Preface ........................................................ ix
1 Recurrence ................................................... 1
1.1 Invariant measures ...................................... 2
1.2 Poincare recurrence theorem ............................. 4
1.3 Examples ............................................... 10
1.4 Induction .............................................. 23
1.5 Multiple recurrence theorems ........................... 29
2 Existence of invariant measures ............................. 35
2.1 Weak* topology ......................................... 36
2.2 Proof of the existence theorem ......................... 45
2.3 Comments in functional analysis ........................ 49
2.4 Skew-products and natural extensions ................... 53
2.5 Arithmetic progressions ................................ 58
3 Ergodic tlieorems ........................................... 64
3.1 Ergodic theorem of von Neumann ......................... 65
3.2 Birkhoff ergodic theorem ............................... 70
3.3 Subadditive ergodic theorem ............................ 78
3.4 Discrete time and continuous time ...................... 87
4 Ergodicity .................................................. 93
4.1 Ergodic systems ........................................ 94
4.2 Examples .............................................. 100
4.3 Properties of ergodic measures ........................ 116
4.4 Comments in conservative dynamics ..................... 120
5 Ergodic decomposition ...................................... 142
5.1 Ergodic decomposition theorem ......................... 142
5.2 Rokhlin disintegration theorem ........................ 150
6 Unique ergodicity .......................................... 157
6.1 Unique ergodicity ..................................... 157
6.2 Minimality ............................................ 159
6.3 Haar measure .......................................... 162
6.4 Theorem of Weyl ....................................... 173
7 Correlations ............................................... 181
7.1 Mixing syslems ........................................ 182
7.2 Markov shifts ......................................... 190
7.3 Interval exchanges .................................... 200
7.4 Decay of correlations ................................. 208
8 Equivalent systems ......................................... 213
8.1 Ergodic equivalence ................................... 214
8.2 Spectral equivalence .................................. 216
8.3 Discrete spectrum ..................................... 222
8.4 Lebesgue spectrum ..................................... 225
8.5 Lebesgue spaces and ergodic isomoфhism ................ 233
9 Entropy .................................................... 242
9.1 Definition of entropy ................................. 243
9.2 Theorem of Kolmogorov-Sinai ........................... 254
9.3 Local entropy ......................................... 262
9.4 Examples .............................................. 268
9.5 Entropy and equivalence ............................... 274
9.6 Entropy and ergodic decomposition ..................... 286
9.7 Jacobians and the Rokhlin formula ..................... 294
10 Variational principle ...................................... 301
10.1 Topological entropy ................................... 302
10.2 Examples .............................................. 313
10.3 Pressure .............................................. 325
10.4 Variational principle ................................. 338
10.5 Equilibrium states .................................... 345
11 Expanding maps ............................................. 352
11.1 Expanding maps on manifolds ........................... 353
11.2 Dynamics of expanding maps ............................ 362
11.3 Entropy and periodic points ........................... 374
12 Thermodynamic formalism .................................... 380
12.1 Theorem of Ruelle ..................................... 381
12.2 Theorem of Livдic ..................................... 399
12.3 Decay of correlations ................................. 402
12.4 Dimension of conformal repellers ...................... 417
Appendix А Topics in measure theory, topology and analysis .. 430
A.1 Measure spaces ........................................ 430
A.2 Integration in measure spaces ......................... 443
A.3 Measures in metric spaces ............................. 453
A.4 Differentiable manifolds .............................. 460
A.5 Lp(μ) spaces .......................................... 469
A.6 Hilbert spaces ........................................ 473
A.7 Spectral theorems ..................................... 477
Hints or solutions for selected exercises ..................... 482
References .................................................... 504
Index of notation ............................................. 511
Index ......................................................... 515
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