Ergodic Theory and Topological Dynamics (eBook)

Front Cover 1
Ergodic Theory and Topological Dynamics 4
Copyright Page 5
Contents 8
Preface 10
Chapter I. Ergodic Theory 12
1. Abstract Dynamical Systems 12
2. Ergodic Theorems 18
3. Ergodicity and Mixing 25
4. Products and Factors 32
5. Inverse Limits 35
6. Induced Systems 39
Exercises 43
Chapter II. Topological Dynamics 54
1. Classical Dynamical Systems 54
2. Minimal and Strictly Ergodic Systems 57
3. Equicontinuous and Distal Systems 62
4. Sums and Products of Dynamical Systems 64
5. Inverse Limits 68
6. The Ellis Semigroup of S 70
7. Expansive Systems 74
Exercises 77
Chapter III. Group Automorphisms and Affine Transformations 82
1. Dynamical Systems on Groups 82
2. Ergodicity 87
3. Discrete and Quasidiscrete Spectrum 95
4. Quasiperiodic Spectrum and the Ergodic Part of t 100
5. Ergodic Automorphisms 103
6. An Affine Transformation Associated with the Dynamical System F 107
Exercises 113
Chapter IV. Entropy 120
1. Conditional Expectation and Kolmogorov Entropy 120
2. The Information Function and Finiteness of h(., A 125
3. Sinai’s Theorem and Generators 130
4. Topological Entropy 138
5. Entropy of Affine Transformations 144
6. McMillan’s Theorem and Entropy of Induced Systems 146
Exercises 153
Chapter V. Bernoulli Systems and Ornstein’s Theorem 161
1. Definitions 162
2. Approximation Lemmas 169
3. The Isomorphism Theorem 184
4. Extensions and Consequences of the Isomorphism Theorem 188
Exercises 190
Bibliography 192
Index 198