Ergodic Theory and Differentiable Dynamics

✍ Scribed by Ricardo Mañé

• Publisher: Springer
• Year: 1987
• Tongue: English
• Leaves: 327
• Category: Library

✦ Synopsis

This book is an introduction to ergodic theory, with emphasis on its relationship with the theory of differentiable dynamical systems, which is sometimes called differentiable ergodic theory. Chapter 0, a quick review of measure theory, is
included as a reference. Proofs are omitted, except for some results on derivatives with respect to sequences of partitions, which are not generally found in standard texts on measure and integration theory and tend to be lost within a much wider framework in more advanced texts. Chapter I starts with a quick and superficial introduction, then presents the main kinds of dynamical systems around which ergodic theory has developed. This development itself starts in chapter II, devoted to the classical concepts and theorems.

Chapters III and IV are devoted to contemporary ergodic theory, born in 1958 with the introduction of the notion of entropy by Kolmogorov and developed primarily by Sinai, Anosov, Bowen and Ornstein, in the sixties and seventies. Chapter III is a typical example of differentiable ergodic theory. It studies ergodic properties of Anosov diffeomorphisms and expanding maps. The techniques used in this analysis have become classical, and remain the conceptual foundation for a good part of today's research. Entropy is the subject of chapter IV; we start with the basic formalism and the calculation of simple examples, then discuss topological entropy, the variational principle of entropy and the construction of the unique
entropy-maximizing measure for hyperbolic homeomorphisms. We conclude with Lyapunov exponents, the Pesin formula for the entropy of volume-preserving diffeomorphisms, and the Brin-Katok local entropy formula.

We have included many advanced results without proof, in the belief that an introductory text does not have to deprive the reader of a comprehensive and up-to-date panorama of the subject. In particular, we state Ornstein's famous classification theorem. There are good and readily accessible expositions of this result (see references in section 1.12), so we see no point in plagiarizing them here. The theorems of Katok and Pesin (sections IV.15 and IV.10) are a different story: there seem to be as yet no pedagogical treatments of them. A third kind of result quoted without proof is exemplified by Manning's theorem on the linearization of Anosov diffeomorphisms (section IV.15): strictly speaking, they are outside the main stream of ideas presented in this work, but familiarity with them is fundamental to
a balanced, global understanding of our subject.

A good part of the information in this book is contained in the exercises. This is intentional, and a careful reading, at least, of all the exercises is essential.

The reader is also encouraged to concentrate on a careful understanding of new ideas and statements, and not so much on proofs, in a first reading. The proofs are often arid and demanding, and a less motivated reader may well be turned away if he attempts to go through all of them.

I would like to thank Elon Lima for asking me to write this book: his insistence during slack periods was decisive in its coming to light. Alexandre Freire helped me immensely, proofreading the original and contributing relevant comments.

✦ Table of Contents

Contents
0 Measure Theory
1 Measures
2 Measurable maps
3 Integrable functions
4 Differentiation and integration
5 Partitions and Derivatives
I Measure-preserving maps
1 Introduction
2 The Poincare Recurrence theorem
3 Volume-preserving diffeomorphisms and flows
4 First integrals
5 Hamiltonians
6 Continued fractions
7 Topological Groups, Lie groups, Haar measure
8 Invariant measures
9 Uniquely ergodic maps
10 Shifts: the probabilistic viewpoint
11 Shifts: the topological viewpoint
12 Equivalent maps
II Ergodicity
1 Birkohff's theorem
2 Ergodicity
3 Ergodicity of homomorphisms and translations of the torus
4 More examples of ergodic maps
5 The theorem of Kolmogorov-Arnold-Moser
6 Ergodic decomposition of invariant measures
7 Furstenberg's example
8 Mixing automorphisms and Lebesgue automorphisms
9 Spectral theory
10 Gaussian shifts
11 Kolmogorov automorphisms
12 Mixing and ergodic Markov shifts
III Expanding maps and Anosov diffeomorphisms
1 Expanding maps
2 Anosov diffeomorphisms
3 Absolute continuity of the stable foliation
IV Entropy
1 Introduction
2 Proof of the Shannon-McMillan-Breiman theorem
3 Entropy
4 The Kolmogorov-Sinari theorem
5 Entropy of expanding maps
6 The Parry measure
7 Topological entropy
8 The variational property of entropy
9 Hyperbolic homeomorphisms
10 Lyapunov exponents. The theorems of Oseledec and Pesin
11 Proof of Oseledec's theorem
12 Proof of Ruelle's inequality
13 Proof of Pesin's formula
14 Entropy of Anosov diffeomorphisms
15 Hyperbolic measures. Katok's theorem
16 The Brin-Katok local entropy formula
Bibliography
Notation index
Subject index