The study of hyperbolic systems is a core theme of modern dynamics. On surfaces the theory of the fine scale structure of hyperbolic invariant sets and their measures can be described in a very complete and elegant way, and is the subject of this book.
The present book, written by well-known experts in smooth dynamical systems and ergodic theory, provides a largely self-contained, rigorous description of this theory. It covers the most important aspects of the subject and is based on several scientific works of the leading research workers in this field. This book fills a gap in the literature of dynamics and is highly recommended for any PhD students interested in this area.
Author(s): Alberto Adrego Pinto, David A. Rand, Flávio Ferreira
Series: Springer Monographs in Mathematics
Edition: 1
Publisher: Springer
Year: 2008
Language: English
Pages: 359
City: Berlin
cover-large.TIF......Page 1
front-matter.pdf......Page 2
Stable and unstable leaves......Page 16
Marking......Page 18
Metric......Page 19
Interval notation......Page 20
Foliated atlas......Page 21
Foliated atlas A(g,)......Page 23
Straightened graph-like charts......Page 25
Orthogonal atlas......Page 32
Further literature......Page 34
Conjugacies......Page 36
HR - Hölder ratios......Page 37
Foliated atlas A(r)......Page 38
Invariants......Page 40
HR Orthogonal atlas......Page 42
Complete invariant......Page 43
Moduli space......Page 48
Further literature......Page 51
Realized solenoid functions......Page 52
Matching condition......Page 53
Boundary condition......Page 54
Scaling function......Page 55
Solenoid functions......Page 56
Further literature......Page 58
Train-tracks......Page 59
Markov maps......Page 61
Exchange pseudo-groups......Page 62
Markings......Page 63
Self-renormalizable structures......Page 65
Explosion of smoothness......Page 66
Further literature......Page 67
Complete sets of holonomies......Page 68
C1,1 diffeomorphisms......Page 71
C1,HD and cross-ratio distortions for ratio functions......Page 72
Fundamental Rigidity Lemma......Page 75
Existence of affine models......Page 78
Proof of the hyperbolic and Anosov rigidity......Page 80
Twin leaves for codimension 1 attractors......Page 81
Non-existence of affine models......Page 83
Non-existence of uniformly C1,HD complete sets of holonomies for codimension 1 attractors......Page 84
Further literature......Page 85
Dual symbolic sets......Page 86
Weighted scaling function and Jacobian......Page 87
Weighted ratio structure......Page 88
Gibbs measure and its dual......Page 89
Further literature......Page 97
Gibbs measures......Page 98
Extended measure scaling function......Page 99
Further literature......Page 105
Measure solenoid functions......Page 106
Cylinder-cylinder condition......Page 107
Measure ratio functions......Page 108
Natural geometric measures......Page 109
Measure ratio functions and self-renormalizable structures......Page 112
Dual measure ratio function......Page 117
Further literature......Page 119
Measure-length ratio cocycle......Page 120
Ratio functions......Page 122
Cocycle-gap pairs......Page 124
Further literature......Page 130
One-dimensional realizations of Gibbs measures......Page 131
Two-dimensional realizations of Gibbs measures......Page 134
Invariant Hausdorff measures......Page 139
Moduli space SOL......Page 143
-bounded solenoid equivalence class of Gibbs measures......Page 144
Further literature......Page 146
Extending the eigenvalues's result of De la Llave, Marco and Moriyon......Page 147
Extending the eigenvalue formula of A. N. Livšic and Ja. G. Sinai......Page 152
Further literature......Page 153
Arc exchange systems......Page 154
Induced arc exchange systems......Page 156
Renormalization of arc exchange systems......Page 159
Renormalization of induced arc exchange systems......Page 161
Markov maps versus renormalization......Page 163
C1+H flexibility......Page 166
C1,HD rigidity......Page 167
Further literature......Page 170
Golden difeomorphisms......Page 171
Golden train-track......Page 172
Golden arc exchange systems......Page 173
Golden renormalization......Page 175
Golden Markov maps......Page 177
Anosov diffeomorphisms......Page 178
Golden diffeomorphisms......Page 179
Arc exchange system......Page 180
Markov maps......Page 182
Exchange pseudo-groups......Page 183
HR structures......Page 184
Fibonacci decomposition......Page 185
Boundary condition......Page 186
Golden tilings......Page 187
Golden tilings versus solenoid functions......Page 188
Golden tilings versus Anosov diffeomorphisms......Page 191
Further literature......Page 192
Affine pseudo-Anosov maps......Page 193
Paper models k......Page 194
Pseudo-linear algebra......Page 196
Pseudo-differentiable maps......Page 201
Cr pseudo-manifolds......Page 204
Pseudo-inner product on k......Page 205
Cr foliations......Page 208
Further literature......Page 209
The grid......Page 210
Cross-ratio distortion of grids......Page 211
Quasisymmetric homeomorphisms......Page 213
Horizontal and vertical translations of ratio distortions......Page 216
Uniformly asymptotically affine (uaa) homeomorphisms......Page 223
C1+r diffeomorphisms......Page 233
C2+r diffeomorphisms......Page 237
Cross-ratio distortion and smoothness......Page 241
Further literature......Page 242
PINTOIND.pdf......Page 0
Smooth structures on trees......Page 244
Examples......Page 245
Basic definitions......Page 248
(1+)-contact equivalence......Page 249
(1+) scale and contact equivalence......Page 250
A refinement of the equivalence property......Page 251
The map Lt......Page 252
The definition of the contact and gap maps......Page 255
The sequence of maps Ln converge......Page 256
The map L......Page 260
Necessary condition for C1+--equivalent......Page 261
Smooth structures with -controlled geometry and bounded geometry......Page 263
Bounded geometry......Page 266
Further literature......Page 268
C1+ H"707Folder structures U for the expanding circle map E......Page 269
Solenoids (E,S)......Page 271
Solenoid functions s:C R+......Page 273
d-Adic tilings and grids......Page 275
Solenoidal charts for the C1+H"707Folder expanding circle map E......Page 277
Smooth properties of solenoidal charts......Page 279
A Teichmüller space......Page 280
Sullivan's solenoidal surfaces......Page 281
(Uaa) structures U for the expanding circle map E......Page 282
Regularities of the solenoidal charts......Page 283
Further literature......Page 285
Cookie-cutters......Page 286
Pronged singularities in pseudo-Anosov maps......Page 287
Train-tracks......Page 288
Train-track obtained by glueing......Page 289
Markov maps......Page 290
The scaling function......Page 293
A Hölder scaling function without a corresponding smooth Markov map......Page 297
Solenoid set......Page 298
Pre-solenoid functions......Page 299
The solenoid property of a cylinder structure......Page 300
The solenoid equivalence between cylinder structures......Page 302
Solenoid functions......Page 304
Matching condition......Page 305
Examples of solenoid functions for Markov maps......Page 306
The horocycle maps and the diffeomorphisms of the circle.......Page 307
Connections of a smooth Markov map.......Page 308
-solenoid functions.......Page 309
Canonical set C of charts......Page 310
One-to-one correspondences......Page 312
Existence of eigenvalues for (uaa) Markov maps......Page 314
Further literature......Page 318
Train-tracks......Page 319
Markov families......Page 320
(Uaa) Markov families......Page 321
Bounded Geometry......Page 324
(Uaa) conjugacies......Page 325
Canonical charts......Page 330
Smooth bounds for Cr Markov families......Page 331
Arzelà-Ascoli Theorem......Page 336
Smooth conjugacies......Page 337
Further literature......Page 340
References......Page 341
Index......Page 353
