Ergodic Theory and Differentiable Dynamics

Title page......Page 1
Preface to english edition......Page 5
Introduction......Page 7
Table of Contents......Page 9
1. Measures......Page 11
3. Integrable Functions......Page 13
4. Differentiation and Integration......Page 16
5. Partitions and Derivatives......Page 17
1. Introduction......Page 25
2. The Poincaré Recurrence Theorem......Page 37
3. Volume-Preserving Diffeomorphisms and Flows......Page 42
4. First Integrals......Page 47
5. Hamiltonians......Page 49
6. Continued Fractions......Page 52
7. Topological Groups, Lie Groups, Haar Measure......Page 55
8. Invariant Measures......Page 62
9. Uniquely Ergodic Maps......Page 68
10. Shifts: the Probabilistic Viewpoint......Page 71
11. Shifts: the Topological Viewpoint......Page 77
12. Equivalent Maps......Page 87
1. Birkhoff\'s Theorem......Page 99
2. Ergodicity......Page 110
3. Ergodicity of Homomorphisms and Translations of the Torus......Page 116
4. More Examples of Ergodic Maps......Page 119
5. The Theorem of Kolmogorov-Arnold-Moser......Page 135
6. Ergodic Decomposition of Invariant Measures......Page 137
7. Furstenberg\'s Example......Page 144
8. Mixing Automorphisms and Lebesgue Automorphisms......Page 151
9. Spectral Theory......Page 158
10. Gaussian Shifts......Page 161
11. Kolmogorov Automorphisms......Page 167
12. Mixing and Ergodic Markov Shifts......Page 171
1. Expanding Maps......Page 176
2. Anosov DitTeomorphisms......Page 188
3. Absolute Continuity of the Stable Foliation......Page 199
1. Introduction......Page 217
2. Proof of the Shannon-McMillan-Breiman Theorem......Page 222
3. Entropy......Page 224
4. The Kolmogorov-Sinai Theorem......Page 228
5. Entropy of Expanding Maps......Page 237
6. The Parry Measure......Page 241
7. Topological Entropy......Page 246
8. The Variational Property of Entropy......Page 254
9. Hyperbolic Homeomorphisms......Page 261
10. Lyapunov Exponents. The Theorems of Oseledec and Pesin......Page 273
11. Proof of Oseledec\'s Theorem......Page 277
12. Proof of Ruelle\'s Inequality......Page 291
13. Proof of Pesin\'s Formula......Page 295
14. Entropy of Anosov Diffeomorphisms......Page 303
15. Hyperbolic Measures. Katok\'s Theorem......Page 306
16. The Brin-Katok Local Entropy Formula......Page 308
Bibliography......Page 315
Notation Index......Page 319
Subject Index......Page 321