Lectures on Ergodic Theory and Pesin Theory on Compact Manifolds

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Paperback | 5 April 1993

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Pesin theory consists of the study of the theory of non-uniformly hyperbolic diffeomorphisms. The aim of this book is to provide the reader with a straightforward account of this theory, following the approaches of Katok and Newhouse. The notes are divided into two parts. The first develops the basic theory, starting with general ergodic theory and introducing Liapunov exponents. Part Two deals with the applications of Pesin theory and contains an account of the existence (and distribution) of periodic points. It closes with a look at stable manifolds, and gives some results on absolute continuity. These lecture notes provide a unique introduction to Pesin theory and its applications. The author assumes that the reader has only a good background of undergraduate analysis and nothing further, so making the book accessible to complete newcomers to the field.

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"...contains numerous simple examples which help the uninitiated reader to get a good idea of the relevance of the theorems..." Nicolai Haydn, Mathematical Reviews

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Introduction	p. 1
The basic theory
Invariant measures and some ergodic theory	p. 5
Invariant measures	p. 5
Poincare recurrence	p. 9
Ergodic measures	p. 9
Ergodic decomposition	p. 10
The ergodic theorem	p. 12
Proof of the ergodic theorem	p. 15
Proof of the ergodic decomposition lemma	p. 18
Notes	p. 19
Ergodic theory for manifolds and Liapunov exponents	p. 21
The subadditive ergodic theorem	p. 21
The subadditive ergodic theorem and diffeomorphisms	p. 22
Oseledec-type theorems	p. 23
Some examples	p. 25
Proof of the Oseledec theorem	p. 31
Further refinements of the Oseledec theorem	p. 36
Proof of the subadditive ergodic theorem	p. 37
Notes	p. 40
Entropy	p. 43
Measure theoretic entropy	p. 43
Measure theoretic entropy and Liapunov exponents	p. 46
Topological entropy	p. 48
Topological entropy and Liapunov exponents	p. 50
Equivalent definitions of measure theoretic entropy	p. 53
Proof of the Pesin-Ruelle inequality	p. 58
Osceledec's theorem, topological entropy and Lie theory	p. 60
Notes	p. 62
The Pesin set and its structure	p. 63
The Pesin set	p. 64
The Pesin set and Liapunov exponents	p. 68
Liapunov metrics on the Pesin set	p. 69
Local distortion	p. 71
Proofs of Propositions 4.1 and 4.2	p. 73
Liapunov exponents with the same sign	p. 76
Notes	p. 77
An interlude	p. 79
Some topical examples	p. 79
Uniformly hyperbolic diffeomorphisms	p. 83
Shadowing
Closing lemma
Stable manifolds
Notes	p. 85
The applications
Closing lemmas and periodic points	p. 87
Liapunov neighborhoods	p. 87
Shadowing lemma	p. 90
Uniqueness of the shadowing point	p. 94
Closing lemmas	p. 95
An application of the closing lemma	p. 96
Notes	p. 98
Structure of "chaotic" diffeomorphisms	p. 99
The distribution of periodic points	p. 99
The number of periodic points	p. 101
Homoclinic points	p. 103
Generalized Smale horse-shoes	p. 105
Entropy stability	p. 108
Entropy, volume growth and Yomdin's inequality	p. 110
Examples of discontinuity of entropy	p. 115
Proofs of propositions 6.1 and 6.2	p. 119
Notes	p. 122
Stable manifolds and more measure theory	p. 123
Stable and unstable manifolds	p. 123
Equality in the Pesin-Ruelle inequality	p. 127
Foliations and absolute continuity	p. 129
Ergodic components	p. 132
Proof of stable manifold theorem	p. 133
Ergodic components and absolute continuity	p. 137
Notes	p. 137
Some preliminary measure theory	p. 139
Some preliminary differential geometry	p. 145
Geodesic flows	p. 151
References	p. 155
Table of Contents provided by Syndetics. All Rights Reserved.

Lectures on Ergodic Theory and Pesin Theory on Compact Manifolds

At a Glance

Paperback

Industry Reviews

Shipping

How to return your order

Defective items

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