Pinto / Rand / Ferreira Fine Structures of Hyperbolic Diffeomorphisms

1. Auflage 2008
ISBN: 978-3-540-87525-3
Verlag: Springer Berlin Heidelberg
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 354 Seiten

Reihe: Springer Monographs in Mathematics

Autoren/Hrsg.

Pinto, Alberto Adrego

Rand, David A.

Ferreira, Flávio

Inhaltsverzeichnis

1;Introduction;16
1.1;Stable and unstable leaves;16
1.2;Marking;18
1.3;Metric;19
1.4;Interval notation;20
1.5;Basic holonomies;21
1.6;Foliated atlas;21
1.7;Foliated atlas A(g,);23
1.8;Straightened graph-like charts;25
1.9;Orthogonal atlas;32
1.10;Further literature;34
2;HR structures;36
2.1;Conjugacies;36
2.2;HR - Hölder ratios;37
2.3;Foliated atlas A(r);38
2.4;Invariants;40
2.5;HR Orthogonal atlas;42
2.6;Complete invariant;43
2.7;Moduli space;48
2.8;Further literature;51
3;Solenoid functions;52
3.1;Realized solenoid functions;52
3.2;Hölder continuity;53
3.3;Matching condition;53
3.4;Boundary condition;54
3.5;Scaling function;55
3.6;Cylinder-gap condition;56
3.7;Solenoid functions;56
3.8;Further literature;58
4;Self-renormalizable structures;59
4.1;Train-tracks;59
4.2;Charts;61
4.3;Markov maps;61
4.4;Exchange pseudo-groups;62
4.5;Markings;63
4.6;Self-renormalizable structures;65
4.7;Hyperbolic diffeomorphisms;66
4.8;Explosion of smoothness;66
4.9;Further literature;67
5;Rigidity;68
5.1;Complete sets of holonomies;68
5.2;C1,1 diffeomorphisms;71
5.3;C1,HD and cross-ratio distortions for ratio functions;72
5.4;Fundamental Rigidity Lemma;75
5.5;Existence of affine models;78
5.6;Proof of the hyperbolic and Anosov rigidity;80
5.7;Twin leaves for codimension 1 attractors;81
5.8;Non-existence of affine models;83
5.9;Non-existence of uniformly C1,HD complete sets of holonomies for codimension 1 attractors;84
5.10;Further literature;85
6;Gibbs measures;86
6.1;Dual symbolic sets;86
6.2;Weighted scaling function and Jacobian;87
6.3;Weighted ratio structure;88
6.4;Gibbs measure and its dual;89
6.5;Further literature;97
7;Measure scaling functions;98
7.1;Gibbs measures;98
7.2;Extended measure scaling function;99
7.3;Further literature;105
8;Measure solenoid functions;106
8.1;Measure solenoid functions;106
8.1.1;Cylinder-cylinder condition;107
8.2;Measure ratio functions;108
8.3;Natural geometric measures;109
8.4;Measure ratio functions and self-renormalizable structures;112
8.5;Dual measure ratio function;117
8.6;Further literature;119
9;Cocycle-gap pairs;120
9.1;Measure-length ratio cocycle;120
9.2;Gap ratio function;122
9.3;Ratio functions;122
9.4;Cocycle-gap pairs;124
9.5;Further literature;130
10;Hausdorff realizations;131
10.1;One-dimensional realizations of Gibbs measures;131
10.2;Two-dimensional realizations of Gibbs measures;134
10.3;Invariant Hausdorff measures;139
10.3.1;Moduli space SOL;143
10.3.2;Moduli space of cocycle-gap pairs;144
10.3.3;-bounded solenoid equivalence class of Gibbs measures;144
10.4;Further literature;146
11;Extended Livšic-Sinai eigenvalue formula;147
11.1;Extending the eigenvalues's result of De la Llave, Marco and Moriyon;147
11.2;Extending the eigenvalue formula of A. N. Livšic and Ja. G. Sinai;152
11.3;Further literature;153
12;Arc exchange systems and renormalization;154
12.1;Arc exchange systems;154
12.1.1;Induced arc exchange systems;156
12.2;Renormalization of arc exchange systems;159
12.2.1;Renormalization of induced arc exchange systems;161
12.3;Markov maps versus renormalization;163
12.4;C1+H flexibility;166
12.5;C1,HD rigidity;167
12.6;Further literature;170
13;Golden tilings (in collaboration with J.P. Almeida and A. Portela);171
13.1;Golden difeomorphisms;171
13.1.1;Golden train-track;172
13.1.2;Golden arc exchange systems;173
13.1.3;Golden renormalization;175
13.1.4;Golden Markov maps;177
13.2;Anosov diffeomorphisms;178
13.2.1;Golden diffeomorphisms;179
13.2.2;Arc exchange system;180
13.2.3;Markov maps;182
13.2.4;Exchange pseudo-groups;183
13.2.5;Self-renormalizable structures;184
13.3;HR structures;184
13.4;Fibonacci decomposition;185
13.4.1;Matching condition;186
13.4.2;Boundary condition;186
13.4.3;The exponentially fast Fibonacci repetitive property;187
13.4.4;Golden tilings;187
13.4.5;Golden tilings versus solenoid functions;188
13.4.6;Golden tilings versus Anosov diffeomorphisms;191
13.5;Further literature;192
14;Pseudo-Anosov diffeomorphisms in pseudo-surfaces;193
14.1;Affine pseudo-Anosov maps;193
14.2;Paper models k;194
14.3;Pseudo-linear algebra;196
14.4;Pseudo-differentiable maps;201
14.4.1;Cr pseudo-manifolds;204
14.4.2;Pseudo-tangent spaces;205
14.4.3;Pseudo-inner product on k;205
14.5;Cr foliations;208
14.6;Further literature;209
15;Appendix A: Classifying C1+ structures on the real line;210
15.1;The grid;210
15.2;Cross-ratio distortion of grids;211
15.3;Quasisymmetric homeomorphisms;213
15.4;Horizontal and vertical translations of ratio distortions;216
15.5;Uniformly asymptotically affine (uaa) homeomorphisms;223
15.6;C1+r diffeomorphisms;233
15.7;C2+r diffeomorphisms;237
15.8;Cross-ratio distortion and smoothness;241
15.9;Further literature;242
16;Appendix B: Classifying C1+ structures on Cantor sets;244
16.1;Smooth structures on trees;244
16.1.1;Examples;245
16.2;Basic definitions;248
16.3;(1+)-contact equivalence;249
16.3.1;(1+) scale and contact equivalence;250
16.3.2;A refinement of the equivalence property;251
16.3.3;The map Lt;252
16.3.4;The definition of the contact and gap maps;255
16.3.5;The map Ln;256
16.3.6;The sequence of maps Ln converge;256
16.3.7;The map L;260
16.3.8;Sufficient condition for C1+--equivalent;261
16.3.9;Necessary condition for C1+--equivalent;261
16.4;Smooth structures with -controlled geometry and bounded geometry;263
16.4.1;Bounded geometry;266
16.5;Further literature;268
17;Appendix C: Expanding dynamics of the circle;269
17.1;C1+ H"707Folder structures U for the expanding circle map E;269
17.2;Solenoids (E,S);271
17.3;Solenoid functions s:C R+;273
17.4;d-Adic tilings and grids;275
17.5;Solenoidal charts for the C1+H"707Folder expanding circle map E;277
17.6;Smooth properties of solenoidal charts;279
17.7;A Teichmüller space;280
17.8;Sullivan's solenoidal surfaces;281
17.9;(Uaa) structures U for the expanding circle map E;282
17.10;Regularities of the solenoidal charts;283
17.11;Further literature;285
18;Appendix D: Markov maps on train-tracks;286
18.1;Cookie-cutters;286
18.2;Pronged singularities in pseudo-Anosov maps;287
18.3;Train-tracks;288
18.3.1;Train-track obtained by glueing;289
18.4;Markov maps;290
18.5;The scaling function;293
18.5.1;A Hölder scaling function without a corresponding smooth Markov map;297
18.6;Smoothness of Markov maps and geometry of the cylinder structures;298
18.6.1;Solenoid set;298
18.6.2;Pre-solenoid functions;299
18.6.3;The solenoid property of a cylinder structure;300
18.6.4;The solenoid equivalence between cylinder structures;302
18.7;Solenoid functions;304
18.7.1;Turntable condition;305
18.7.2;Matching condition;305
18.8; Examples of solenoid functions for Markov maps;306
18.8.1;The horocycle maps and the diffeomorphisms of the circle.;307
18.8.2;Connections of a smooth Markov map.;308
18.9;-solenoid functions.;309
18.10;Canonical set C of charts;310
18.11;One-to-one correspondences;312
18.12;Existence of eigenvalues for (uaa) Markov maps;314
18.13;Further literature;318
19;Appendix E: Explosion of smoothness for Markov families;319
19.1;Markov families on train-tracks;319
19.1.1;Train-tracks;319
19.1.2;Markov families;320
19.1.3;(Uaa) Markov families;321
19.1.4;Bounded Geometry;324
19.2;(Uaa) conjugacies;325
19.3;Canonical charts;330
19.4;Smooth bounds for Cr Markov families;331
19.4.1;Arzelà-Ascoli Theorem;336
19.5;Smooth conjugacies;337
19.6;Further literature;340
20;References;341
21;Index;353