Ergodic Theory

This book provides an introduction to the ergodic theory and topological dynamics of actions of countable groups. It is organized around the theme of probabilistic and combinatorial independence, and highlights the complementary roles of the asymptotic and the perturbative in its comprehensive treatment of the core concepts of weak mixing, compactness, entropy, and amenability. The more advanced material includes Popa's cocycle superrigidity, the Furstenberg-Zimmer structure theorem, and sofic entropy.

The structure of the book is designed to be flexible enough to serve a variety of readers. The discussion of dynamics is developed from scratch assuming some rudimentary functional analysis, measure theory, and topology, and parts of the text can be used as an introductory course. Researchers in ergodic theory and related areas will also find the book valuable as a reference.

Preface. - Introduction. - General Framework and Notational Conventions. - Part 1 Weak Mixing Comactness. - Basic Concepts in Ergodic Theory. - Structure Theory for P. M. P. Actions. - Amenability. - Property (T). - Orbit Equivalence Beyond Amenability. - Topological Dynamics. - Tameness and Independence. - Part 2 Entropy. - Entropy for Actions of Amenable Groups. - Entropy for Actions of Sofic Groups. - The f-invariant. - Entropy and Independence. - Algebraic Actions: Expansiveness, Homoclinicity, and Entropy. - Algebraic Actions: Entropy and the Fuglede-Kadison Determinant. - Appendix A. Polish Spaces and Standard Borel Spaces. - Appendix B. Positive Definite Functions and Weak Containment. - Appendix C. Hilbert Modules. - Appendix D. Weakly Almost Periodic Functions. - Appendix E. Gaussian Actions.