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| Petersen K.E. — Ergodic theory |
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| Предметный указатель |
Higher-dimensional van der Waerden Theorem 172
Hilbert transform, ergodic 90 108 113 116 119
Hilbert transform, maximal inequality for 110 113 115
Hilbert transform, real-variable 90 107 108 115
Hilbert, D. 173
Hindman’s theorem 172 173 187 195 196
hole 120 121 128
Homogeneous pair 168
Homogeneous space 10
Homogeneous subset 168
Homomorphism 11 21
Homomorphism of cascades 159
Homomorphism of measure algebras 15
Hopf decomposition 123 125 dissipative
Hopf Maximal Ergodic Theorem 75 123 130 131
Hopf, Eberhard 74 75 123
Horocycle cascade 153
Horocycle flow 9 63 211
Horowitz, J. 103
Hurewicz’ Ergodic Theorem 131
Idempotent 158
Incompressible transformation 38 39 126
Independence 239 242 275
Index set (of a skeleton) 289
Induced transformation 12 34 39 40 41 45 56 210 257 259
Infinite product transformation (entropy of) 248
Infinitely recurrent transformation 38 39
Information function 235 238 243
Initary 249 273 282
Invariant set 42 125 126
Inverse limit 12 21 160
Inverse limit , entropy 248
Inverse limit of cascades 160 162
Inversion formula 20
Ionescu Tulcea, Alexandra 260 (see also Bellow)
IP-set 173 195 196
Irreducible stochastic matrix 52 59
Ising model 280
Isometric extension 160
Isomorphism 4 16 58 60 61 63 234 246 254 273 274 275 276 277 278 279 280 301
Isomorphism Theorem for Lebesgue spaces 16 71 187
Jacobs, Konrad 12 187 301
Jensen’s Inequality 18 22
Jewett - Krieger Theorem 175 187 188
Jewett, Robert I. 186
Join of open covers 264
Join of partitions 233 237
joining 281 293 301
Jones, Lee Kenneth 70 163
Jones, R. L. 89 100
Kac, M. 46
Kac’ Theorem 37 46
Kakutani - Hahn Theorem 155
Kakutani - Rokhlin Lemma 48 94 95 tower)
Kakutani equivalence 273
Kakutani fixed point theorem 154
Kakutani skyscraper decomposition 45 48
Kakutani, Shizuo 12 26 27 39 41 64 70 76 82 91 94 154 163 187 210 216
Katok, A. B. 72 211
Katznelson, Yitzhak 27 61 133 165 186 253 279
Keane, Michael 211 281 282
Keplerian element 157
Keynes, Harvey B. 152 153 157 211
Khintchine Recurrence Theorem 37 195
Khintchine, A. I. 6 34 37 98 162 231
Kolmogorov - Sinai theorem 234 244 248
Kolmogorov consistency theorem 22 36
Kolmogorov, A. N. 4 27 62 90 94 98 152 211 227 234 246
Konheim, A. G. 264 267 268 273
Koopman, B. O. 64 65
Krengel, Ulrich 99 258
Krickeberg decomposition 104
Krieger Generator Theorem 244
Krieger, Wolfgang 187 (see also Jewett - Krieger Theorem)
Kronecker system 182
Kronecker Weyl Theorem 250
Kronecker, Leopold 158
Krylov, Nicolas 45
Kuipers, L. 161
Lagrange, J. L. 157
Law of Random Signs 94
Law of the iterated logarithm 98
Least common refinement 233 264
Lebesgue Differentiation Theorem 101 102
Lebesgue set 102
Lebesgue space 16
Lebesgue spectrum 61 62 63
Lebesgue, H. 101
Ledrappier, F. 280
Lehrer, E. 211
Liberto, Francesco di 280
Lind, D. A. 22 61 279
Liou ville’s Theorem 5 22 186
Liouville number 94
Littlewood, J. E. 89 90 91
Local ergodic theorem 79 100 102 114
Loomis, Lynn H. 107
Loosely Bernoulli 211 274
Mackey, George W. 55
Marcus, Brian 76 211 282
Marker Lemma 285
Marker process 283
markers 283
Markov operator 33 120 121
Markov partition 252 253
Markov process 119
Markov shift 7 35 52 56 60
Markov shift, entropy of 246 248
Markov shift, ergodicity of 51
Markov shift, higher mixing of 64
Markov shift, isomorphism to Bernoulli shifts 276 279
Markov shift, strong mixing of 59
Markov shift, weak Bernoulli property of 276
Markov shift, weak mixing of 64
Markov, A. A. 42
Marriage lemmas 284 292 295 296 297 299
Martin, Nathaniel F. G. 73
Martingale 103
Martingale convergence theorem 103 262
Maximal almost periodic factor 182
Maximal equality 74 76 80 81 85 87
Maximal ergodic theorem 27 31 91 92 93
Maximal Ergodic Theorem for flows 76
Maximal Ergodic Theorem for operators 75 130
Maximal function 89 91 119
Maximal function, ergodic 74 75
Maximal function, ergodic Hilbert 113
Maximal function, eventual 119
Maximal function, Hardy - Littlewood 90 100
Maximal function, Hilbert 107
Maximal function, nontangential 90
Maximal inequality 74 76 90 91 100 103 107 110 113 115 116 260
Maximum spectral type 19 61
Maxwell - Boltzmann velocity distribution 228
McAndrew, M. H. 264 267 268 273
McMillan, Brockway 260
Mean ergodic theorem 3 23 37 176 177
Mean Ergodic Theorem in Banach space 26
Mean Ergodic Theorem in Hilbert space 24
Mean motion, problem of 157
Mean sojourn time 44 45
Mean value (of almost-periodic sequence) 137 139
Measure algebra 15
Measure algebra of a measure space 15
Measure-preserving homeomorphism 72
Measure-preserving transformation 2
Meshalkin, L. D. 282
Metric indecomposability 42
Metric space of a measure algebra 16
Metrically prime 221
| Metrically transitive 42
Miles, G. 61 279
Minimal 42 136 150
Misiurewicz, M. 269
Mixing 73 210 strong topological uniform weak
Monothetic group 154
Monotone equivalence 274
Morse sequence (or set) 210 273
Multidimensional Szemer di Theorem 165
Multiple recurrence 46 163 164 186 Furstenberg Furstenberg Ergodic
Natural extension 13 21 280
Nemytskii, V. V. 161
Neveu, J. 104 258
Newton, D. 211
Niederreiter, H. 161
Nonatomic 16
Nonsingular 2 41 131
Nonwandering set 151
One-parameter group 132 (see also Flow)
Optional sampling 104 105
Orbit 2 150
Oren, Ishai 211
Oresme, Nicole 156 157
Ornstein, Donald S. 4 27 59 63 73 88 119 211 273 274 275 276 278 279 280 284
Oscillation set 80
oscillations 80 82
Oxtoby, J. C. 45 72
Parry, William 10 55 211
Parseval’s formula 149
Parthasarathy, K. L. 161
Partition 232
Partition, countable 234 242
Partition, distribution of 275
Petersen, Karl E. 55 76 87 89 94 152 153 157 187 210 211
Phase space 5 228
Physical system 34
Pigeonhole Principle 157
Pinsker’s formula 243
Plancherel theorem 20
Plante, J. 6
Poincar Recurrence Theorem 34 35 37 151 163 229
Poincar, Henri 161 186
Point homomorphism mod 0 17
Point-distal 161
Pointwise almost periodic 154 159
Poisson integral 108
Polit, Steve 233
Pontryagin duality theorem 20
Positive contraction 89 120
Positive upper density 163
Prasad, V. S. 72
Prasjuk, O. S. 63
Prime transformation 211 216 217
PRODUCT 11
Prohorov, Ju. V. 161
Promiscuity number 293
Proximal 154 158 161 173
Quasi-discrete spectrum 55
Quasi-eigenfunction 55
Quasi-ergodic hypothesis 42
Rnyi, A. 58
Rado, R. 162
Rahe, M. 211 224
Random Ergodic Theorem 99
Random walk 85 86
Rank decomposition (of a skeleton) 287
Raoult, J. P. 187
Rationally independent 51
Ratner, Marina 211 280
Realization problem 4 186
Recurrence 4 33 41
Recurrence, regional 151
Recurrence, topological 150
Recurrent point 34 151
Recurrent transformation 38 39
Refinement of a partition 233
Refinement of a society 293
Refinement of an open cover 264
Refining sequence of open covers 265
Regional transitivity 152
Relatively compact extension 184
Relatively dense 37 136 150
Relatively ergodic extension 183
Relatively weakly mixing extension 183 185
Return time 56
Reverse maximal inequality 88
Riesz, Frderic 77 90 91 100
Rising Sun Lemma 77 78 85 90
Robertson, James B. 152 153
Rokhlin, V. A. 17 63 72 95 254
Rosenblatt, J. 94
Rost, H. 131
Rota, Gian - Carlo 284
Rotation of the circle 8 49 156 157 translation)
Rotations of compact abelian groups 8 133 translation)
Roth, Klaus 162
Rothschild, Bruce L. 172
Rudin, Walter 21
Rudolph, Daniel J. 211 233 273 282
Russo, Lucio 280
Ryll - Nardzewski Fixed Point Theorem 155
Ryll - Nardzewski, Czeslaw 155
Sand 120 121 128
Sawyer, S. 91
Scheller, H. 258
Schreier, O. 162
Schur, Issai 162
Second law of thermodynamics 35 229
Semialgebra 14 58
Semilocal operator 114
Separable measure algebra 16
Shannon - McMillan - Breiman Theorem 245 261 263 284 285 289 291
Shannon, С. E. 227 230 232 260
Shapiro, Leonard 157 210 211
Shields, Paul C. 274 280
Shift transformation 7
Sierpinski, Waclaw 108
Sigmund, Karl 161 244 258
Silverstein, M. L. 89
Sinai, Ya. G. 4 42 234 248 249 252 279
Singular integral 107
Skeleton 283 286
Skeleton Lemma 287
Skeleton, decomposition of 287
Skew product 11 21 53 55 100 254
Skyscraper 40 45 48 200 Tower)
Smorodinsky, Meir 211 280 281 282
Society 284 292 301
Society, dual of 293
Source 230 231 242 259
Space mean 42 44
Spectral measure 19 69
Spectral theorem 19 65 68 133 148
Spectral type 55
Speed of approximation 72
Speed of convergence in Ergodic Theorem 90 93
Stationary stochastic process 6 85 87 91 98 248
Stein, Elias 88 89 91 107 119
Stepanov, V. V. 161
Stepin, A. M. 72 221
Stochastic matrix 36 51 52 59
Stong mixing 4 57 58 59 60 61 62 210 211 mixing)
Stong mixing, failure of, with weak mixing 217 219 220
Stong mixing, of a sequence of sets 73
Stong mixing, Ornstein’s condition for 73
Stong mixing, topological 151 152 210 212 214
Stopping time 104 105 106
Strassen, V. 301
Strictly ergodic 152 156 187 194 217
Strong Law of Large Numbers 3
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