Foliations on Surfaces

By: Igor Nikolaev

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Hardcover | 1 December 2000

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480 Pages

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23.5 x 16.51 x 3.18

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Foliations is one of the major concepts of modern geometry and topology meaning a partition of topological space into a disjoint sum of leaves. This book is devoted to geometry and topology of surface foliations and their links to ergodic theory, dynamical systems, complex analysis, differential and noncommutative geometry. This comprehensive book addresses graduate students and researchers and will serve as a reference book for experts in the field.

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Published: 9th December 2010

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Index of Notation	p. XXV
Foliations on 2-Manifolds	p. 1
Notations	p. 1
Examples	p. 3
Smooth Functions	p. 3
1-Forms	p. 4
Line Elements	p. 5
Curvature Lines	p. 6
A-Diffeomorphisms	p. 7
Constructions	p. 10
Suspension	p. 10
Measured Foliations	p. 11
Affine Foliations	p. 14
Labyrinths	p. 15
Gluing Together	p. 16
General Theory
Local Theory	p. 21
Introduction	p. 21
Symmetry	p. 22
Normal Forms	p. 22
Typical Normal Forms	p. 22
Degenerate Normal Forms	p. 24
Structurally Stable Singularities	p. 28
Blowing-up Method	p. 28
Fundamental Lemma	p. 30
Classification	p. 33
Bifurcations	p. 34
Morse-Smale Foliations	p. 37
Rough Foliations	p. 37
Main Theorem	p. 37
Structural Stability	p. 38
Density	p. 42
Classification of Morse-Smale Foliations	p. 43
Rotation Systems	p. 43
Equivalence Criterion	p. 47
Realization of the Graphs	p. 50
Example	p. 52
Gradient-like Foliations	p. 53
Lyapunov Function	p. 53
Lyapunov Graph	p. 56
Connected Components of Morse-Smale Foliations	p. 57
Degrees of Stability	p. 59
Foliations Without Holonomy	p. 67
Periodic Components	p. 67
Quasiminimal Sets	p. 68
Structure of a Quasiminimal Set	p. 69
Blowing-Down	p. 73
Decomposition	p. 81
Surgery	p. 86
Surgery of Labyrinths	p. 86
Surgery of Measured Foliations	p. 94
Number of Quasiminimal Sets	p. 104
Application: Smoothing Theorem	p. 117
Invariants of Foliations	p. 125
Torus	p. 125
Minimal Foliations	p. 125
Foliations With a Cantor Minimal Set	p. 131
Foliations With Cherry Cells	p. 135
Analytic Classification	p. 138
Homotopy Rotation Class	p. 144
Surfaces of Genus g ≥ 2	p. 144
Classification	p. 148
Properties of the Homotopy Rotation Class	p. 155
Non Orientable Surfaces	p. 156
Torus With the Cross-Cap	p. 156
Surfaces of Genus p ≥ 4	p. 157
Discrete Invariants	p. 157
Regular Foliations on the Sphere	p. 157
Orbit Complex	p. 158
Foliations Without Holonomy	p. 160
Cells	p. 161
Classification of Elementary Cells	p. 161
Amalgamation of Elementary Cells	p. 162
Conley-Lyapunov-Peixoto Graph	p. 164
Classification	p. 165
Foliations With Symmetry	p. 167
Cayley Graph	p. 167
Isomorphism	p. 169
Realization	p. 170
Homology and Cohomology Invariants	p. 171
Asymptotic Cycles	p. 172
Fundamental Class	p. 175
Cycles of A. Zorich	p. 178
Smooth Classification	p. 182
Torus and Klein Bottle	p. 182
Surfaces of Genus g ≥ 2	p. 184
Curves on Surfaces	p. 191
Curves and the Absolute	p. 191
Notations	p. 191
Background	p. 192
Proof of Weil's Conjectures	p. 194
Theorems of D. V. Anosov	p. 195
Asymptotic Directions	p. 197
Of Recurrent Semi-Trajectory	p. 197
Of Analytic Flow	p. 198
Of Foliation	p. 199
Of Curves With Restriction on the Geodesic Curvature	p. 199
Approximation of a Curve	p. 200
Limit Sets at the Absolute	p. 202
Geodesic Deviation	p. 203
Deviation Property of Trajectories	p. 203
Deviation From the Geodesic Framework	p. 206
Ramified Coverings	p. 207
Swing of Trajectories	p. 209
Unbounded Deviation	p. 213
Surfaces of Genus g ≥ 2	p. 213
Irrational Direction on Torus	p. 215
Rational Direction on Torus	p. 216
Family of Curves	p. 216
Non-compact Surfaces	p. 223
Foliations in the Plane	p. 223
Non Singular Case	p. 223
Singular Case	p. 225
Level Set of Harmonic Functions	p. 227
Structural Stability	p. 230
Examples	p. 232
Depth of the Centre	p. 232
Minimal Sets	p. 233
Minimal Flows	p. 235
Transitive Flows	p. 236
Applications
Ergodic Theory	p. 241
Notations	p. 241
Existence of Invariant Measures	p. 242
Liouville's Theorem	p. 242
Torus	p. 244
Surfaces of Genus g ≥ 2	p. 246
Ergodicity	p. 247
Torus	p. 247
Surfaces of Genus g ≥ 2	p. 251
Mixing	p. 251
Torus	p. 252
Surfaces of Genus g ≥ 2	p. 255
Entropy	p. 256
Homeomorphisms of the Unit Circle	p. 261
Denjoy Flow	p. 261
Cherry Class	p. 263
Cherry Example	p. 263
Flows With One Cell	p. 265
Flows With Several Cells	p. 270
Foliations on the Sphere	p. 281
Notations	p. 281
Main Result	p. 283
Application to the Labyrinths	p. 288
Appendix: The Dulac Functions	p. 288
Addendum: Bendixson's Theorem	p. 291
Diffeomorphisms of Surfaces	p. 295
A-diffeomorphisms	p. 295
Attractors of R. V. Plykin	p. 295
One-Dimensional Basic Sets on the Sphere	p. 296
Surfaces of Genus g ≥ 1	p. 297
Singularity Data	p. 298
Isotopy Classes of Diffeomorphisms	p. 300
C*-Algebras	p. 305
Irrational Rotation Algebra	p. 305
Dimension Groups	p. 305
Continued Fractions	p. 307
Effros-Shen's Theorem	p. 308
Projections of A_¿	p. 309
Morita Equivalence	p. 310
Embedding of A_¿	p. 312
Artin Rotation Algebra	p. 313
Approximationssatz	p. 313
Artin Numbers	p. 315
Applications	p. 322
K Theory	p. 324
Foliation With Reeb Components	p. 326
Baum-Connes Conjecture	p. 326
C^*-Algebras of Morse-Smale Flows	p. 328
Quadratic Differentials	p. 331
Notations	p. 331
Local Theory	p. 331
Normal Forms	p. 331
Finite Critical Points	p. 334
Pole of Order 2	p. 335
Higher Order Poles	p. 336
Global Behaviour of the Trajectories	p. 337
Flat Structures	p. 341
Flat Metric With Cone Singularities	p. 341
Notations	p. 341
Classification of Closed Flat Surfaces	p. 343
Connection With Quadratic Differentials and Measured Foliations	p. 345
Rational Billiards	p. 346
Veech Dichotomy	p. 348
Principal Curvature Lines	p. 353
Local Theory	p. 353
Notations	p. 353
Invariants of the 2-Jets	p. 354
Stability Lemma	p. 356
Classification of Simple Umbilics	p. 360
Carathéodory Conjecture	p. 364
Notations	p. 364
ϕ-Geodesics	p. 365
CMC-Surfaces	p. 367
Proof of Theorem 13.2.1	p. 368
Elements of Global Theory	p. 371
Structural Stability	p. 371
Bifurcation of Umbilical Connections	p. 372
Differential Equations	p. 375
Characteristic Curve	p. 375
Background and Notations	p. 375
Theorem of Hartman and Wintner	p. 376
Classification	p. 378
Generic Singularities	p. 378
Theorem of A. G. Kuzmin	p. 379
Positive Differential 2-Forms	p. 383
Notations	p. 383
Local Theory	p. 384
Normal Forms	p. 384
Classification	p. 386
Space of Forms	p. 388
Stable Subspace	p. 388
Theorem of V. Guinez	p. 388
Control Theory	p. 391
Introduction	p. 391
Optimal Control	p. 392
Optimal Flows	p. 393
Generic Optimal Flows on the Plane	p. 395
Optimal Flows on 2 Manifolds	p. 397
Appendix
Riemann Surfaces	p. 401
Uniformization Theorem	p. 401
Discrete Groups	p. 403
Möbius Transformations	p. 403
Fuchsian Group	p. 406
Limit Set of Fuchsian Groups	p. 408
Modular Group	p. 410
Examples	p. 411
Teichmuller Space	p. 412
Conformal Invariants	p. 412
Quasiconformal Mappings	p. 413
Beltrami Equation	p. 414
Ahlfors-Bers' Theorem	p. 415
Geometry of Quadratic Differentials	p. 416
Associated Metric	p. 418
Isothermal Coordinates	p. 420
Complex Curves	p. 421
Projective Curves	p. 421
Degree-Genus Formula	p. 424
Elliptic Curves	p. 425
Divisors and the Riemann-Roch Theorem	p. 427
Application: Dimension of the Teichmuller Space	p. 429
Bibliography	p. 431
Index	p. 447
Table of Contents provided by Publisher. All Rights Reserved.

Foliations on Surfaces

At a Glance

Hardcover

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