Fine Structures of Hyperbolic Diffeomorphisms
by Pinto, Alberto Adrego; Rand, David A.; Ferreira, FlavioFormat: Hardcover
Pub. Date: 2008-11-30
Publisher(s): Springer Verlag
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Summary
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 Biography
Alberto Pinto Awards& Honours: 2008 Recognition of the scientific merit of the works done in Psychology Sciences by the Psychology School of University of Minho2007 Espinho Town hall Golden Medal by scientific merit 1984Prize Augusto Martins, University of Porto.1981 One of the six winners of the Mini-Olympics of Mathematics in Portugal. David Rand Awards& Honours1986 Whitehead Prize of the London Mathematical Society.1988 Wolfson Research Award. 1988 Founding editor of Nonlinearity2004 Fellow, Institute of Mathematics and Its Applications (by invitation)2005 Britten Lecturer. McMaster University2006 EPSRC Senior Research Fellowship
Table of Contents
| Introduction | p. 1 |
| Stable and unstable leaves | p. 1 |
| Marking | p. 3 |
| Metric | p. 4 |
| Interval notation | p. 5 |
| Basic holonomies | p. 6 |
| Foliated atlas | p. 6 |
| Foliated atlas A[superscript l] (g, [rho]) | p. 8 |
| Straightened graph-like charts | p. 10 |
| Orthogonal atlas | p. 17 |
| Further literature | p. 19 |
| HR structures | p. 21 |
| Conjugacies | p. 21 |
| HR - Holder ratios | p. 22 |
| Foliated atlas A(r) | p. 23 |
| Invariants | p. 25 |
| HR Orthogonal atlas | p. 27 |
| Complete invariant | p. 28 |
| Moduli space | p. 33 |
| Further literature | p. 36 |
| Solenoid functions | p. 37 |
| Realized solenoid functions | p. 37 |
| Holder continuity | p. 38 |
| Matching condition | p. 38 |
| Boundary condition | p. 39 |
| Scaling function | p. 40 |
| Cylinder-gap condition | p. 41 |
| Solenoid functions | p. 41 |
| Further literature | p. 43 |
| Self-renormalizable structures | p. 45 |
| Train-tracks | p. 45 |
| Charts | p. 47 |
| Markov maps | p. 47 |
| Exchange pseudo-groups | p. 48 |
| Markings | p. 49 |
| Self-renormalizable structures | p. 51 |
| Hyperbolic diffeomorphisms | p. 52 |
| Explosion of smoothness | p. 52 |
| Further literature | p. 53 |
| Rigidity | p. 55 |
| Complete sets of holonomies | p. 55 |
| C[superscript 1,1] diffeomorphisms | p. 58 |
| C[superscript 1,HD superscript l] and cross-ratio distortions for ratio functions | p. 59 |
| Fundamental Rigidity Lemma | p. 62 |
| Existence of affine models | p. 65 |
| Proof of the hyperbolic and Anosov rigidity | p. 67 |
| Twin leaves for codimension 1 attractors | p. 68 |
| Non-existence of affine models | p. 70 |
| Non-existence of uniformly C[superscript 1,HD superscript l] complete sets of holonomies for codimension 1 attractors | p. 71 |
| Further literature | p. 72 |
| Gibbs measures | p. 73 |
| Dual symbolic sets | p. 73 |
| Weighted scaling function and Jacobian | p. 74 |
| Weighted ratio structure | p. 75 |
| Gibbs measure and its dual | p. 76 |
| Further literature | p. 84 |
| Measure scaling functions | p. 85 |
| Gibbs measures | p. 85 |
| Extended measure scaling function | p. 86 |
| Further literature | p. 92 |
| Measure solenoid functions | p. 93 |
| Measure solenoid functions | p. 93 |
| Cylinder-cylinder condition | p. 94 |
| Measure ratio functions | p. 95 |
| Natural geometric measures | p. 96 |
| Measure ratio functions and self-renormalizable structures | p. 99 |
| Dual measure ratio function | p. 104 |
| Further literature | p. 106 |
| Cocycle-gap pairs | p. 107 |
| Measure-length ratio cocycle | p. 107 |
| Gap ratio function | p. 109 |
| Ratio functions | p. 109 |
| Cocycle-gap pairs | p. 111 |
| Further literature | p. 117 |
| Hausdorff realizations | p. 119 |
| One-dimensional realizations of Gibbs measures | p. 119 |
| Two-dimensional realizations of Gibbs measures | p. 122 |
| Invariant Hausdorff measures | p. 127 |
| Moduli space SOL[superscript l] | p. 131 |
| Moduli space of cocycle-gap pairs | p. 132 |
| [delta subscript l]-bounded solenoid equivalence class of Gibbs measures | p. 132 |
| Further literature | p. 134 |
| Extended Livsic-Sinai eigenvalue formula | p. 135 |
| Extending the eigenvalues's result of De la Llave, Marco and Moriyon | p. 135 |
| Extending the eigenvalue formula of A. N. Livsic and Ja. G. Sinai | p. 140 |
| Further literature | p. 141 |
| Arc exchange systems and renormalization | p. 143 |
| Arc exchange systems | p. 143 |
| Induced arc exchange systems | p. 145 |
| Renormalization of arc exchange systems | p. 148 |
| Renormalization of induced arc exchange systems | p. 150 |
| Markov maps versus renormalization | p. 152 |
| C[superscript 1+H] flexibility | p. 155 |
| C[superscript 1,HD] rigidity | p. 156 |
| Further literature | p. 159 |
| Golden tilings (in collaboration with J.P. Almeida and A. Portela) | p. 161 |
| Golden difeomorphisms | p. 161 |
| Golden train-track | p. 162 |
| Golden arc exchange systems | p. 163 |
| Golden renormalization | p. 165 |
| Golden Markov maps | p. 167 |
| Anosov diffeomorphisms | p. 168 |
| Golden diffeomorphisms | p. 169 |
| Arc exchange system | p. 170 |
| Markov maps | p. 172 |
| Exchange pseudo-groups | p. 173 |
| Self-renormalizable structures | p. 174 |
| HR structures | p. 174 |
| Fibonacci decomposition | p. 175 |
| Matching condition | p. 176 |
| Boundary condition | p. 176 |
| The exponentially fast Fibonacci repetitive property | p. 177 |
| Golden tilings | p. 177 |
| Golden tilings versus solenoid functions | p. 178 |
| Golden tilings versus Anosov diffeomorphisms | p. 181 |
| Further literature | p. 182 |
| Pseudo-Anosov diffeomorphisms in pseudo-surfaces | p. 183 |
| Affine pseudo-Anosov maps | p. 183 |
| Paper models [Sigma subscript k] | p. 184 |
| Pseudo-linear algebra | p. 186 |
| Pseudo-differentiable maps | p. 191 |
| C[superscript r] pseudo-manifolds | p. 194 |
| Pseudo-tangent spaces | p. 195 |
| Pseudo-inner product on [Sigma subscript k] | p. 195 |
| C[superscript r] foliations | p. 198 |
| Further literature | p. 199 |
| Appendix A: Classifying C[superscript 1+] structures on the real line | p. 201 |
| The grid | p. 201 |
| Cross-ratio distortion of grids | p. 202 |
| Quasisymmetric homeomorphisms | p. 204 |
| Horizontal and vertical translations of ratio distortions | p. 207 |
| Uniformly asymptotically affine (uaa) homeomorphisms | p. 214 |
| C[superscript 1+r] diffeomorphisms | p. 224 |
| C[superscript 2+r] diffeomorphisms | p. 228 |
| Cross-ratio distortion and smoothness | p. 232 |
| Further literature | p. 233 |
| Appendix B: Classifying C[superscript 1+] structures on Cantor sets | p. 235 |
| Smooth structures on trees | p. 235 |
| Examples | p. 236 |
| Basic definitions | p. 239 |
| (1 + [alpha])-contact equivalence | p. 240 |
| (1 + [alpha]) scale and contact equivalence | p. 241 |
| A refinement of the equivalence property | p. 242 |
| The map L[subscript t] | p. 243 |
| The definition of the contact and gap maps | p. 246 |
| The map L[subscript n] | p. 247 |
| The sequence of maps L[subscript n] converge | p. 247 |
| The map L[subscript infinity] | p. 251 |
| Sufficient condition for C[superscript 1+ alpha superscript -]-equivalent | p. 252 |
| Necessary condition for C[superscript 1+ alpha superscript -]-equivalent | p. 252 |
| Smooth structures with [alpha]-controlled geometry and bounded geometry | p. 254 |
| Bounded geometry | p. 257 |
| Further literature | p. 259 |
| Appendix C: Expanding dynamics of the circle | p. 261 |
| C[superscript 1+Holder] structures U for the expanding circle map E | p. 261 |
| Solenoids (E,S) | p. 263 |
| Solenoid functions s: C [right arrow] R[superscript +] | p. 265 |
| d-Adic tilings and grids | p. 267 |
| Solenoidal charts for the C[superscript 1+Holder] expanding circle map E | p. 269 |
| Smooth properties of solenoidal charts | p. 271 |
| A Teichmuller space | p. 272 |
| Sullivan's solenoidal surfaces | p. 273 |
| (Uaa) structures U for the expanding circle map E | p. 274 |
| Regularities of the solenoidal charts | p. 275 |
| Further literature | p. 277 |
| Appendix D: Markov maps on train-tracks | p. 279 |
| Cookie-cutters | p. 279 |
| Pronged singularities in pseudo-Anosov maps | p. 280 |
| Train-tracks | p. 281 |
| Train-track obtained by glueing | p. 282 |
| Markov maps | p. 283 |
| The scaling function | p. 286 |
| A Holder scaling function without a corresponding smooth Markov map | p. 290 |
| Smoothness of Markov maps and geometry of the cylinder structures | p. 291 |
| Solenoid set | p. 291 |
| Pre-solenoid functions | p. 292 |
| The solenoid property of a cylinder structure | p. 293 |
| The solenoid equivalence between cylinder structures | p. 295 |
| Solenoid functions | p. 297 |
| Turntable condition | p. 298 |
| Matching condition | p. 298 |
| Examples of solenoid functions for Markov maps | p. 299 |
| The horocycle maps and the diffeomorphisms of the circle | p. 300 |
| Connections of a smooth Markov map | p. 301 |
| [alpha]-solenoid functions | p. 302 |
| Canonical set C of charts | p. 303 |
| One-to-one correspondences | p. 305 |
| Existence of eigenvalues for (uaa) Markov maps | p. 307 |
| Further literature | p. 311 |
| Appendix E: Explosion of smoothness for Markov families | p. 313 |
| Markov families on train-tracks | p. 313 |
| Train-tracks | p. 313 |
| Markov families | p. 314 |
| (Uaa) Markov families | p. 315 |
| Bounded Geometry | p. 318 |
| (Uaa) conjugacies | p. 319 |
| Canonical charts | p. 324 |
| Smooth bounds for C[superscript r] Markov families | p. 325 |
| Arzela-Ascoli Theorem | p. 330 |
| Smooth conjugacies | p. 331 |
| Further literature | p. 334 |
| References | p. 335 |
| Index | p. 347 |
| Table of Contents provided by Ingram. All Rights Reserved. |
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