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Ergodic Theory Via Joinings Mathematical Surveys and Monographs Volume 101 http://dx.doi.org/10.1090/surv/101 Ergodic Theory via Joinings Mathematical Surveys and Monographs Volume 101 Ergodic Theory via Joinings Eli Glasner American Mathematical Society EDITORIAL COMMITTEE Peter S. Landweber Tudor Stefan Ratiu Michael P. Loss, Chair J. T. Stafford 2000 Mathematics Subject Classification. Primary 37Axx, 28Dxx, 37Bxx, 54H20, 20Cxx. Library of Congress Cataloging-in-Publication Data Glasner, Eli, 1945- Ergodic theory via joinings / Eli Glasner. p. cm. — (Mathematical surveys and monographs ; v. 101) Includes bibliographical references and indexes. ISBN 0-8218-3372-3 (alk. paper) 1. Ergodic theory. 2. Topological dynamics. 3. Measure theory. I. Title. II. Mathematical surveys and monographs ; no. 101. QA611.5.G53 2003 515/.42—dc21 2002043617 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to [email protected]. © 2003 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. @ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http: //www. ams. org/ 10 9 8 7 6 5 4 3 2 1 08 07 06 05 04 03 To Hillel Furstenberg Contents Introduction Part 1. General Group Actions 11 Chapter 1. Topological Dynamics 13 1. Topological transitivity, minimality 13 2. Equicontinuity and distality 18 3. Proximality and weak mixing 22 4. The enveloping semigroup 28 5. Pointed systems and their corresponding algebras 32 6. Ellis' joint continuity theorem 33 7. Furstenberg's distal structure theorem 34 8. Almost equicontinuity 34 9. Weak almost periodicity 38 10. The unique invariant mean on WAP functions 42 11. Van der Waerden's theorem 46 12. Notes 47 Chapter 2. Dynamical Systems on Lebesgue Spaces 49 1. Lebesgue spaces 49 2. Dynamical systems and their factors 53 3. The automorphism group and some basic constructions 56 4. Poincare's recurrence theorem 58 5. Notes 59 Chapter 3. Ergodicity and Mixing Properties 61 1. Unitary representations 62 2. The Koopman representation 67 3. Rohlin's skew-product theorem 69 4. The ergodic decomposition 71 5. Group and homogeneous skew-products 72 6. Amenable groups 79 7. Ergodicity and weak mixing for Z-systems 80 8. The pointwise ergodic theorem 83 9. Mixing and the Kolmogorov property for Z-systems 86 10. Stationary stochastic processes and dynamical systems 89 11. Gaussian dynamical systems 90 12. Weak mixing of Gaussian systems 91 13. Notes 93 viii CONTENTS Chapter 4. Invariant Measures on Topological Systems 95 1. Invariant probability measures 96 2. Generic points 98 3. Unique ergodicity 99 4. Examples of strictly ergodic systems 101 5. Minimal Heisenberg nil-systems are strictly ergodic 103 6. The geodesic and horocycle flows 105 7. ^-systems 110 8. Notes 114 Chapter 5. Spectral Theory 115 1. The spectral theorem for a unitary operator 115 2. The spectral invariants of a dynamical system 118 3. The spectral type of a if-system 120 4. Irreducible Koopman representations 121 5. Notes 123 Chapter 6. Joinings 125 1. Joinings of two systems 125 2. Composition of joinings and the semigroup of Markov operators 129 3. Group extensions and Veech's theorem 133 4. A joining characterization of homogeneous skew-products 136 5. Finite type joinings 139 6. Disjointness and the relative independence theorem 140 7. Joinings and spectrum 144 8. Notes 144 Chapter 7. Some Applications of Joinings 147 1. The Halmos-von Neumann theorem 147 2. A joining characterization of mixing 148 3. Mixing of all orders of horocycle flows 150 4. a-weak mixing 153 5. Rudolph's counterexamples machine 156 6. Notes 157 Chapter 8. Quasifactors 159 1. Factors and quasifactors 160 2. A proof of the ergodic decomposition theorem 163 3. The order of orthogonality of a quasifactor 164 4. The de Finetti-Hewitt-Savage theorem 166 5. Quasifactors and infinite order symmetric self joinings 168 6. Joining quasifactors 170 7. The symmetric product quasifactors 172 8. A weakly mixing system with a non-weakly mixing quasifactor 175 9. Notes 176 Chapter 9. Isometric and Weakly Mixing Extensions 177 1. £2(X) as a direct integral of the Hilbert bundle i)(Y) 177 2. Generalized eigenfunctions and isometric extensions 181 3. The Hilbert-Schmidt bundle 184 CONTENTS ix 4. The Y-eigenfunctions of a relative product 190 5. Weakly mixing extensions 192 6. Notes 193 Chapter 10. The Furstenberg-Zimmer Structure Theorem 195 1. /-extensions 196 2. Separating sieves, distal and /-extensions 197 3. The structure theorem 201 4. Factors and quasifactors of distal extensions 203 5. Notes 204 Chapter 11. Host's Theorem 205 1. Pairwise independent joinings 205 2. Mandrekar-Nadkarni's theorem 207 3. Proof of the purity theorem 209 4. Mixing systems of singular type are mixing of all orders 212 5. Notes 213 Chapter 12. Simple Systems and Their Self-Joinings 215 1. Group systems 216 2. Factors of simple systems 216 3. Joinings of simple systems I 218 4. JQFs of simple systems 220 5. Joinings of simple systems II 221 6. Pairwise independent joinings of simple Z-systems 223 7. Simplicity of higher orders 226 8. About 2-simple but not 3-simple systems 228 9. Notes 229 Chapter 13. Kazhdan's Property and the Geometry of Mr(X) 231 1. Strong ergodicity and property T 232 2. A theorem of Bekka and Valette 235 3. A topological characterization of property T 240 4. The Bauer Poulsen dichotomy 241 5. A characterization of the Haagerup property 243 6. Notes 244 Part 2. Entropy Theory for Z-systems 245 Chapter 14. Entropy 247 1. Topological entropy 249 2. Measure entropy 254 3. Applications of the martingale convergence theorem 260 4. Kolmogorov-Sinai theorem 263 5. Shannon-McMillan-Breiman theorem 264 6. Examples 266 7. Notes 267 Chapter 15. Symbolic Representations 269 1. Symbolic systems 269 CONTENTS Kakutani, Rohlin and K-R towers 271 Partitions and symbolic representations 273 (a, e, 7V)-generic points 277 An ergodic theorem for towers 279 A SMB theorem for towers 281 The J-metric 283 The Jewett-Krieger theorem 291 Notes 296 Chapter 16. Constructions 299 1. Rank one systems 299 2. Chacon's transformation 301 3. A rank one a-weakly mixing system 302 4. Notes 305 Chapter 17. The Relation Between Measure and Topological Entropy 307 1. The variational principle 307 2. A combinatorial lemma 310 3. A variational principle for open covers 312 4. An application to expansive systems 315 5. Entropy capacity 316 6. Notes 317 Chapter 18. The Pinsker Algebra, CPE and Zero Entropy Systems 319 1. The Pinsker algebra 319 2. The Rohlin-Sinai theorem 322 3. Zero entropy 325 4. Notes 327 Chapter 19. Entropy Pairs 329 1. Topological entropy pairs 330 2. Measure entropy pairs 332 3. A measure entropy pair is an entropy pair 334 4. A characterization of E^ 336 5. A measure /i with E^ — Ex 337 6. Entropy pairs and the ergodic decomposition 338 7. Measure entropy pairs and factors 340 8. Topological Pinsker factors 341 9. The entropy pairs of a product system 342 10. An application to the proximal relation 344 11. Notes 345 Chapter 20. Krieger's and Ornstein's Theorems 347 1. Ornstein's fundamental lemma 347 2. Krieger's finite generator theorem 352 3. Finitely determined processes 354 4. Bernoulli processes are finitely determined 357 5. Sinai's factor theorem and Ornstein's isomorphism theorem 360 6. Notes 361 CONTENTS xi Appendix A. Prerequisite Background and Theorems 363 Bibliography 369 Index of Symbols 379 Index of Terms 381 APPENDIX A Prerequisite Background and Theorems A.l. THEOREM (The subadditivity lemma). Let {an}^L1 be a sequence of non- negative numbers which is subadditive; i.e. for every m and n, am+n < am + an, then the limit limn^00 an/n exists and equals infn an/n. (See e.g. [257], Theorem 4.9.) Call a commutative Banach algebra B over R, with unit and satisfying ||/2|| = ||/||2,V/ G B, a real commutative C* algebra. A.2. THEOREM. 1. There is a 1-1 correspondence between real commuta­ tive C* algebras B and compact Hausdorff spaces X. Under this correspon­ dence B is isometrically isomorphic with C(X) (the Banach algebra of real valued continuous functions on X with sup norm: \\f\\ — supxeX\f(x)\). The space X is the Gelfand space or the spectrum of B; i.e. the set of all continuous multiplicative homomorphisms of the algebra B into R. The topology on X can be described as the weak* topology induced on X when it is considered as a subset of the dual Banach space B*. The algebra B is separable iff X is metrizable. 2. There is a 1-1 correspondence between closed subalgebras containing the unit 1 G D C B, and continuous onto maps (j) : X —» Y from the compact space X, the spectrum of B, onto the compact space Y, the spectrum of D, where for a functional x G X, (j){x) G Y is the restriction of x to the subalgebra D.
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