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Afraimovich V., Ugalde E. — Fractal Dimensions for Poincare Recurrences
Afraimovich V., Ugalde E. — Fractal Dimensions for Poincare Recurrences



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Название: Fractal Dimensions for Poincare Recurrences

Авторы: Afraimovich V., Ugalde E.

Аннотация:

In this book, the authors introduce and study a new characteristic of complexity of motions in dynamical systems that has recently appeared. The characteristic measures an average return time and can be expressed in the framework of the dimension theory in dynamical systems.To show it, the authors summarized the current state of the modern dynamical systems theory and the theory of fractal dimensions. They use ideas and notions from symbolic and smooth dynamics, like topological entropy and topological pressure, Lyapunov exponents, etc. Therefore the book can serve as a text for special topic courses for graduate students.

Being written on a high level of mathematical rigorousness, the book can be useful for people from applied sciences as well. An "euristic" chapter and examples of applications of the developed theory show how to use dimensions for Poincare recurrences in specific problems of nonlinear dynamics. In particular, an impressive result is related to distributions of Poincare recurrences in a non purely chaotic Hamiltonian systems when the asymptotic law has polynomial tails. The exponent in the law is expressed by means of the dimension for Poincare recurremces.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2006

Количество страниц: 260

Добавлена в каталог: 22.05.2008

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
Adding dyadic      28
Adding machine      17 18 21 23
Adding p-adic      50 104
Adding polyadic      19 28
Adding polyadic, minimal      23
Adding simple      23
Admissible      9 10
Amalgamation      23
Axiom-A      162 187
Birkhoff's theorem      65 146 155 221
Bowen equation      44—47 87 102 110 116 135 175 178
Bowen set      54
Box dimension      9
Brudno's theorem      140
Cantor set      12
Cantori      185
Caratheodory      54
Caratheodory, capacities      44
Caratheodory, construction      53
Caratheodory, structure      53 162
Chebyshev inequality      224
Coding, function      84
Coding, map      35 40 102 164
Coding, procedure      26
Conjugacy      12
Continued fraction      67
critical exponent      192
Critical value      57
CYCLE      17 23—25
Cycle, associated integral      24
Cycle, branching ratio      24—26 28
Cycle, hierarchy      28
Cycle, multiplicities      23
Cycle, successor      24—26
Cylinder      9
Dimension, AP      57
Dimension, Billingsley      56
Dimension, box      43 44 201
Dimension, Caratheodory      54 149 163 167
Dimension, correlation      56
Dimension, for Hamiltonian system      185
Dimension, for Poincare recurrences      48 57 63 64 77 87 102 190 191
Dimension, fractal      16 44 54 191
Dimension, Hausdorff      43 54 73 79 87 91 95 111 157 177
Dimension, Hausdorff, measure      149 161
Dimension, horizontal      129
Dimension, local      136
Dimension, measure      135 149 151 159 161 163
Dimension, measure, theoretical      167
Dimension, minimal sets      65
Dimension, pointwise      135 145 146 151 161 177
Dimension, pointwise, suspended flows      153
Dimension, topological      167
Diophantine approximations      67
Distance      10 117
Distance, Bowen — Walters      117 118
Doubly stochastic operator      221
Duffing      199
Dynamical partition      10
Eigenfunction      20 21 114
Eigenvalue      21
Encoding function      33
Entropy, measure theoretical      171
Entropy, topological      12 29
Feigenbaum attractor      51
Fragmentation      23
Function sign      12
Gauge function      56 57 59 60 63 66 69 95 96
Geometric construction      135
Geometric construction, basic sets      36
Geometric construction, controlled packing      48 77 79 91 96
Geometric construction, gap condition      48 77 91 96
Geometric construction, main      35
Geometric construction, Moran type      35 39 40 43 45 77 83 84 90 101 157 163 171
Geometric construction, Moran type, generalized      37 41 42 46 164
Geometric construction, Moran type, strong      48 95
Geometric construction, Moran-like      10
Geometric construction, one-dimensional      48 78
Geometric construction, sticky set      51 188
Hausdorff      54
Hausdorff, dimension      9
Hopftheorem      223
Invariant manifold      195
Invariant set      9
Isometry      221
Jewett — Krieger theorem      65
K-system      144
Kac's theorem      144 187 219
Kolmogorov complexity      137 138
Koopman's operator      221
Legendre-transform      79 87 161 181
Linking socket      14
Local rule      9
Lyapunov exponent      144 146 149 162 165 176 179
Manneville — Pomeau map      89
Map standard      186 193
Map volume-preserving      220
Markov chain      12 14 15 30 40 45 87 109 110
Markov chain, golden mean      47
Markov chain, mixing      13 14 16
Markov chain, ordered      40 116
Markov chain, topological pressure      29
Markov map      12 40 109 144 160
Markov map, invariant subsets      40
Markov map, non-uniformly hyperbolic      110
Markov partition      87 109
Martingale      136
Maximal cylinder      48
Measure, Borel probability      136
Measure, Hausdorff      43 54
Measure, invariant      135
Measure, outer      77
Measure, uniquely ergodic      20
Metric      10
Metric, standard      10
Metric, ultra-      11
Metric, usual      84
Mixing      9
Mixing, property      13
Mixing, system      14
Mixing, time      9 14
Moran cover      43 45
Moran equation      44 47 51 159 162
Multifractal analysis Poincare recurrences      185 190
Multipermutative minimal      18
Neural network      99
Order lexicographic      12 94
Order relation      12
Partition function      34 92 191
Partition refined      10
Path non-traversing      122
Periodic orbit      15
Periodic sequence      15
Poincare recurrence      56 57
Poincare theorem      217
Polyadic expansion      20
Potential function      29 33 45 56 92 94 97 102 110 113 180
Potential function, Holder continuous      16
Prefix transient      13
Repeller      93
Repeller, conformal      78 83 87 157 160 162 163 178 181
Repeller, conformal, ergodic      175
Repeller, hyperbolic      98 168
Repeller, non-ergodic      178
Repeller, non-uniformly hyperbolic      87
Repeller, uniformly hyperbolic      77
Return time      57 77 99 101 102 136 138 144 164 189
Rhythm function      100
Rhythmical dynamics      99
Rokhlin tower      219
Rotation      66 69 72
Rotation, ofthe circle      63 66 141
Ruelle — Perron — Frobenius operator      113
Separation condition      35
Sequence cycling      25—28
Sequence equivalence      22
Set, approximating      90 94
Set, basic      191
Set, Cantor      35 36 40 69 77 90 101 109 110 189
Set, critical      87—90 93 94
Set, fractal      1 83
Set, minimal      27 59 65 69 77
Set, nonwandering      23
Set, ordered      12
Shannon — McMillan — Breiman theorem      136 137
Shift operator      9
Sierpinsky carpet      188
Socket word      15
Space ultra-metric      11
Specification      10 12
Specification, property      39 80 91 101 102 116 145 169
Specified subshift      9 87 90 92 94 110 117 135 138 153 168
Spectral radius      29 47
Spectrum, for Poincare recurrences      79 83 87 90 94 101 110 119 120 177 181 185 191
Spectrum, for Poincare recurrences, measure      135 171
Spectrum, for sticky set      95
Spectrum, Hentschel — Procaccia      56 160
Spectrum, local      136
Spectrum, Lyapunov exponents      179 183
Spectrum, measure      136 163
Spectrum, multifractal      160
Spectrum, of Lyapunov exponents      78
Spectrum, ofdimensions for Poincare recurrences      53 56 57
Spectrum, variational principle      171
Statistical sum      33 46 97 101 104 130 190
Stickiness      187
Sticky riddle      50
Sticky set      49 95 99 156 185 187 190 192
Sub-additivity      32 137
Subshift      9 10 17
Substitution rule      14
Suspended flow      117 118
Suspended flow, variational principle      172
Symbolic dynamics      9
Synchronization      195 196
Synchronization, dimension for Poincare recurrences      204 206
Synchronization, dimension Hausdorff      205
Synchronization, identical      195
Synchronization, manifold      196
Synchronization, topological      201
System, Anosov      185
System, dissipative      195
System, Hamiltonian      185 190 193
System, hyperbolic      87
System, minimal      18 95
System, multipermutative      17 19 21 28 33 49 95 104
System, multipermutative, distal      27
System, multipermutative, minimal      17 20—23 97 101 161 190
System, multipermutative, nonminimal      23
System, mutually coupled Lorenz      209
System, non-hyperbolic      87
System, polysymbolic      19
System, polysymbolic, minimal      20
System, specified      77
System, symbolic      17 41
System, topologically synchronized      208
System, uniformly hyperbolic      109
Tent map      41
Thermodynamical formalism      167
Time asymptotics      187 192
Time asymptotics, anomalous      188
Topological discrete spectrum      20 21
Topological entropy      39 54 63 64 77 104 150
Topological invariants      20
Topological pressure      16 28 33 56 80 88 102 104 110 114 131 149 167 171 180
Topological pressure, Bowen      28 34
Topological pressure, dimension-like definition      32
Topological pressure, non-invariant sets      33
Topologically conjugate      19
Transition graph      219
Transition matrix      13 29 40 47 109
Traversing path      121
Turing machine      137
Ultrametric      135 136 146 151
Uniquely ergodic      20
Variational Principle      135 167
Variational principle, suspended flow      172
Wandering point      61
Word admissible      10
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