Ergodic Theory

Name: Ergodic Theory

Author(s): e Ricardo Mañé

Pages: 388

Publication: IMPA, 1983

ISBN: 978-85-244-0137-4

Edition: 1

Preface

Chapter 0 – Measurement Theory
I Measures
II Measurable Functions
III Integrable Functions
IV. Differentiation and Integration
V Partitions and Derivation

Chapter 1 – Transformations that preserve measure
1. Introduction
2. Poincaré's Recurrence Theorem
3 Diffeomorphisms and Volume-Preserving Fields
4 First Integrals
5 Hamiltonians
6. Decomposition into Continued Fractions
7 Topological and Lie Groups. Haar measurement
8 Invariant Measures
9 Uniquely Ergodic Transformations
10 Shifts. The probabilistic point of view.
11 Shifts. The topological viewpoint.
12 Equivalent Transformations

Chapter II – Ergodicity
1. Birkhoff's Theorem
2 Ergodicity
3 Ergodicity of Tn homomorphisms and translations
4 Other examples of ergodicity
5 The Arnold-Kolmogorov-Moser Theorem
6 Ergodic Decomposition of Invariant Measures
7 The Example of Furstenberg
8 Transformations Mixing and Lebesgue
9 Spectral Theory
10 Gaussian Shifts
11 Kolmogorov Automorphisms
12 Mixing and Ergodicity of Markov Shifts

Chapter III Expanding Transformations and Anosov Diffeomorphisms
1. Expanding Transformations
2 Anosov Diffeomorphisms
3. Absolute continuity of stable foliation.
4 Applications of the absolute continuity of stable foliation

Chapter IV – Entropy
1. Introduction
2 Proof of the Shannon-Mc Millan-Breiman Theorem
3 Entropy
4 The Kolmogorov-Sinai Theorem
5 Entropy of Expanding Transformations
6 Parry's 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. Ruelle's proof of inequality.
13. Proof of Pesin's formula
14 Entropy of Anosov diffeomorphisms
15 Hyperbolic measurements. Katok's Theorem

Bibliography
Index