 | Barreira L. Dimension and recurrence in hyperbolic dynamics / ed. by Bass H., Oesterle' J., Weinstein A. - Basel: Birkhäuser, 2008. - xiv, 300 p. - (Progress in mathematics; 272). - ISBN 978-3-7643-8881-2
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Preface ...................................................... xiii
1. Introduction ................................................. 1
1.1. Dimension and recurrence in hyperbolic dynamics ......... 1
1.2. Contents of the book: a brief tour ...................... 3
2. Basic Notions ................................................ 7
2.1. Dimension theory ........................................ 7
2.2. Ergodic theory ......................................... 13
2.3. Thermodynamic formalism ................................ 14
I Dimension Theory ............................................. 17
3. Dimension Theory and Thermodynamic Formalism ................ 19
3.1. Dimension theory of geometric constructions ............ 19
3.2. Thermodynamic formalism and dimension theory ........... 24
3.2.1. Thermodynamic formalism for symbolic dynamics ... 24
3.2.2. Dimension of limit sets of geometric
constructions ................................... 27
3.3. Nonstationary geometric constructions .................. 31
3.3.1. Nonadditive thermodynamic formalism ............. 31
3.3.2. Dimension of limit sets of nonstationary
constructions ................................... 35
4. Repellers and Hyperbolic Sets ............................... 41
4.1. Dimension of repellers of conformal maps ............... 41
4.2. Hyperbolic sets and Markov partitions .................. 49
4.2.1. Basic notions and product structure ............. 49
4.2.2. Boundaries of Markov partitions ................. 52
4.2.3. Product structure of Gibbs measures ............. 55
4.3. Dimension of hyperbolic sets of conformal maps ......... 59
4.4. Dimension for nonconformal maps: brief notes ........... 64
5. Measures of Maximal Dimension ............................... 67
5.1. Basic notions and basic properties ..................... 67
5.2. Existence of measures of maximal dimension ............. 70
II. Multifractal Analysis: Core Theory ......................... 79
6. Multifractal Analysis of Equilibrium Measures ............... 81
6.1. Dimension spectrum for repellers ....................... 81
6.2. Dimension spectrum for hyperbolic sets ................. 93
7. General Concept of Multifractal Analysis ................... 101
7.1. General concept and basic notions ..................... 101
7.2. The notion of u-dimension ............................. 103
7.3. Multifractal analysis of u-dimension .................. 108
7.4. Domain of the spectra ................................. 1ll
7.5. Existence of spectra with prescribed data ............. 113
7.6. Nondegeneracy of the spectra .......................... 120
8. Dimension of Irregular Sets ................................ 127
8.1. Introduction .......................................... 127
8.2. Irregular sets and distinguishing measures ............ 130
8.3. Existence of distinguishing measures .................. 138
8.4. Topological Markov chains ............................. 139
8.5. Repellers ............................................. 141
8.6. Hyperbolic sets ....................................... 144
9. Variational Principles in Multifractal Analysis ............ 147
9.1. Conditional variational principle ..................... 147
9.2. Topological Markov chains ............................. 156
9.3. Dimension of irregular sets ........................... 160
9.4. Repellers and mixed spectra ........................... 161
III. Multifractal Analysis: Further Developments .............. 165
10. Multidimensional Spectra and Number Theory ................ 167
10.1.Conditional variational principle ..................... 167
10.2.Geometry of the domains ............................... 173
10.3.Regularity of the multifractal spectra ................ 177
10.4.New phenomena in multidimensional spectra ............. 179
10.5.Topological Markov chains ............................. 182
10.6.Finer structure of the spectrum ....................... 184
10.7.Applications to number theory ......................... 186
11.Multifractal Rigidity ...................................... 191
11.1.Multifractal classification of dynamical systems ...... 191
11.2.Entropy spectrum and topological Markov chains ........ 193
11.2.1.Equivalence classes of functions ............... 193
11.2.2.Locally constant functions ..................... 194
11.2.3.Multifractal rigidity for locally constant
functions ...................................... 197
11.2.4.Failure of multifractal rigidity ............... 206
12.Hyperbolic Sets: Past and Future ........................... 209
12.1.A model case: the Smale horseshoe ..................... 209
12.2.Dimension spectra ..................................... 210
12.3.Existence of full measures ............................ 213
12.4.Formula for the spectrum .............................. 219
12.5.Conditional variational principle ..................... 220
IV.Hyperbolicity and Recurrence ............................... 221
13.Pointwise Dimension for Hyperbolic Dynamics ................ 223
13.1.Repellers of conformal maps ........................... 223
13.1.1.Formula for the pointwise dimension ............ 223
13.1.2.Dimension along ergodic decompositions ......... 228
13.2.Hyperbolic sets of conformal maps ..................... 230
13.2.1.Formula for the pointwise dimension ............ 230
13.2.2.Dimension along ergodic decompositions ......... 234
14.Product Structure of Hyperbolic Measures ................... 237
14.1.Nonuniform hyperbolicity .............................. 237
14.2.Dynamical systems with nonzero Lyapunov exponents ..... 239
14.3.Product structure of hyperbolic measures .............. 241
14.4.Product structure of measures in hyperbolic sets ...... 244
15.Quantitative Recurrence and Dimension Theory ............... 255
15.1.Basic notions ......................................... 255
15.2.Upper bounds for recurrence rates ..................... 256
15.3.Recurrence rate and pointwise dimension ............... 261
15.4.Product structure and recurrence ...................... 264
15.4.1.Preliminary results ............................ 264
15.4.2.Stable and unstable recurrence rates ........... 271
15.4.3.Product structure of recurrence ................ 281
Bibliography .................................................. 285
Index ......................................................... 297
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