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Lectures on Ergodic Theory and Pesin Theory on Compact Manifolds : London Mathematical Society Lecture Note Series - Mark Pollicott

Lectures on Ergodic Theory and Pesin Theory on Compact Manifolds

London Mathematical Society Lecture Note Series

Paperback Published: 5th April 1993
ISBN: 9780521435932
Number Of Pages: 172

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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. Emphasis is placed on generality and on the crucial role of measure theory, although no specialist knowledge of this subject is required.

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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

Introductionp. 1
The basic theory
Invariant measures and some ergodic theoryp. 5
Invariant measuresp. 5
Poincare recurrencep. 9
Ergodic measuresp. 9
Ergodic decompositionp. 10
The ergodic theoremp. 12
Proof of the ergodic theoremp. 15
Proof of the ergodic decomposition lemmap. 18
Notesp. 19
Ergodic theory for manifolds and Liapunov exponentsp. 21
The subadditive ergodic theoremp. 21
The subadditive ergodic theorem and diffeomorphismsp. 22
Oseledec-type theoremsp. 23
Some examplesp. 25
Proof of the Oseledec theoremp. 31
Further refinements of the Oseledec theoremp. 36
Proof of the subadditive ergodic theoremp. 37
Notesp. 40
Entropyp. 43
Measure theoretic entropyp. 43
Measure theoretic entropy and Liapunov exponentsp. 46
Topological entropyp. 48
Topological entropy and Liapunov exponentsp. 50
Equivalent definitions of measure theoretic entropyp. 53
Proof of the Pesin-Ruelle inequalityp. 58
Osceledec's theorem, topological entropy and Lie theoryp. 60
Notesp. 62
The Pesin set and its structurep. 63
The Pesin setp. 64
The Pesin set and Liapunov exponentsp. 68
Liapunov metrics on the Pesin setp. 69
Local distortionp. 71
Proofs of Propositions 4.1 and 4.2p. 73
Liapunov exponents with the same signp. 76
Notesp. 77
An interludep. 79
Some topical examplesp. 79
Uniformly hyperbolic diffeomorphismsp. 83
Closing lemma
Stable manifolds
Notesp. 85
The applications
Closing lemmas and periodic pointsp. 87
Liapunov neighborhoodsp. 87
Shadowing lemmap. 90
Uniqueness of the shadowing pointp. 94
Closing lemmasp. 95
An application of the closing lemmap. 96
Notesp. 98
Structure of "chaotic" diffeomorphismsp. 99
The distribution of periodic pointsp. 99
The number of periodic pointsp. 101
Homoclinic pointsp. 103
Generalized Smale horse-shoesp. 105
Entropy stabilityp. 108
Entropy, volume growth and Yomdin's inequalityp. 110
Examples of discontinuity of entropyp. 115
Proofs of propositions 6.1 and 6.2p. 119
Notesp. 122
Stable manifolds and more measure theoryp. 123
Stable and unstable manifoldsp. 123
Equality in the Pesin-Ruelle inequalityp. 127
Foliations and absolute continuityp. 129
Ergodic componentsp. 132
Proof of stable manifold theoremp. 133
Ergodic components and absolute continuityp. 137
Notesp. 137
Some preliminary measure theoryp. 139
Some preliminary differential geometryp. 145
Geodesic flowsp. 151
Referencesp. 155
Table of Contents provided by Syndetics. All Rights Reserved.

ISBN: 9780521435932
ISBN-10: 0521435935
Series: London Mathematical Society Lecture Note Series
Audience: Professional
Format: Paperback
Language: English
Number Of Pages: 172
Published: 5th April 1993
Publisher: Cambridge University Press
Country of Publication: GB
Dimensions (cm): 23.5 x 15.88  x 1.27
Weight (kg): 0.25