Ergodic Theory

Karl Petersen

Contents

1. Introduction and preliminaries

1. The basic questions of ergodic theory
2. The basic examples
3. The basic constructions
4. Some useful facts from measure theory and functional analysis

2. The fundamentals of ergodic theory

1. The mean ergodic theorem
2. The pointwise ergodic theorem
3. Recurrence
4. Ergodicity
5. Strong mixing
6. Weak mixing

3. More about almost everywhere convergence

1. More about the maximal ergodic theorem
2. More about the pointwise ergodic theorem
3. Differentiation of integrals and the local ergodic theorem
4. The martingale convergence theorems
5. The maximal inequality for the Hilbert transform
6. The ergodic Hilbert transform
7. The filling scheme
8. The Chacon-Ornstein theorem

4. More about recurrence

1. Construction of eigenfunctions
2. Some topological dynamics
3. The Szmeredi theorem
4. The topological representation of ergodic transformations
5. Two examples: metric weak mixing without topological strong mixing, a prime transformation

5. Entropy

1. Entropy in physics, information theory, and ergodic theory
2. Information and conditioning
3. Generators and the Kolmogorov-Sinai theorem

6. More about entropy

1. More examples of the computation of entropy
2. The Shannon-McMillan-Breiman theorem
3. Topological entropy
4. Introduction to Ornstein theory
5. Finitary coding between Bernoulli shifts