The main objective of this book is to give a broad unified introduction to the study of dimension and recurrence in hyperbolic dynamics. It includes the discussion of the foundations, main results, and main techniques in the rich interplay of four main areas of research: hyperbolic dynamics, dimension theory, multifractal analysis, and quantitative recurrence. It also gives a panorama of several selected topics of current research interest. All the results are included with detailed proofs, many of them simplified or rewritten on purpose for the book. The text is self-contained.
Author(s): Luis Barreira
Series: Progress in Mathematics
Edition: 1
Publisher: Birkhäuser Boston
Year: 2008
Language: English
Pages: 315
Cover......Page 1
Series: Progress in Mathematics Volume 272......Page 3
Title: Dimension and Recurrence in Hyperbolic Dynamics......Page 4
Copyright - ISBN: 3764388811......Page 5
Ferran Sunyer i Balaguer 2008 prize......Page 6
Contents......Page 10
Preface......Page 14
1.1 Dimension and recurrence in hyperbolic dynamics......Page 16
1.2 Contents of the book: a brief tour......Page 18
2.1 Dimension theory......Page 22
2.2 Ergodic theory......Page 28
2.3 Thermodynamic formalism......Page 29
I. Dimension Theory......Page 32
3.1 Dimension theory of geometric constructions......Page 34
3.2 Thermodynamic formalism and dimension theory......Page 39
3.3 Nonstationary geometric constructions......Page 46
4.1 Dimension of repellers of conformal maps......Page 56
4.2 Hyperbolic sets and Markov partitions......Page 64
4.3 Dimension of hyperbolic sets of conformal maps......Page 74
4.4 Dimension for nonconformal maps: brief notes......Page 79
5.1 Basic notions and basic properties......Page 82
5.2 Existence ofmeasures of maximal dimension......Page 85
II. Multifractal Analysis: Core Theory......Page 94
6.1 Dimension spectrum for repellers......Page 96
6.2 Dimension spectrum for hyperbolic sets......Page 108
7.1 General concept and basic notions......Page 116
7.2 The notion of u-dimension......Page 118
7.3 Multifractal analysis of u-dimension......Page 123
7.4 Domain of the spectra......Page 126
7.5 Existence of spectra with prescribed data......Page 128
7.6 Nondegeneracy of the spectra......Page 135
8.1 Introduction......Page 142
8.2 Irregular sets and distinguishing measures......Page 145
8.3 Existence of distinguishing measures......Page 153
8.4 Topological Markov chains......Page 154
8.5 Repellers......Page 156
8.6 Hyperbolic sets......Page 159
9.1 Conditional variational principle......Page 162
9.2 Topological Markov chains......Page 171
9.3 Dimension of irregular sets......Page 175
9.4 Repellers and mixed spectra......Page 176
III. Multifractal Analysis: Further Developments......Page 180
10.1 Conditional variational principle......Page 182
10.2 Geometry of the domains......Page 188
10.3 Regularity of the multifractal spectra......Page 192
10.4 New phenomena in multidimensional spectra......Page 194
10.5 Topological Markov chains......Page 197
10.6 Finer structure of the spectrum......Page 199
10.7 Applications to number theory......Page 201
11.1 Multifractal classification of dynamical systems......Page 206
11.2 Entropy spectrum and topological Markov chains......Page 208
12.1 A model case: the Smale horseshoe......Page 224
12.2 Dimension spectra......Page 225
12.3 Existence of full measures......Page 228
12.4 Formula for the spectrum......Page 234
12.5 Conditional variational principle......Page 235
IV. Hyperbolicity and Recurrence......Page 236
13.1 Repellers of conformal maps......Page 238
13.2 Hyperbolic sets of conformal maps......Page 245
14.1 Nonuniform hyperbolicity......Page 252
14.2 Dynamical systems with nonzero Lyapunov exponents......Page 254
14.3 Product structure of hyperbolic measures......Page 256
14.4 Product structure of measures in hyperbolic sets......Page 259
15.1 Basic notions......Page 270
15.2 Upper bounds for recurrence rates......Page 271
15.3 Recurrence rate and pointwise dimension......Page 276
15.4 Product structure and recurrence......Page 279
Bibliography......Page 300
F......Page 312
M......Page 313
U......Page 314
V......Page 315
