Fine structures of hyperbolic diffeomorphisms:
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| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin ; Heidelberg
Springer-Verlag
2009
|
| Series: | Springer monographs in mathematics
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XVI, 353 Seiten Diagramme |
| ISBN: | 9783540875246 |
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MARC
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| 100 | 1 | |a Pinto, Alberto Adrego |d 1964- |0 (DE-588)105394408X |4 aut | |
| 245 | 1 | 0 | |a Fine structures of hyperbolic diffeomorphisms |c Alberto A. Pinto ; David A. Rand ; Flávio Ferreira |
| 264 | 1 | |a Berlin ; Heidelberg |b Springer-Verlag |c 2009 | |
| 300 | |a XVI, 353 Seiten |b Diagramme | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Springer monographs in mathematics | |
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| 650 | 0 | 7 | |a Dynamisches System |0 (DE-588)4013396-5 |2 gnd |9 rswk-swf |
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| 689 | 0 | |5 DE-604 | |
| 700 | 1 | |a Rand, David A. |0 (DE-588)114855331 |4 aut | |
| 700 | 1 | |a Ferreira, Flávio P. |0 (DE-588)1157391117 |4 aut | |
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Record in the Search Index
| _version_ | 1804138617822511104 |
|---|---|
| adam_text | Contents
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 Al(g, p)
.................................... 8
1.8
Straightened graph-like charts
............................. 10
1.9
Orthogonal atlas
........................................ 17
1.10
Further literature
........................................ 19
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
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
XII Contents
4
Self-renormałłzable
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
C1 1 diffeomorphisms
.................................... 58
5.3
C1 11^ 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 C1 1101 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 and Jacobian
.................... 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
Contents XIII
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
........................................
Ill
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 SOL1
................................131
10.3.2
Moduli space of cocycle-gap pairs
...................132
10.3.3
¿¿.-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.
Livšic
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
C1+H flexibility
.........................................155
12.5
Сг но
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
XIV Contents
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
Ą
........................................184
14.3
Pseudo-linear algebra
....................................186
14.4
Pseudo-differentiable maps
...............................191
14.4.1
Cr pseudo-manifolds
...............................194
14.4.2
Pseudo-tangent spaces
.............................195
14.4.3
Pseudo-inner product
onĄ
........................195
14.5
Cr foliations
............................................198
14.6
Further literature
........................................199
A Appendix A: Classifying C1+ structures on the real line
... 201
A.I 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 C1+r diffeomorphisms
....................................224
A.7 C2+r diffeomorphisms
....................................228
A.
8
Cross-ratio distortion and smoothness
......................232
A.9 Further literature
........................................233
В
Appendix B: Classifying C1+ structures on Cantor sets
.... 235
B.I Smooth structures on trees
...............................235
B.I.I Examples
........................................236
B.2 Basic definitions
.........................................239
B.3
(1 +
a)-contact equivalence
...............................240
B.3.1
(1 +
a) scale and contact equivalence
................241
B.3.2 A refinement of the equivalence property
.............242
B.3.3 The map Lt
......................................243
B.3.4 The definition of the contact and gap maps
...........246
B.3.5 The map Ln
......................................247
B.3.
6
The sequence of maps Ln converge
..................247
B.3.7 The map
!„ .....................................251
Contents
XV
В.
3.8
Sufficient condition for C1+a~-equivalent
.............252
B.3.9 Necessary condition for
CXJra
-equivalent
............252
B.4 Smooth structures with
α
-controlled geometry and bounded
geometry
...............................................254
B.4.1 Bounded geometry
................................257
B.5 Further literature
........................................259
С
Appendix C: Expanding dynamics of the circle
............261
C.I C1+Hblder structures
U
for the expanding circle map
E
.......261
C.2 Solenoids
(Ě,Š)
..........................................263
C.3 Solenoid functions s:C~^R+
............................265
C.4 d-Adic tilings and grids
..................................267
C.5 Soienoidal charts for the
C1+Hölder
expanding circle map
E
... 269
С.
6
Smooth properties of soienoidal charts
.....................271
C.7
A
Teichmüller
space
.....................................272
C.8 Sullivan s soienoidal surfaces
..............................273
C.9 (Uaa) structures
U
for the expanding circle map
E
..........274
С.
10
Regularities of the soienoidal charts
........................275
C.ll Further literature
........................................277
D
Appendix D: Markov maps on train-tracks
................279
D.I 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 «-solenoid functions
......................................302
D.10 Canonical set
С
of charts
.................................303
D.ll One-to-one correspondences
...............................305
XVI Contents
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.I Markov families on train-tracks
............................313
E.I.I 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 Cr Markov families
.....................325
E.4.1
Arzelà-Ascoli
Theorem
.............................330
E.5 Smooth conjugacies
......................................331
E.6 Further literature
........................................334
References
.....................................................335
Index
..........................................................347
|
| any_adam_object | 1 |
| author | Pinto, Alberto Adrego 1964- Rand, David A. Ferreira, Flávio P. |
| author_GND | (DE-588)105394408X (DE-588)114855331 (DE-588)1157391117 |
| author_facet | Pinto, Alberto Adrego 1964- Rand, David A. Ferreira, Flávio P. |
| author_role | aut aut aut |
| author_sort | Pinto, Alberto Adrego 1964- |
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| building | Verbundindex |
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| dewey-full | 515.39 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515.39 |
| dewey-search | 515.39 |
| dewey-sort | 3515.39 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV035312800 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T21:31:02Z |
| institution | BVB |
| isbn | 9783540875246 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-017117514 |
| oclc_num | 271643833 |
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| owner | DE-19 DE-BY-UBM DE-355 DE-BY-UBR DE-83 |
| owner_facet | DE-19 DE-BY-UBM DE-355 DE-BY-UBR DE-83 |
| physical | XVI, 353 Seiten Diagramme |
| publishDate | 2009 |
| publishDateSearch | 2009 |
| publishDateSort | 2009 |
| publisher | Springer-Verlag |
| record_format | marc |
| series2 | Springer monographs in mathematics |
| spelling | Pinto, Alberto Adrego 1964- (DE-588)105394408X aut Fine structures of hyperbolic diffeomorphisms Alberto A. Pinto ; David A. Rand ; Flávio Ferreira Berlin ; Heidelberg Springer-Verlag 2009 XVI, 353 Seiten Diagramme txt rdacontent n rdamedia nc rdacarrier Springer monographs in mathematics Hyperbolizität (DE-588)4710615-3 gnd rswk-swf Dynamisches System (DE-588)4013396-5 gnd rswk-swf Diffeomorphismus (DE-588)4149767-3 gnd rswk-swf Dynamisches System (DE-588)4013396-5 s Diffeomorphismus (DE-588)4149767-3 s Hyperbolizität (DE-588)4710615-3 s DE-604 Rand, David A. (DE-588)114855331 aut Ferreira, Flávio P. (DE-588)1157391117 aut Erscheint auch als Online-Ausgabe 978-3-540-87525-3 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017117514&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Pinto, Alberto Adrego 1964- Rand, David A. Ferreira, Flávio P. Fine structures of hyperbolic diffeomorphisms Hyperbolizität (DE-588)4710615-3 gnd Dynamisches System (DE-588)4013396-5 gnd Diffeomorphismus (DE-588)4149767-3 gnd |
| subject_GND | (DE-588)4710615-3 (DE-588)4013396-5 (DE-588)4149767-3 |
| title | Fine structures of hyperbolic diffeomorphisms |
| title_auth | Fine structures of hyperbolic diffeomorphisms |
| title_exact_search | Fine structures of hyperbolic diffeomorphisms |
| title_full | Fine structures of hyperbolic diffeomorphisms Alberto A. Pinto ; David A. Rand ; Flávio Ferreira |
| title_fullStr | Fine structures of hyperbolic diffeomorphisms Alberto A. Pinto ; David A. Rand ; Flávio Ferreira |
| title_full_unstemmed | Fine structures of hyperbolic diffeomorphisms Alberto A. Pinto ; David A. Rand ; Flávio Ferreira |
| title_short | Fine structures of hyperbolic diffeomorphisms |
| title_sort | fine structures of hyperbolic diffeomorphisms |
| topic | Hyperbolizität (DE-588)4710615-3 gnd Dynamisches System (DE-588)4013396-5 gnd Diffeomorphismus (DE-588)4149767-3 gnd |
| topic_facet | Hyperbolizität Dynamisches System Diffeomorphismus |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017117514&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT pintoalbertoadrego finestructuresofhyperbolicdiffeomorphisms AT randdavida finestructuresofhyperbolicdiffeomorphisms AT ferreiraflaviop finestructuresofhyperbolicdiffeomorphisms |