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Ergodic Theory via Joinings
Eli Glasner, Tel Aviv University, Israel
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Mathematical Surveys and Monographs
2003; 384 pp; hardcover
Volume: 101
ISBN-10: 0-8218-3372-3
ISBN-13: 978-0-8218-3372-8
List Price: US$98
Member Price: US$78.40
Order Code: SURV/101
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See also:

Lyapunov Exponents and Smooth Ergodic Theory - Luis Barreira and Yakov B Pesin

This book introduces modern ergodic theory. It emphasizes a new approach that relies on the technique of joining two (or more) dynamical systems. This approach has proved to be fruitful in many recent works, and this is the first time that the entire theory is presented from a joining perspective.

Another new feature of the book is the presentation of basic definitions of ergodic theory in terms of the Koopman unitary representation associated with a dynamical system and the invariant mean on matrix coefficients, which exists for any acting groups, amenable or not. Accordingly, the first part of the book treats the ergodic theory for an action of an arbitrary countable group.

The second part, which deals with entropy theory, is confined (for the sake of simplicity) to the classical case of a single measure-preserving transformation on a Lebesgue probability space.

Topics treated in the book include:

  • The interface between topological dynamics and ergodic theory;
  • The theory of distal systems due to H. Furstenberg and R. Zimmer--presented for the first time in monograph form;
  • B. Host's solution of Rohlin's question on the mixing of all orders for systems with singular spectral type;
  • The theory of simple systems;
  • A dynamical characterization of Kazhdan groups;
  • Weiss's relative version of the Jewett-Krieger theorem;
  • Ornstein's isomorphism theorem;
  • A local variational principle and its applications to the theory of entropy pairs.

The book is intended for graduate students who have a good command of basic measure theory and functional analysis and who would like to master the subject. It contains many detailed examples and many exercises, usually with indications of solutions. It can serve equally well as a textbook for graduate courses, for independent study, supplementary reading, or as a streamlined introduction for non-specialists who wish to learn about modern aspects of ergodic theory.

Readership

Graduate students and research mathematicians interested in ergodic theory.

Reviews

"The first book which presents the foundations of ergodic theory in such generality contains a selection of more specialized topics so far only available in research papers. It also includes a good dose of abstract topological dynamics ... a very valuable source of information ... the writing is very clear and precise ... There is an excellent, wide-ranging bibliography ... among books on abstract measure-theoretic ergodic theory, Glasner's is the most ambitious in scope ... there are many topics which are available here for the first time in a book. This is a very impressive achievement which I look forward to returning to often."

-- Mathematical Reviews

Table of Contents

  • Introduction
General group actions
  • Topological dynamics
  • Dynamical systems on Lebesgue spaces
  • Ergodicity and mixing properties
  • Invariant measures on topological systems
  • Spectral theory
  • Joinings
  • Some applications of joinings
  • Quasifactors
  • Isometric and weakly mixing extensions
  • The Furstenberg-Zimmer structure theorem
  • Host's theorem
  • Simple systems and their self-joinings
  • Kazhdan's property and the geometry of \(M_{\Gamma}(\mathbf{X})\)
Entropy theory for \(\mathbb{Z}\)-systems
  • Entropy
  • Symbolic representations
  • Constructions
  • The relation between measure and topological entropy
  • The Pinsker algebra, CPE and zero entropy systems
  • Entropy pairs
  • Krieger's and Ornstein's theorems
  • Prerequisite background and theorems
  • Bibliography
  • Index of symbols
  • Index of terms
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