Foliations on surfaces:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin [u.a.]
Springer
2001
|
| Series: | Ergebnisse der Mathematik und ihrer Grenzgebiete
3. Folge ; 41 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Literaturverz. S. [431] - 446 |
| Physical Description: | XXVI, 450 S. Ill., graph. Darst. |
| ISBN: | 3540675248 |
Staff View
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| 245 | 1 | 0 | |a Foliations on surfaces |c Igor Nikolaev |
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 2001 | |
| 300 | |a XXVI, 450 S. |b Ill., graph. Darst. | ||
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| 490 | 1 | |a Ergebnisse der Mathematik und ihrer Grenzgebiete : 3. Folge |v 41 | |
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Record in the Search Index
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|---|---|
| adam_text | Contents
Index of Notation XXV
0. Foliations on 2 Manifolds 1
0.1 Notations 1
0.2 Examples 3
0.2.1 Smooth Functions 3
0.2.2 1 Forms 4
0.2.3 Line Elements 5
0.2.4 Curvature Lines 6
0.2.5 A Diffeomorphisms 8
0.3 Constructions 10
0.3.1 Suspension 10
0.3.2 Measured Foliations 11
0.3.3 Affine Foliations 14
0.3.4 Labyrinths 15
0.3.5 Gluing Together 17
Part I. General Theory
1. Local Theory 21
1.1 Introduction 21
1.2 Symmetry 22
1.3 Normal Forms 22
1.3.1 Typical Normal Forms 22
1.3.2 Degenerate Normal Forms 24
1.4 Structurally Stable Singularities 28
1.4.1 Blowing up Method 28
1.4.2 Fundamental Lemma 30
1.4.3 Classification 33
1.5 Bifurcations 34
XVIII Contents
2. Morse Smale Foliations 37
2.1 Rough Foliations 37
2.1.1 Main Theorem 37
2.1.2 Structural Stability 38
2.1.3 Density 42
2.2 Classification of Morse Smale Foliations 43
2.2.1 Rotation Systems 43
2.2.2 Equivalence Criterion 47
2.2.3 Realization of the Graphs 50
2.2.4 Example 52
2.3 Gradient like Foliations 53
2.3.1 Lyapunov Function 53
2.3.2 Lyapunov Graph 56
2.4 Connected Components of Morse Smale Foliations 57
2.5 Degrees of Stability 59
3. Foliations Without Holonomy 67
3.1 Periodic Components 67
3.2 Quasiminimal Sets 68
3.2.1 Structure of a Quasiminimal Set 69
3.2.2 Blowing Down 73
3.3 Decomposition 81
3.4 Surgery 86
3.4.1 Surgery of Labyrinths 86
3.4.2 Surgery of Measured Foliations 94
3.5 Number of Quasiminimal Sets 104
3.6 Application: Smoothing Theorem 117
4. Invariants of Foliations 125
4.1 Torus 125
4.1.1 Minimal Foliations 125
4.1.2 Foliations With a Cantor Minimal Set 131
4.1.3 Foliations With Cherry Cells 135
4.1.4 Analytic Classification 138
4.2 Homotopy Rotation Class 144
4.2.1 Surfaces of Genus g 2 144
4.2.2 Classification 148
4.2.3 Properties of the Homotopy Rotation Class 155
4.3 Non Orientable Surfaces 156
4.3.1 Torus With the Cross Cap 156
4.3.2 Surfaces of Genus p 4 157
4.4 Discrete Invariants 157
4.4.1 Regular Foliations on the Sphere 157
4.4.2 Orbit Complex 158
4.5 Foliations Without Holonomy 160
Contents XIX
4.5.1 Cells 161
4.5.2 Classification of Elementary Cells 161
4.5.3 Amalgamation of Elementary Cells 162
4.5.4 Conley Lyapunov Peixoto Graph 164
4.5.5 Classification 165
4.6 Foliations With Symmetry 167
4.6.1 Cayley Graph 167
4.6.2 Isomorphism 169
4.6.3 Realization 170
4.7 Homology and Cohomology Invariants 171
4.7.1 Asymptotic Cycles 172
4.7.2 Fundamental Class 175
4.7.3 Cycles of A. Zorich 178
4.8 Smooth Classification 182
4.8.1 Torus and Klein Bottle 182
4.8.2 Surfaces of Genus g 2 184
5. Curves on Surfaces 191
5.1 Curves and the Absolute 191
5.1.1 Notations 191
5.1.2 Background 192
5.1.3 Proof of Weil s Conjectures 194
5.1.4 Theorems of D. V. Anosov 195
5.2 Asymptotic Directions 197
5.2.1 Of Recurrent Semi Trajectory 197
5.2.2 Of Analytic Flow 198
5.2.3 Of Foliation 199
5.2.4 Of Curves With Restriction
on the Geodesic Curvature 199
5.3 Approximation of a Curve 200
5.4 Limit Sets at the Absolute 202
5.5 Geodesic Deviation 203
5.5.1 Deviation Property of Trajectories 203
5.5.2 Deviation From the Geodesic Framework 206
5.5.3 Ramified Coverings 207
5.5.4 Swing of Trajectories 209
5.6 Unbounded Deviation 213
5.6.1 Surfaces of Genus g 2 213
5.6.2 Irrational Direction on Torus 215
5.6.3 Rational Direction on Torus 216
5.7 Family of Curves 216
XX Contents
6. Non compact Surfaces 223
6.1 Foliations in the Plane 223
6.1.1 Non Singular Case 223
6.1.2 Singular Case 225
6.1.3 Level Set of Harmonic Functions 227
6.2 Structural Stability 230
6.3 Examples 232
6.3.1 Depth of the Centre 232
6.3.2 Minimal Sets 233
6.3.3 Minimal Flows 235
6.3.4 Transitive Flows 236
Part II. Applications
7. Ergodic Theory 241
7.1 Notations 241
7.2 Existence of Invariant Measures 242
7.2.1 Liouville s Theorem 242
7.2.2 Torus 244
7.2.3 Surfaces of Genus g 2 246
7.3 Ergodicity 247
7.3.1 Torus 247
7.3.2 Surfaces of Genus g 2 251
7.4 Mixing 251
7.4.1 Torus 252
7.4.2 Surfaces of Genus g 2 255
7.5 Entropy 256
8. Homeomorphisms of the Unit Circle 261
8.1 Denjoy Flow 261
8.2 Cherry Class 263
8.2.1 Cherry Example 263
8.2.2 Flows With One Cell 265
8.2.3 Flows With Several Cells 270
8.3 Foliations on the Sphere 281
8.3.1 Notations 281
8.3.2 Main Result 283
8.3.3 Application to the Labyrinths 288
8.3.4 Appendix: The Dulac Functions 288
8.4 Addendum: Bendixson s Theorem 291
Contents XXI
9. Diffeomorphisms of Surfaces 295
9.1 A diffeomorphisms 295
9.1.1 Attractors of R. V. Plykin 295
9.1.2 One Dimensional Basic Sets on the Sphere 296
9.1.3 Surfaces of Genus g 297
9.2 Singularity Data 298
9.3 Isotopy Classes of Diffeomorphisms 300
10. C* Algebras 305
10.1 Irrational Rotation Algebra 305
10.1.1 Dimension Groups 305
10.1.2 Continued Fractions 307
10.1.3 Effros Shen s Theorem 308
10.1.4 Projections of Aa 309
10.1.5 Morita Equivalence 310
10.1.6 Embedding of Aa 312
10.2 Artin Rotation Algebra 313
10.2.1 Approximationssatz 313
10.2.2 Artin Numbers 315
10.2.3 Applications 322
10.3 K Theory 324
10.3.1 Foliation With Reeb Components 326
10.3.2 Baum Connes Conjecture 326
10.4 C* Algebras of Morse Smale Flows 328
11. Quadratic Differentials 331
11.1 Notations 331
11.2 Local Theory 331
11.2.1 Normal Forms 331
11.2.2 Finite Critical Points 334
11.2.3 Pole of Order 2 335
11.2.4 Higher Order Poles 336
11.3 Global Behaviour of the Trajectories 337
12. Flat Structures 341
12.1 Flat Metric With Cone Singularities 341
12.1.1 Notations 341
12.1.2 Classification of Closed Flat Surfaces 343
12.2 Connection With Quadratic Differentials and
Measured Foliations 345
12.3 Rational Billiards 346
12. 1 Veech Dichotomy 348
XXII Contents
13. Principal Curvature Lines 353
13.1 Local Theory 353
13.1.1 Notations 353
13.1.2 Invariants of the 2 Jets 354
13.1.3 Stability Lemma 356
13.1.4 Classification of Simple Umbilics 360
13.2 Caratheodory Conjecture 364
13.2.1 Notations 364
13.2.2 ^ Geodesies 365
13.2.3 CMC Surfaces 367
13.2.4 Proof of Theorem 13.2.1 368
13.3 Elements of Global Theory 371
13.3.1 Structural Stability 371
13.3.2 Bifurcation of Umbilical Connections 372
14. Differential Equations 375
14.1 Characteristic Curve 375
14.1.1 Background and Notations 375
14.1.2 Theorem of Hartman and Wintner 376
14.2 Classification 378
14.2.1 Generic Singularities 378
14.2.2 Theorem of A. G. Kuzmin 379
15. Positive Differential 2 Forms 383
15.1 Notations 383
15.2 Local Theory 384
15.2.1 Normal Forms 384
15.2.2 Classification 386
15.3 Space of Forms 388
15.3.1 Stable Subspace 388
15.3.2 Theorem of V. Guinez 388
16. Control Theory (by B. Piccoli) 391
16.1 Introduction 391
16.2 Optimal Control 392
16.3 Optimal Flows 393
16.4 Generic Optimal Flows on the Plane 395
16.5 Optimal Flows on 2 Manifolds 397
Contents XXIII
Part III. Appendix
17. Riemann Surfaces 401
17.1 Uniformization Theorem 401
17.2 Discrete Groups 403
17.2.1 Mobius Transformations 403
17.2.2 Fuchsian Group 406
17.2.3 Limit Set of Fuchsian Groups 408
17.2.4 Modular Group 410
17.2.5 Examples 411
17.3 Teichmuller Space 412
17.3.1 Conformal Invariants 412
17.3.2 Quasiconformal Mappings 413
17.3.3 Beltrami Equation 414
17.3.4 Ahlfors Bers Theorem 415
17.3.5 Geometry of Quadratic Differentials 416
17.3.6 Associated Metric 418
17.3.7 Isothermal Coordinates 420
17.4 Complex Curves 421
17.4.1 Projective Curves 421
17.4.2 Degree Genus Formula 424
17.4.3 Elliptic Curves 425
17.4.4 Divisors and the Riemann Roch Theorem 427
17.4.5 Application: Dimension of the Teichmuller Space 429
Bibliography 431
Index 447
|
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| illustrated | Illustrated |
| indexdate | 2024-07-09T18:44:49Z |
| institution | BVB |
| isbn | 3540675248 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-009127196 |
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| series2 | Ergebnisse der Mathematik und ihrer Grenzgebiete : 3. Folge |
| spelling | Nikolaev, Igor 1961- Verfasser (DE-588)12105358X aut Foliations on surfaces Igor Nikolaev Berlin [u.a.] Springer 2001 XXVI, 450 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Ergebnisse der Mathematik und ihrer Grenzgebiete : 3. Folge 41 Literaturverz. S. [431] - 446 Differentialtopologie - Kompakte Fläche - Blätterung Foliations (Mathematics) Manifolds (Mathematics) Fläche (DE-588)4129864-0 gnd rswk-swf Differentialtopologie (DE-588)4012255-4 gnd rswk-swf Blätterung (DE-588)4007006-2 gnd rswk-swf Differentialtopologie (DE-588)4012255-4 s Fläche (DE-588)4129864-0 s Blätterung (DE-588)4007006-2 s DE-604 Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Folge ; 41 (DE-604)BV000899194 41 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009127196&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Nikolaev, Igor 1961- Foliations on surfaces Ergebnisse der Mathematik und ihrer Grenzgebiete Differentialtopologie - Kompakte Fläche - Blätterung Foliations (Mathematics) Manifolds (Mathematics) Fläche (DE-588)4129864-0 gnd Differentialtopologie (DE-588)4012255-4 gnd Blätterung (DE-588)4007006-2 gnd |
| subject_GND | (DE-588)4129864-0 (DE-588)4012255-4 (DE-588)4007006-2 |
| title | Foliations on surfaces |
| title_auth | Foliations on surfaces |
| title_exact_search | Foliations on surfaces |
| title_full | Foliations on surfaces Igor Nikolaev |
| title_fullStr | Foliations on surfaces Igor Nikolaev |
| title_full_unstemmed | Foliations on surfaces Igor Nikolaev |
| title_short | Foliations on surfaces |
| title_sort | foliations on surfaces |
| topic | Differentialtopologie - Kompakte Fläche - Blätterung Foliations (Mathematics) Manifolds (Mathematics) Fläche (DE-588)4129864-0 gnd Differentialtopologie (DE-588)4012255-4 gnd Blätterung (DE-588)4007006-2 gnd |
| topic_facet | Differentialtopologie - Kompakte Fläche - Blätterung Foliations (Mathematics) Manifolds (Mathematics) Fläche Differentialtopologie Blätterung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009127196&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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