Ergodic Theory : Independence and Dichotomies / by David Kerr, Hanfeng Li.

Format:

Book

Author/Creator:

Kerr, David, 1971- Author.

Li, Hanfeng., Author.

Series:

Springer Monographs in Mathematics, 1439-7382

Language:

English

Subjects (All):

Dynamics.

Ergodic theory.

Group theory.

Functional analysis.

Dynamical Systems and Ergodic Theory.

Group Theory and Generalizations.

Functional Analysis.

Local Subjects:

Dynamical Systems and Ergodic Theory.

Group Theory and Generalizations.

Functional Analysis.

Physical Description:

1 online resource (455 pages) : illustrations.

Edition:

1st ed. 2016.

Place of Publication:

Cham : Springer International Publishing : Imprint: Springer, 2016.

Summary:

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.

Contents:

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.

Notes:

Includes bibliographical references.