Ergodic theory via joinings

Cover......Page 1
Title page......Page 2
Contents......Page 6
Introduction......Page 12
Part 1. General Group Actions......Page 22
1. Topological transitivity, minimality......Page 24
2. Equicontinuity and distality......Page 29
3. Proximality and weak mixing......Page 33
4. The enveloping semigroup......Page 39
5. Pointed systems and their corresponding algebras......Page 43
6. Ellis\' joint continuity theorem......Page 44
8. Almost equicontinuity......Page 45
9. Weak almost periodicity......Page 49
10. The unique invariant mean on WAP functions......Page 53
11. Van der Waerden\'s theorem......Page 57
12. Notes......Page 58
1. Lebesgue spaces......Page 60
2. Dynamical systems and their factors......Page 64
3. The automorphism group and some basic constructions......Page 67
4. Poincaré\'s recurrence theorem......Page 69
5. Notes......Page 70
Chapter 3. Ergodicity and Mixing Properties......Page 72
1. Unitary representations......Page 73
2. The Koopman representation......Page 78
3. Rohlin\'s skew-product theorem......Page 80
4. The ergodic decomposition......Page 82
5. Group and homogeneous skew-products......Page 83
6. Amenable groups......Page 90
7. Ergodicity and weak mixing for Z-systems......Page 91
8. The pointwise ergodic theorem......Page 94
9. Mixing and the Kolmogorov property for Z-systems......Page 97
10. Stationary stochastic processes and dynamical systems......Page 100
11. Gaussian dynamical systems......Page 101
12. Weak mixing of Gaussian systems......Page 102
13. Notes......Page 104
Chapter 4. Invariant Measures on Topological Systems......Page 106
1. Invariant probability measures......Page 107
2. Generic points......Page 109
3. Unique ergodicity......Page 110
4. Examples of strictly ergodic systems......Page 112
5. Minimal Heisenberg nil-systems are strictly ergodic......Page 114
6. The geodesic and horocycle flows......Page 116
7. E-systems......Page 121
8. Notes......Page 125
1. The spectral theorem for a unitary operator......Page 126
2. The spectral invariants of a dynamical system......Page 129
3. The spectral type of a K-system......Page 131
4. Irreducible Koopman representations......Page 132
5. Notes......Page 134
1. Joinings of two systems......Page 136
2. Composition of joinings and the semigroup of Markov operators......Page 140
3. Group extensions and Veech\'s theorem......Page 144
4. A joining characterization of homogeneous skew-products......Page 147
5. Finite type joinings......Page 150
6. Disjointness and the relative independence theorem......Page 151
8. Notes......Page 155
1. The Halmos-von Neumann theorem......Page 158
2. A joining characterization of mixing......Page 159
3. Mixing of all orders of horocycle flows......Page 161
4. a-weak mixing......Page 164
5. Rudolph\'s counterexamples machine......Page 167
6. Notes......Page 168
Chapter 8. Quasifactors......Page 170
1. Factors and quasifactors......Page 171
2. A proof of the ergodic decomposition theorem......Page 174
3. The order of orthogonality of a quasifactor......Page 175
4. The de Finetti-Hewitt-Savage theorem......Page 177
5. Quasifactors and infinite order symmetric self joinings......Page 179
6. Joining quasifactors......Page 181
7. The symmetric product quasifactors......Page 183
8. A weakly mixing system with a non-weakly mixing quasifactor......Page 186
9. Notes......Page 187
1. L²(X) as a direct integral of the Hilbert bundle H(Y)......Page 188
2. Generalized eigenfunctions and isometric extensions......Page 192
3. The Hilbert-Schmidt bundle......Page 195
4. The Y-eigenfunctions of a relative product......Page 201
5. Weakly mixing extensions......Page 203
6. Notes......Page 204
Chapter 10. The Furstenberg-Zimmer Structure Theorem......Page 206
1. I-extensions......Page 207
2. Separating sieves, distal and I -extensions......Page 208
3. The structure theorem......Page 212
4. Factors and quasifactors of distal extensions......Page 214
5. Notes......Page 215
1. Pairwise independent joinings......Page 216
2. Mandrekar-Nadkarni\'s theorem......Page 218
3. Proof of the purity theorem......Page 220
4. Mixing systems of singular type are mixing of all orders......Page 223
5. Notes......Page 224
Chapter 12. Simple Systems and Their Self-Joinings......Page 226
2. Factors of simple systems......Page 227
3. Joinings of simple systems I......Page 229
4. JQFs of simple systems......Page 231
5. Joinings of simple systems II......Page 232
6. Pairwise independent joinings of simple Z-systems......Page 234
7. Simplicity of higher orders......Page 237
8. About 2-simple but not 3-simple systems......Page 239
9. Notes......Page 240
Chapter 13. Kazhdan\'s Property and the Geometry of etc......Page 242
1. Strong ergodicity and property T......Page 243
2. A theorem of Bekka and Valette......Page 246
3. A topological characterization of property T......Page 251
4. The Bauer Poulsen dichotomy......Page 252
5. A characterization of the Haagerup property......Page 254
6. Notes......Page 255
Part 2. Entropy Theory for Z-systems......Page 256
Chapter 14. Entropy......Page 258
1. Topological entropy......Page 260
2. Measure entropy......Page 265
3. Applications of the martingale convergence theorem......Page 271
4. Kolmogorov-Sinai theorem......Page 274
5. Shannon-McMillan-Breiman theorem......Page 275
6. Examples......Page 277
7. Notes......Page 278
1. Symbolic systems......Page 280
2. Kakutani, Rohlin and K-R towers......Page 282
3. Partitions and symbolic representations......Page 284
4. (a,€,N)-generic points......Page 288
5. An ergodic theorem for towers......Page 290
6. A SMB theorem for towers......Page 292
7. The d-metric......Page 294
8. The Jewett-Krieger theorem......Page 302
9. Notes......Page 307
1. Rank one systems......Page 310
2. Chacon\'s transformation......Page 312
3. A rank one a-weakly mixing system......Page 313
4. Notes......Page 316
1. The variational principle......Page 318
2. A combinatorial lemma......Page 321
3. A variational principle for open covers......Page 323
4. An application to expansive systems......Page 326
5. Entropy capacity......Page 327
6. Notes......Page 328
1. The Pinsker algebra......Page 330
2. The Rohlin-Sinai theorem......Page 333
3. Zero entropy......Page 336
4. Notes......Page 338
Chapter 19. Entropy Pairs......Page 340
1. Topological entropy pairs......Page 341
2. Measure entropy pairs......Page 343
3. A measure entropy pair is an entropy pair......Page 345
4. A characterization of etc......Page 347
5. A measure mu with etc......Page 348
6. Entropy pairs and the ergodic decomposition......Page 349
7. Measure entropy pairs and factors......Page 351
8. Topological Pinsker factors......Page 352
9. The entropy pairs of a product system......Page 353
10. An application to the proximal relation......Page 355
11. Notes......Page 356
1. Ornstein\'s fundamental lemma......Page 358
2. Krieger\'s finite generator theorem......Page 363
3. Finitely determined processes......Page 365
4. Bernoulli processes are finitely determined......Page 368
5. Sinai\'s factor theorem and Ornstein\'s isomorphism theorem......Page 371
6. Notes......Page 372
Appendix A. Prerequisite Background and Theorems......Page 374
Bibliography......Page 380
Index of Symbols......Page 390
Index of Terms......Page 392