Basic Ergodic Theory (3/E)

This is an introductory text on ergodic theory. The presentation has a slow pace and the book can be read by any person with a background in basic measure theory and metric topology. A new feature of the book is that the basic topics of ergodic theory such as the Poincaré recurrence lemma, induced automorphisms and Kakutani towers, compressibility and E. Hopf's theorem, the theorem of Ambrose on representation of flows, are treated at the descriptive set -theoretic level before their measure -theoretic or topological versions are presented. In addition, topics centering around the Glimm-Effros theorem, which have so far not found a place in texts on ergodic theory are discussed in this book.

The third edition has, among other improvements, a new chapter on Additional Topics that include Liouville's theorem of classical mechanics, basics of Shannon Entropy and the Kolmogorov-Sinai theorem, and van der Waerden's theorem on arithmetical progressions.

Contents

1. The Poincar´e Recurrence Lemma
2. Ergodic Theorems of Birkhoff and von Neumann
3. Ergodicity
4. Mixing Conditions and Their Characterisations
5. Bernoulli Shift and Related Concepts
6. Discrete Spectrum Theorem
7. Induced Automorphisms and Related Concepts
8. Borel Automorphisms are Polish Homeomorphisms
9. The Glimm-Effros Theorem
10. E. Hopf 's Theorem
11. H. Dye's Theorem
12. Flows and Their Representations
13.Additional Topics
Bibliography
Index