Fine Structures of Hyperbolic Diffeomorphisms / Edition 1

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Overview

Fine Structures of Hyperbolic Diffeomorphisms
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Editorial Reviews

From the Publisher
From the reviews:

“The book is a mostly self contained text on the theory of the fine scale structures of hyperbolic diffeomorphisms on surfaces. It is aimed at researchers and Ph. D students interested in the topic. Most of the text is based on the research work of the authors but it also contains related topics and background material. It is clearly written and very well structured.” (Isabel Lugão Rios, Mathematical Reviews, Issue 2010 e)

“The main theme of the book Fine Structures of Hyperbolic Diffeomorphisms, by Pinto, Rand and Ferreira, is the rigidity and flexibility of two-dimensional diffeomorphisms on hyperbolic basic sets and properties of invariant measures that are related to the geometry of these invariant sets. … The book under review is based on a series of articles by the authors and is aimed at experts in the field. The theorems are clearly stated and complete proofs are provided.” (W. De Melo, Bulletin of the American Mathematical Society, Vol. 48 (1), January, 2011)

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Product Details

  • ISBN-13: 9783540875246
  • Publisher: Springer Berlin Heidelberg
  • Publication date: 12/25/2008
  • Series: Springer Monographs in Mathematics Series
  • Edition description: 2009
  • Edition number: 1
  • Pages: 354
  • Product dimensions: 6.20 (w) x 9.40 (h) x 1.00 (d)

Meet the Author

Alberto Pinto Awards& Honours:

2008 Recognition of the scientific merit of the works done in Psychology Sciences by the Psychology School of University of Minho

2007 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& Honours

1986 Whitehead Prize of the London Mathematical Society.

1988 Wolfson Research Award.

1988 Founding editor of Nonlinearity

2004 Fellow, Institute of Mathematics and Its Applications (by invitation)

2005 Britten Lecturer. McMaster University

2006 EPSRC Senior Research Fellowship

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Table of Contents

1 Introduction 1

1.1 Stable and unstable leaves 1

1.2 Marking 3

1.3 Metric 4

1.4 Interval notation 5

1.5 Basic holonomies 6

1.6 Foliated atlas 6

1.7 Foliated atlas A[superscript l] (g, [rho]) 8

1.8 Straightened graph-like charts 10

1.9 Orthogonal atlas 17

1.10 Further literature 19

2 HR structures 21

2.1 Conjugacies 21

2.2 HR - Holder ratios 22

2.3 Foliated atlas A(r) 23

2.4 Invariants 25

2.5 HR Orthogonal atlas 27

2.6 Complete invariant 28

2.7 Moduli space 33

2.8 Further literature 36

3 Solenoid functions 37

3.1 Realized solenoid functions 37

3.2 Holder continuity 38

3.3 Matching condition 38

3.4 Boundary condition 39

3.5 Scaling function 40

3.6 Cylinder-gap condition 41

3.7 Solenoid functions 41

3.8 Further literature 43

4 Self-renormalizable structures 45

4.1 Train-tracks 45

4.2 Charts 47

4.3 Markov maps 47

4.4 Exchange pseudo-groups 48

4.5 Markings 49

4.6 Self-renormalizable structures 51

4.7 Hyperbolic diffeomorphisms 52

4.8 Explosion of smoothness 52

4.9 Further literature 53

5 Rigidity 55

5.1 Complete sets of holonomies 55

5.2 C[superscript 1,1] diffeomorphisms 58

5.3 C[superscript 1,HD superscript l] and cross-ratio distortions for ratio functions 59

5.4 Fundamental Rigidity Lemma 62

5.5 Existence of affine models 65

5.6 Proof of the hyperbolic and Anosov rigidity 67

5.7 Twin leaves for codimension 1 attractors 68

5.8 Non-existence of affine models 70

5.9 Non-existence of uniformly C[superscript 1,HD superscript l] complete sets of holonomies for codimension 1 attractors 71

5.10 Further literature 72

6 Gibbs measures 73

6.1 Dual symbolic sets 73

6.2 Weighted scaling function andJacobian 74

6.3 Weighted ratio structure 75

6.4 Gibbs measure and its dual 76

6.5 Further literature 84

7 Measure scaling functions 85

7.1 Gibbs measures 85

7.2 Extended measure scaling function 86

7.3 Further literature 92

8 Measure solenoid functions 93

8.1 Measure solenoid functions 93

8.1.1 Cylinder-cylinder condition 94

8.2 Measure ratio functions 95

8.3 Natural geometric measures 96

8.4 Measure ratio functions and self-renormalizable structures 99

8.5 Dual measure ratio function 104

8.6 Further literature 106

9 Cocycle-gap pairs 107

9.1 Measure-length ratio cocycle 107

9.2 Gap ratio function 109

9.3 Ratio functions 109

9.4 Cocycle-gap pairs 111

9.5 Further literature 117

10 Hausdorff realizations 119

10.1 One-dimensional realizations of Gibbs measures 119

10.2 Two-dimensional realizations of Gibbs measures 122

10.3 Invariant Hausdorff measures 127

10.3.1 Moduli space SOL[superscript l] 131

10.3.2 Moduli space of cocycle-gap pairs 132

10.3.3 [delta subscript l]-bounded solenoid equivalence class of Gibbs measures 132

10.4 Further literature 134

11 Extended Livsic-Sinai eigenvalue formula 135

11.1 Extending the eigenvalues's result of De la Llave, Marco and Moriyon 135

11.2 Extending the eigenvalue formula of A. N. Livsic and Ja. G. Sinai 140

11.3 Further literature 141

12 Arc exchange systems and renormalization 143

12.1 Arc exchange systems 143

12.1.1 Induced arc exchange systems 145

12.2 Renormalization of arc exchange systems 148

12.2.1 Renormalization of induced arc exchange systems 150

12.3 Markov maps versus renormalization 152

12.4 C[superscript 1+H] flexibility 155

12.5 C[superscript 1,HD] rigidity 156

12.6 Further literature 159

13 Golden tilings (in collaboration with J.P. Almeida and A. Portela) 161

13.1 Golden difeomorphisms 161

13.1.1 Golden train-track 162

13.1.2 Golden arc exchange systems 163

13.1.3 Golden renormalization 165

13.1.4 Golden Markov maps 167

13.2 Anosov diffeomorphisms 168

13.2.1 Golden diffeomorphisms 169

13.2.2 Arc exchange system 170

13.2.3 Markov maps 172

13.2.4 Exchange pseudo-groups 173

13.2.5 Self-renormalizable structures 174

13.3 HR structures 174

13.4 Fibonacci decomposition 175

13.4.1 Matching condition 176

13.4.2 Boundary condition 176

13.4.3 The exponentially fast Fibonacci repetitive property 177

13.4.4 Golden tilings 177

13.4.5 Golden tilings versus solenoid functions 178

13.4.6 Golden tilings versus Anosov diffeomorphisms 181

13.5 Further literature 182

14 Pseudo-Anosov diffeomorphisms in pseudo-surfaces 183

14.1 Affine pseudo-Anosov maps 183

14.2 Paper models [Sigma subscript k] 184

14.3 Pseudo-linear algebra 186

14.4 Pseudo-differentiable maps 191

14.4.1 C[superscript r] pseudo-manifolds 194

14.4.2 Pseudo-tangent spaces 195

14.4.3 Pseudo-inner product on [Sigma subscript k] 195

14.5 C[superscript r] foliations 198

14.6 Further literature 199

A Appendix A: Classifying C[superscript 1+] structures on the real line 201

A.1 The grid 201

A.2 Cross-ratio distortion of grids 202

A.3 Quasisymmetric homeomorphisms 204

A.4 Horizontal and vertical translations of ratio distortions 207

A.5 Uniformly asymptotically affine (uaa) homeomorphisms 214

A.6 C[superscript 1+r] diffeomorphisms 224

A.7 C[superscript 2+r] diffeomorphisms 228

A.8 Cross-ratio distortion and smoothness 232

A.9 Further literature 233

B Appendix B: Classifying C[superscript 1+] structures on Cantor sets 235

B.1 Smooth structures on trees 235

B.1.1 Examples 236

B.2 Basic definitions 239

B.3 (1 + [alpha])-contact equivalence 240

B.3.1 (1 + [alpha]) scale and contact equivalence 241

B.3.2 A refinement of the equivalence property 242

B.3.3 The map L[subscript t] 243

B.3.4 The definition of the contact and gap maps 246

B.3.5 The map L[subscript n] 247

B.3.6 The sequence of maps L[subscript n] converge 247

B.3.7 The map L[subscript infinity] 251

B.3.8 Sufficient condition for C[superscript 1+ alpha superscript -]-equivalent 252

B.3.9 Necessary condition for C[superscript 1+ alpha superscript -]-equivalent 252

B.4 Smooth structures with [alpha]-controlled geometry and bounded geometry 254

B.4.1 Bounded geometry 257

B.5 Further literature 259

C Appendix C: Expanding dynamics of the circle 261

C.1 C[superscript 1+Holder] structures U for the expanding circle map E 261

C.2 Solenoids (E,S) 263

C.3 Solenoid functions s: C [right arrow] R[superscript +] 265

C.4 d-Adic tilings and grids 267

C.5 Solenoidal charts for the C[superscript 1+Holder] expanding circle map E 269

C.6 Smooth properties of solenoidal charts 271

C.7 A Teichmuller space 272

C.8 Sullivan's solenoidal surfaces 273

C.9 (Uaa) structures U for the expanding circle map E 274

C.10 Regularities of the solenoidal charts 275

C.11 Further literature 277

D Appendix D: Markov maps on train-tracks 279

D.1 Cookie-cutters 279

D.2 Pronged singularities in pseudo-Anosov maps 280

D.3 Train-tracks 281

D.3.1 Train-track obtained by glueing 282

D.4 Markov maps 283

D.5 The scaling function 286

D.5.1 A Holder scaling function without a corresponding smooth Markov map 290

D.6 Smoothness of Markov maps and geometry of the cylinder structures 291

D.6.1 Solenoid set 291

D.6.2 Pre-solenoid functions 292

D.6.3 The solenoid property of a cylinder structure 293

D.6.4 The solenoid equivalence between cylinder structures 295

D.7 Solenoid functions 297

D.7.1 Turntable condition 298

D.7.2 Matching condition 298

D.8 Examples of solenoid functions for Markov maps 299

D.8.1 The horocycle maps and the diffeomorphisms of the circle 300

D.8.2 Connections of a smooth Markov map 301

D.9 [alpha]-solenoid functions 302

D.10 Canonical set C of charts 303

D.11 One-to-one correspondences 305

D.12 Existence of eigenvalues for (uaa) Markov maps 307

D.13 Further literature 311

E Appendix E: Explosion of smoothness for Markov families 313

E.1 Markov families on train-tracks 313

E.1.1 Train-tracks 313

E.1.2 Markov families 314

E.1.3 (Uaa) Markov families 315

E.1.4 Bounded Geometry 318

E.2 (Uaa) conjugacies 319

E.3 Canonical charts 324

E.4 Smooth bounds for C[superscript r] Markov families 325

E.4.1 Arzela-Ascoli Theorem 330

E.5 Smooth conjugacies 331

E.6 Further literature 334

References 335

Index 347

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