Softcover ISBN:  9781470419516 
Product Code:  SURV/101.S 
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AMS Member Price:  $78.40 
Electronic ISBN:  9781470413286 
Product Code:  SURV/101.E 
List Price:  $98.00 
MAA Member Price:  $88.20 
AMS Member Price:  $78.40 

Book DetailsMathematical Surveys and MonographsVolume: 101; 2003; 384 ppMSC: Primary 37; 28; 54; 20;
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 measurepreserving 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 JewettKrieger 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 nonspecialists who wish to learn about modern aspects of ergodic theory.ReadershipGraduate students and research mathematicians interested in ergodic theory.

Table of Contents

Part 1. General group actions

1. Topological dynamics

2. Dynamical systems on Lebesgue spaces

3. Ergodicity and mixing properties

4. Invariant measures on topological systems

5. Spectral theory

6. Joinings

7. Some applications of joinings

8. Quasifactors

9. Isometric and weakly mixing extensions

10. The FurstenbergZimmer structure theorem

11. Host’s theorem

12. Simple systems and their selfjoinings

13. Kazhdan’s property and the geometry of $M_\Gamma (X)$

Part 2. Entropy Theory for Zsystems

14. Entropy

15. Symbolic representations

16. Constructions

17. The relation between measure and topological entropy

18. The Pinsker algebra, CPE and zero entropy systems

19. Entropy pairs

20. Krieger’s and Ornstein’s theorems

Appendix A. Prerequisite background and theorems


Additional Material

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, wideranging bibliography … among books on abstract measuretheoretic 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


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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 measurepreserving 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 JewettKrieger 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 nonspecialists who wish to learn about modern aspects of ergodic theory.
Graduate students and research mathematicians interested in ergodic theory.

Part 1. General group actions

1. Topological dynamics

2. Dynamical systems on Lebesgue spaces

3. Ergodicity and mixing properties

4. Invariant measures on topological systems

5. Spectral theory

6. Joinings

7. Some applications of joinings

8. Quasifactors

9. Isometric and weakly mixing extensions

10. The FurstenbergZimmer structure theorem

11. Host’s theorem

12. Simple systems and their selfjoinings

13. Kazhdan’s property and the geometry of $M_\Gamma (X)$

Part 2. Entropy Theory for Zsystems

14. Entropy

15. Symbolic representations

16. Constructions

17. The relation between measure and topological entropy

18. The Pinsker algebra, CPE and zero entropy systems

19. Entropy pairs

20. Krieger’s and Ornstein’s theorems

Appendix A. Prerequisite background and theorems

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, wideranging bibliography … among books on abstract measuretheoretic 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