Entropy in Dynamical Systems

Preface......Page 8
Acknowledgments......Page 13
0.1 The leitmotiv......Page 14
0.2 A few words about the history of entropy......Page 16
0.3.1 Entropy in physics......Page 17
0.3.2 Shannon entropy......Page 20
0.3.3 Connection between Shannon and Boltzmann entropy......Page 22
0.3.4 Dynamical entropy......Page 24
0.3.5 Dynamical entropy as data compression rate......Page 26
0.3.6 Entropy as disorder......Page 30
0.4 Conventions......Page 31
Part I Entropy in ergodic theory......Page 34
1.1 Information and entropy of probability vectors......Page 36
1.2 Partitions and sigma-algebras......Page 43
1.3 Information and static entropy of a partition......Page 45
1.4 Conditional static entropy......Page 46
1.5 Conditional entropy via probabilistic tools*......Page 48
1.6 Basic properties of static entropy......Page 49
1.7 Metrics on the space of partitions......Page 55
1.8 Mutual information*......Page 59
1.9 Non-Shannon inequalities*......Page 61
Exercises......Page 64
2.1 Subadditivity......Page 66
2.2 Preliminaries on dynamical systems......Page 70
2.3 Dynamical entropy of a process......Page 73
2.4 Properties of dynamical entropy......Page 78
2.5 Affinity of dynamical entropy......Page 81
2.6 Conditional dynamical entropy via disintegration*......Page 82
2.7 Summary of the properties of entropy......Page 85
2.8 Combinatorial entropy......Page 86
Exercises......Page 91
3.1 Independence and ε-independence......Page 93
3.2 The Pinsker sigma-algebra in a process......Page 98
3.3 The Shannon–McMillan–Breiman Theorem......Page 102
3.4 The Ornstein–Weiss Return Times Theorem......Page 107
3.5 Horizontal data compression......Page 110
Exercises......Page 113
4.1 Entropy of a dynamical system......Page 115
4.2 Generators......Page 118
4.3 The natural extension......Page 124
4.4 Joinings......Page 129
4.5 Ornstein Theory*......Page 133
Exercises......Page 143
5.1 History of the Law of Series......Page 145
5.2 Attracting and repelling in signal processes......Page 148
5.3 Decay of repelling in positive entropy......Page 152
5.3.1 The idea of the proof and the basic lemma......Page 155
5.3.2 The proof of Theorem 5.3.3......Page 157
5.4 Typicality of attracting for long cylinders......Page 165
Part II Entropy in topological dynamics......Page 170
6.1 Three definitions of topological entropy......Page 172
6.1.1 The metric definition via separated orbits......Page 173
6.1.2 The metric definition via spanning orbits......Page 174
6.1.3 The topological definition via covers......Page 175
6.1.4 Relations between the above notions......Page 176
6.2 Properties of topological entropy......Page 178
6.3 Topological conditional and tail entropies......Page 180
6.4 Properties of topological conditional entropy......Page 184
6.5 Topological joinings......Page 185
6.6 The simplex of invariant measures......Page 188
6.7 Topological fiber entropy......Page 192
6.8 The major Variational Principles......Page 194
6.9 Determinism in topological systems......Page 203
6.9.1 Topological analogs of determinism......Page 204
6.9.2 Hierarchy of maximal factors......Page 208
6.10 Topological preimage entropy*......Page 210
Exercises......Page 212
7.1 Zero-dimensional dynamical systems......Page 214
7.2 Topological entropy in dimension zero......Page 215
7.3 The invariant measures in dimension zero......Page 216
7.4 The Variational Principle in dimension zero......Page 218
7.5 Tail entropy and asymptotic h-expansiveness in dimension zero......Page 219
7.6 Principal zero-dimensional extensions......Page 225
Exercises......Page 238
8.1 The type of convergence......Page 240
8.1.1 Introduction to defect theory......Page 241
8.1.2 Repair functions and superenvelopes......Page 247
8.1.3 The transfinite sequence......Page 249
8.1.4 Uniform equivalence......Page 251
8.1.5 The order of accumulation*......Page 254
8.2.1 Alternative computation of EH......Page 257
8.2.2 The case of a Bauer simplex......Page 260
8.2.3 Lifting and push-down techniques......Page 262
8.3.1 Three definitions......Page 267
8.3.2 Properties of the above notions......Page 271
8.4 Entropy structure......Page 276
Exercises......Page 283
9.1 What are symbolic extensions?......Page 285
9.2 The Symbolic Extension Entropy Theorem......Page 287
9.3 Properties of symbolic extension entropy......Page 300
9.4 Symbolic extensions of interval maps......Page 306
Exercises......Page 314
10.1 Margulis–Ruelle Inequality and Pesin Entropy Formula......Page 316
10.2 Tail entropy estimate......Page 320
10.3 Symbolic extensions of smooth systems......Page 321
Part III Entropy theory for operators......Page 324
11.1 A few words on operator dynamics......Page 326
11.2.1 The axioms......Page 329
11.2.2 Elementary properties of operator entropy......Page 333
11.2.3 Asymptotic lattice stability......Page 335
11.2.4 Uniqueness of entropy for doubly stochastic operators......Page 340
11.3 An explicit measure-theoretic definition......Page 342
11.4 Not so bad properties of the operator entropy......Page 345
Exercises......Page 348
12.1 Three definitions......Page 349
12.2 Properties of the topological operator entropy......Page 352
12.3 Half of the variational principle......Page 354
Exercises......Page 356
13.1 Questions on doubly stochastic operators......Page 357
13.2 Questions concerning Markov operators......Page 358
A.1.1 The rectangle rule......Page 360
A.1.3 Nets......Page 361
A.1.4 Semicontinuous functions......Page 362
A.1.5 Exchanging suprema and infima......Page 365
A.2 Convex analysis tools......Page 366
A.2.1 Concave upper semicontinuous functions......Page 367
A.2.3 The barycenter map, harmonic functions......Page 368
A.2.4 Choquet simplices......Page 370
A.2.5 Bauer simplices......Page 373
A.2.6 Separation theorems......Page 374
A.3 Miscellanies......Page 375
Appendix B Conditional S–M–B......Page 379
References......Page 387
Index......Page 399