**Mathematical Surveys and Monographs**

Volume: 101;
2003;
384 pp;
Softcover

MSC: Primary 37; 28; 54; 20;

Print ISBN: 978-1-4704-1951-6

Product Code: SURV/101.S

List Price: $98.00

AMS Member Price: $78.40

MAA member Price: $88.20

**Electronic ISBN: 978-1-4704-1328-6
Product Code: SURV/101.E**

List Price: $98.00

AMS Member Price: $78.40

MAA member Price: $88.20

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#### Supplemental Materials

# Ergodic Theory via Joinings

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*Eli Glasner*

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 & Endorsements

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

# Table of Contents

## Ergodic Theory via Joinings

- Contents vii8 free
- Introduction 114 free
- Part 1. General Group Actions 1124 free
- Chapter 1. Topological Dynamics 1326
- 1. Topological transitivity, minimality 1326
- 2. Equicontinuity and distality 1831
- 3. Proximality and weak mixing 2235
- 4. The enveloping semigroup 2841
- 5. Pointed systems and their corresponding algebras 3245
- 6. Ellis' joint continuity theorem 3346
- 7. Furstenberg's distal structure theorem 3447
- 8. Almost equicontinuity 3447
- 9. Weak almost periodicity 3851
- 10. The unique invariant mean on WAP functions 4255
- 11. Van der Waerden's theorem 4659
- 12. Notes 4760

- Chapter 2. Dynamical Systems on Lebesgue Spaces 4962
- Chapter 3. Ergodicity and Mixing Properties 6174
- 1. Unitary representations 6275
- 2. The Koopman representation 6780
- 3. Rohlin's skew-product theorem 6982
- 4. The ergodic decomposition 7184
- 5. Group and homogeneous skew-products 7285
- 6. Amenable groups 7992
- 7. Ergodicity and weak mixing for Z-systems 8093
- 8. The pointwise ergodic theorem 8396
- 9. Mixing and the Kolmogorov property for Z-systems 8699
- 10. Stationary stochastic processes and dynamical systems 89102
- 11. Gaussian dynamical systems 90103
- 12. Weak mixing of Gaussian systems 91104
- 13. Notes 93106

- Chapter 4. Invariant Measures on Topological Systems 95108
- Chapter 5. Spectral Theory 115128
- Chapter 6. Joinings 125138
- 1. Joinings of two systems 125138
- 2. Composition of joinings and the semigroup of Markov operators 129142
- 3. Group extensions and Veech's theorem 133146
- 4. A joining characterization of homogeneous skew-products 136149
- 5. Finite type joinings 139152
- 6. Disjointness and the relative independence theorem 140153
- 7. Joinings and spectrum 144157
- 8. Notes 144157

- Chapter 7. Some Applications of Joinings 147160
- Chapter 8. Quasifactors 159172
- 1. Factors and quasifactors 160173
- 2. A proof of the ergodic decomposition theorem 163176
- 3. The order of orthogonality of a quasifactor 164177
- 4. The de Finetti-Hewitt-Savage theorem 166179
- 5. Quasifactors and infinite order symmetric self joinings 168181
- 6. Joining quasifactors 170183
- 7. The symmetric product quasifactors 172185
- 8. A weakly mixing system with a non-weakly mixing quasifactor 175188
- 9. Notes 176189

- Chapter 9. Isometric and Weakly Mixing Extensions 177190
- Chapter 10. The Furstenberg-Zimmer Structure Theorem 195208
- Chapter 11. Host's Theorem 205218
- Chapter 12. Simple Systems and Their Self-Joinings 215228
- 1. Group systems 216229
- 2. Factors of simple systems 216229
- 3. Joinings of simple systems I 218231
- 4. JQFs of simple systems 220233
- 5. Joinings of simple systems II 221234
- 6. Pairwise independent joinings of simple Z-systems 223236
- 7. Simplicity of higher orders 226239
- 8. About 2-simple but not 3-simple systems 228241
- 9. Notes 229242

- Chapter 13. Kazhdan's Property and the Geometry of M[sub(Γ)](X) 231244

- Part 2. Entropy Theory for Z-systems 245258
- Chapter 14. Entropy 247260
- Chapter 15. Symbolic Representations 269282
- 1. Symbolic systems 269282
- 2. Kakutani, Rohlin and K-R towers 271284
- 3. Partitions and symbolic representations 273286
- 4. (α, ε, N)-generic points 277290
- 5. An ergodic theorem for towers 279292
- 6. A SMB theorem for towers 281294
- 7. The d-metric 283296
- 8. The Jewett-Krieger theorem 291304
- 9.Notes 296309

- Chapter 16. Constructions 299312
- Chapter 17. The Relation Between Measure and Topological Entropy 307320
- Chapter 18. The Pinsker Algebra, CPE and Zero Entropy Systems 319332
- Chapter 19. Entropy Pairs 329342
- 1. Topological entropy pairs 330343
- 2. Measure entropy pairs 332345
- 3. A measure entropy pair is an entropy pair 334347
- 4. A characterization of E[sub(μ)] 336349
- 5. A measure μ with E[sub(μ)] = E[sub(x)] 337350
- 6. Entropy pairs and the ergodic decomposition 338351
- 7. Measure entropy pairs and factors 340353
- 8. Topological Pinsker factors 341354
- 9. The entropy pairs of a product system 342355
- 10. An application to the proximal relation 344357
- 11. Notes 345358

- Chapter 20. Krieger's and Ornstein's Theorems 347360

- Appendix A. Prerequisite Background and Theorems 363376
- Bibliography 369382
- Index of Symbols 379392 free
- Index of Terms 381394