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Petersen K.E. — Ergodic theory

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Название: Ergodic theory

Автор: Petersen K.E.

Аннотация:

The author presents the fundamentals of the ergodic theory of point transformations and several advanced topics of intense research. The study of dynamical systems forms a vast and rapidly developing field even when considering only activity whose methods derive mainly from measure theory and functional analysis. Each of the basic aspects of ergodic theory — examples, convergence theorems, recurrence properties, and entropy — receives a basic and a specialized treatment. The author's accessible style and the profusion of exercises, references, summaries, and historical remarks make this a useful book for graduate students or self study.

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Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1990

Количество страниц: 329

Добавлена в каталог: 31.10.2010

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Предметный указатель

- Theorem      35
-automorphism      62 63 187 273
-recurrent point      186
$\alpha$ , -name      274
$\alpha$ , -name      274
$\alpha$ -generic      274
$\alpha$ -name      274
$\bar{d}$ -distance      221 277 281
$\beta$ -transformation      280
$\mathcal{U}$ , -name      270
$\sigma$ -ideal      15
$\varepsilon$ -entropy      267
$\varepsilon$ -independent      275
$\varepsilon$ -period      136
(, $\varepsilon$ )-separated      266
(, $\varepsilon$ )-spanning      267
Abramov, L. M.      55 254 258
Adler, Roy L.      253 264 267 268 273 280 282
Affine map      10 154
Akcoglu, M. A.      127
Almost periodic function      139 133
Almost periodic point      136 150
Almost periodic pointwise      159
Almost periodic sequence      136 139 140 148
Almost periodic system      182
Almost periodic uniformly      136 137 154
Alpern, Steve      72
Ambrose, W.      12 64
Anosov flow      280
Anosov, D. V.      72
Anzai, Hirotada      53 55
Aperiodic matrix      59
Aperiodic transformation      48
Approximate eigenvalue      56
Approximation proof of the Ergodic Theorem      92
Artin, E.      162
ASSIGNMENT      284
Assignment Lemma      298 299
atom      16
Auslander, L.      10
Autocorrelation      184
Automorphism of a compact group      8 21 60 153
Automorphism of a compact group, Bernoulli property of      279
Automorphism of a compact group, higher mixing of      64
Automorphism of a nilmanifold      61 279
Automorphism of a torus, entropy of      249 259
Automorphism of a torus, ergodicity of      60 61
Automorphism of a torus, strong mixing and Bernoulli properties of      61 254 279
Baker’s transformation      21
Ballot problem      85
Banach, M. Stefan      91
Banach’s Principle      91 99
Baudet, Pierre Joseph Henry      162
Bellow - Furstenberg Theorem      196 200 210
Bellow, A.      187 194
Berg, Kenneth R.      249
Bernoulli flow      279
Bernoulli shift (or scheme)      7 21 60 61 85 86 94 232 248 278
Bernoulli shift (or scheme), classification up to isomorphism      273
Bernoulli shift (or scheme), closure under factors      279
Bernoulli shift (or scheme), conditions for isomorphism to      275 276 278 279
Bernoulli shift (or scheme), construction of strictly ergodic models for      187
Bernoulli shift (or scheme), countable Lebesgue spectrum of      62
Bernoulli shift (or scheme), ergodicity of      49
Bernoulli shift (or scheme), higher mixing of      64
Bernoulli shift (or scheme), isomorphism with      254 277 279 280
Bernoulli shift (or scheme), relation to -automorphisms      62 63
Bernoulli shift (or scheme), strong mixing of      58
Bernoulli shift (or scheme), unilateral      21
Besicovitch, A. S.      142
Bessel’s inequality      148
Billiards      279
Billingsley, Patrick      17
Binary bits      230
Birkhoff Recurrence Theorem      151 168
Birkhoff, George D.      3 30 90 119 151 161 168 186
Blum, J. R.      282
Bochner, S.      139
Bogoliouboff, Nicolas      45
Bohl, P.      157
Bohr - Fourier series      149
Bohr, Harald      139 142 148
Boltzmann, L.      3 42 186 228
Boltzmann’s -Theorem      35
Boltzmann’s constant      228
Boltzmann’s ergodic hypothesis      42
Boole, G.      107 109
Borel, E.      3
Bounded gaps      136 (see also Relatively dense)
Bowen definition of topological entropy      266 267
Bowen, Rufus      168 252 264 267 273
Breiman, Leo      260
Brownian motion      280
Bunimovitch, L. A.      280
Burkholder, D. L.      88 89
Calder $\acute{o}$ n - Zygmund decomposition      107
Calder $\acute{o}$ n. A. P.      91 107 108 114
Carleson, Lennart      260
Cascade      150
Cav $\ddot{a}$ llin      157
Center      151
Chacon - Ornstein Theorem      76 119 121 132
Chacon, R. V.      119 127 211 216
Character group      20
Clausius, Rudolf      227
Coboundary      92
Code      274
Conditional distribution      275
Conditional entropy      235 238
Conditional expectation      17 235
Conditional information function      235 237 242 243 263
Conditional probability      18 235
Conjugate function      90 107
Conservative      38 39 41 121 123 125 126
Conservative part      41 125
Contraction      24 75 120
Converse dominated theorems      88
Cotlar, M.      108
Countable partition      234 242
Covering lemma      107 108
Crossing set      80
Crossings      82
Cutting and stacking      219 223
CYCLE      295
C’arath $\acute{e}$ odory, Constantin      16
Dani, S. G.      61 279
Decomposition (of a skeleton)      287
Dekking, F. M.      211
Del Junco, Andres      94 211 221 224
Denker, Manfred      161 187 244 258 268 282
Density 0      65 70 73 163
Derivative transformation      47 200
Derriennic, Y.      87 89
Diagonal measure      176 178
Diffeomorphism (realization of m      186
Dinaburg, E. I.      264 267
Dirichlet, P. G. Lejeune      157
Discrepancy      94
Discrete spectrum      44 55 64 154
Disintegration of a measure      183 298
Disjointness of skeletons      286
Dissipative      126
Dissipative part      41 123 125
Distal      154 158 159 160
Distribution (of a partition)      275
Dominated ergodic theorem      75 87
Doob, J. L.      7 91 103
Dual (of a society)      292 293
Eberlein, Ernst      187
Ehrenfest, Paul and Tatiana      35 41

Eigenfunction      65 133 148
Eigenfunction, and weak mixing      64 72
Eigenfunction, continuous      153 160
Eigenfunction, generalized      55
Eigenfunction, of a group action      182
Eigenvalue      43 65
Eigenvalue, approximate      56
Ellis semigroup      158 173
Ellis, Robert      158 159 160 161
Endomorphisms of shift dynamical system      221
Engel, David      76 81 82
England, James W.      73
entropy      4 186 187 227 244 variational
Entropy computations, automorphism of the torus      249 259
Entropy computations, Bernoulli shift      245 248
Entropy computations, discrete spectrum system      245
Entropy computations, induced transformation      257 259
Entropy computations, infinite product      248
Entropy computations, inverse limit      248
Entropy computations, Markov shift      246 248
Entropy computations, product transformation      247
Entropy computations, rotation of the circle      245
Entropy computations, skew product      254 259
Entropy computations, strictly ergodic system      273
Entropy computations, subshift of finite type      273
Entropy metric      244 248
Entropy, as a function of the measure      243
Entropy, as an isomorphism invariant      233
Entropy, convergence theorem for      241
Entropy, infinite      259
Entropy, of a -automorphism      63
Entropy, of a Lebesgue spectrum system      63
Entropy, of a partition      232
Entropy, of a source      231 242 259
Entropy, of a transformation with respect to a partition      241 243
Entropy, of a transformation with respect to a partition, definition      233
Entropy, of a transformation with respect to a partition, existence      240
Entropy, of a transformation with respect to a partition, inequality for      242
Entropy, of a transformation, convergence theorem for      247
Entropy, of a transformation, definition      233
Entropy, properties of      238
Entropy, topological      264 265 266 267
Entropy, zero      244
Enveloping semigroup      see Ellis semigroup
Equicontinuous      153 154 156
Equidistribution      156
Equilibrium      34
Equimeasurable functions      114
Equivalence of fillers      284 289
Erdos, P.      162 167
Ergodic decomposition      81
Ergodic Hilbert transform      see Hilbert transform ergodic
Ergodic hypothesis      3 42 186
Ergodic Szemeredi theorem      163 (see also Furstenberg - Katznelson Theorem)
Ergodic Theorem (individual or pointwise)      3 23 27 30 44 52 91 92 93 103 119 198 222 262 Chacon Dominated Hilbert Maximal Mean Random and
Ergodic Theorem (individual or pointwise), approximation proof of      92
Ergodic Theorem (individual or pointwise), for Markov operators      33
Ergodicity      3 41 42 56 58 60 61
Ergodicity, conditions for      57
Ergodicity, examples of      49-54
Ergodicity, relation to topological ergodicity      152
Ergodicity, topological      151 152
Expected return time      35 46
Extension      11
Extension of a cascade      159
Extension with respect to a group action      182
Factor      10 21
Factor of a Bernoulli shift      279
Factor of a cascade      159 162
Factor with respect to a group action      182
Fathi, Albert      72
Fefferman, C.      89 91
Feldman, J.      274
Feller, William      85
Fiber entropy      254
Fiber measures      183
Fiber product      183
Fieldsteel, A.      211
Filler      283
Filler alphabet      283 288
Filler Bernoulli shift      284 288
Filler entropy      288
Filler equivalence      289
Filler Lemma      291 299 300
Filler measure      283 288
Filler set (of a skeleton)      289
Filling scheme      76 119 120 122 123 126 129 130
Finite code      221
Finitely determined      279
Flow      1 76 150
Flow built under a function      11 21
Foguel, Shaul R.      33 123 127
Fourier conjugate      107
Fourier transform      20
Friedman, N. A.      59 276 279
Fundamental theorem of calculus      90 91
Furstenberg - Katznelson Theorem      165 166 185
Furstenberg - Weiss Theorem      164 170 171
Furstenberg Structure Theorem for distal cascades      160 162
Furstenberg, H.      55 133 151 157 159 161 163 164 165 173 186 187 194 211
Gallavotti, Giovanni      279 280
Garsia, Adriano M.      75 89 91
Gauge (of a partition)      196
Gaussian system      8 63
Gelfand, T. M.      189
Gelfand’s problem      50
Generalized discrete spectrum      55
Generalized eigenfunction      55
Generating semialgebra      14
Generator      234 243 244 248 274
GENERIC      4 72 175 274
Geodesic cascade      153
Geodesic flow      9 280
Girsanov, I. V.      63
Glasner, Shmuel      154 155
Goodman, T. N. T.      264
Goodwyn, L. Wayne      264
Gottschalk, W. H.      9 10 136 150 161
Gottschalk’s Theorem      168
Gr $\ddot{u}$ nwald’s Theorem      172 186
Graham, Ronald L.      172
Grant, Edward      156 157
Green, L.      10
Grillenberger, Christian      161 187 244 258
Group of automorphisms of [0, 1]      71
Group rotation      64 181 translations discrete
Gundy, R. F.      88 89
Gurevich, B. M.      63
Gyld $\acute{e}$ n      157
Haar measure      20
Hahn, Frank      10 154 155 186
Hajian, Arshag B.      41
Halmos, Paul R.      17 34 39 48 60 64 71 72
Hamiltonian function      5
Hamilton’s equations      5
Hamming metric      277
Hansel, G.      187
Hanson, D. L.      282
Hard-sphere gas      42 279
Hardy - Littlewood Maximal Function      91
Hardy - Littlewood Maximal Theorem      90
Hardy spaces      89 90
Hardy, G. H.      89 90
Harmonic function      108
Harper, L. H.      284
Hartman, Philip      77
Hecke, E.      162
Hedlund, G. A.      9 10 161 221
Herman, Michael R.      72
Higher-degree mixing      63 64 73

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