. Isomorphism of Dynamical Systems. 28
Generators of Dynamical Systems
Lecture 4. Dynamical Systems with Pure Point 36
Spectra
Lecture 5. General Properties of Eigenfunctions and 43
Eigenvalues of Ergodic Automorphisms. Isomorphism
of Dynamical Systems with Pure Point Spectrum
II. Entropy Theory of Dynamical Systems
Lecture 6. Entropy Theory of Dynamical Systems 55
Lecture 7. Breiman Theorem. Pinsker Partition. 69
K-Systems. Exact Endomorphisms. Gibbs Measures
Lecture 8. Entropy of Dynamical Systems with 77
Multidimensional Time. Systems of Cellular Automata
as Dynamical Systems
Ill. One-Dimensional Dynamics
Lecture 9. Continued Fractions and Farey Fractions 87
Lecture I0 . Homeomorphisms and Diffeomorphisms 95
of the Circle
VI CONTENTS
Lecture 11. Sharkovski's Ordering and Feigenbaum's 111
Universality
Lecture 12. Expanding Mappings of the Circle 123
IV. Two-Dimensional Dynamics
Lecture 13. Standard Map. Twist Maps. Periodic 137
Orbits. Aubry-Mather Theory
Lecture 14. Periodic Hyperbolic Points, Their Stable 147
and Unstable Manifolds. Homoclinic and Heteroclinic
Orbits
Lecture 15. Homoclinic and Heteroclinic Points and 167
Stochastic Layers
V. Elements of the Theory of Hyperbolic
Dynamical Systems
Lecture 16. Geodesic Flows and Their 177
Generalizations. Discontinuous Dynamical Systems.
Stable and Unstable Manifolds
Lecture 17. Existence of Local Manifolds. Gibbs 194
Measures
Lecture 18. Markov Partitions. d/t'-Theorem for 204
Dynamical Systems. Elements of Thermodynamic
Formalism
Index 217
PREFACE
This book may be considered a continuation of Introduction to Ergodic
Theory published by Princeton University Press in 1977.
The main notions of ergodic theory are recounted-together with some
elements of spectral theory of dynamical systems, including the theory of dynamical
systems with pure point spectrum-in the first of five parts.
Part II is dedicated to the entropy theory of dynamical systems; this
theory may be considered more or less complete now. We discuss only its part
needed for applications. In Lecture 8 of this part, we consider systems of cellular
automata and study some of their properties connected with entropy.
Lecture I 0 of Part Ill contains the proof of a version of M. Herman's
theorem found by K. Khanin and me. Lecture 11 is dedicated to Sharkovski's
theorem and the main ideas of Feigenbaum's universality. Lecture 12 concerns
expanding mappings. The proof of the main theorem here is given in the spirit of
thermodynamic formalism, which is discussed mainly in Part V.
In Part IV we consider some properties of the standard or Chirikov map,
present some ideas connected with the Aubry-Mather theory, and introduce the
notions of homoclinic and heteroclinic points and stochastic layers. We also prove
a theorem that shows in which sense these layers are stochastic. Lecture 14
contains the proof of a theorem that gives the estimation from above of the angle
determining the splitting of separatrices. This proof, due to I. Cornfeld and me, has
not been published before. The estimation is slightly worse than that obtained by
some other methods, but the ideas of the proof can be useful elsewhere. This
lecture may be omitted during the first reading.
In Part V, the theory of hyperbolic dynamical systems is considered. The
presentation is new in some aspects. In several cases, we explain the main ideas,
discussing only important examples. This part can be a source for deeper studies.
E. I. Dinaburg read the manuscript very attentively and made many useful
remarks. C. Series improved the English version of the manuscript. R. MacKay read
the final version of the manuscript and suggested many significant improvements.
It is my pleasure to thank them for their help.