Lecture Notes in Mathematics

An Overview

By: Igor Nikolaev, Evgeny Zhuzhoma

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Paperback | 15 July 1999

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324 Pages

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Time-evolution in low-dimensional topological spaces is a subject of puzzling vitality. This book is a state-of-the-art account, covering classical and new results. The volume comprises Poincar©-Bendixson, local and Morse-Smale theories, as well as a carefully written chapter on the invariants of surface flows. Of particular interest are chapters on the Anosov-Weil problem, C*-algebras and non-compact surfaces. The book invites graduate students and non-specialists to a fascinating realm of research. It is a valuable source of reference to the specialists.

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Non-Fiction Mathematics Topology Calculus & Mathematical Analysis Differential Calculus & Equations Algebra Medicine Medicine in General Geometry Science Physics Combinatorics & Graph Theory

Discrete Mathematics Reference, Information & Interdisciplinary Subjects Research & Information Information theory Cybernetics & Systems Theory

Definitions and examples	p. 1
Preliminaries	p. 1
Basic constructions	p. 4
The projection method	p. 5
The universal covering method	p. 8
The suspension method	p. 6
Whitney theorem	p. 11
Connected sum of flows	p. 11
The branch covering method	p. 12
Basic examples	p. 12
Gradient and Morse-Smale flows	p. 12
Transitive flows	p. 13
Flows with Cantor type limit sets	p. 14
Area preserving and Hamiltonian flows	p. 17
Harmonic and geodesic vector fields	p. 19
Poincare-Bendixson's theory	p. 23
Existence of closed transversal	p. 23
Absence of non-trivial recurrent trajectories on some surfaces	p. 27
Hilmy's and Cherry's theorems on quasiminimal sets	p. 28
Maier's theorems on quasiminimal sets	p. 30
Gutierrez's structure theorem	p. 34
Limit set of individual trajectory	p. 35
List of limit and minimal sets	p. 35
Results of Solntzev and Vinograd	p. 37
On the existence of minimal sets	p. 38
Decomposition of flows	p. 41
Decomposition theorems	p. 41
Irreducible flows on torus	p. 41
Canonical neighborhood	p. 42
Gardiner - Levitt's decomposition	p. 42
Pants decomposition	p. 45
Decomposition of area preserving and Hamiltonian flows	p. 47
Center of flow	p. 50
Blowing-down of flows	p. 53
Regular flows	p. 55
Singular trajectories	p. 55
Cells	p. 56
Application: smoothing of flows	p. 59
Local theory	p. 63
Topological normal forms	p. 63
Analytical normal forms	p. 64
Smooth normal forms	p. 66
Finitely smooth normal forms	p. 66
Degenerate critical points	p. 67
C¹ normal forms of degenerate singularities	p. 70
Space of flows and vector fields	p. 73
Structural stability	p. 73
Peixoto's graphs. Classification of Morse-Smale flows	p. 76
Rotation systems	p. 76
Peixoto theorems	p. 79
Peixoto's counterexample revisited	p. 81
Lyapunov's method	p. 83
Lyapunov functions	p. 83
Lyapunov graphs	p. 86
Connected components of Morse-Smale flows	p. 87
Degrees of non-stability	p. 89
Typical properties of non-stable flows	p. 92
Ergodic theory	p. 95
Liouville's theorem	p. 96
Kolmogorov's theorem for flows on torus	p. 98
Non-trivial invariant measures	p. 100
Ergodicity	p. 101
Mixing	p. 106
Entropy	p. 111
Invariants of surface flows	p. 115
Topological classification of torus flows	p. 116
Rotation numbers	p. 116
Classification of minimal flows	p. 119
Classification of the Denjoy flows	p. 120
Classification of flows of the Cherry type	p. 121
Oriented surfaces of genus ≥ 2	p. 128
Aranson-Grines homotopy rotation class	p. 125
Homotopy rotation orbit	p. 126
Equivalence of irrational flows	p. 127
Properties of the homotopy rotation classes	p. 128
Application of geodesic laminations	p. 129
Transitive flows on non-orientable surfaces	p. 132
Torus with a cross-cup	p. 132
Non-orientable surfaces of genus ≥ 4	p. 134
Classification of exceptional minimal sets	p. 134
Classification of the regular flows	p. 135
Leontovich-Maier's theorem for sphere flows	p. 136
Neumann-O'Brien's orbit complex	p. 136
Bolsinov-Fomenko's classification of Hamiltonian flows	p. 139
Classification of non-wandering flows	p. 141
Elementary cells of non-wandering flows	p. 142
Conley-Lyapunov-Peixoto graphs	p. 144
Equivalence Problem	p. 145
Realization Problem	p. 146
Cayley graph of a flow	p. 147
Finite groups and Cayley graphs	p. 147
Isomorphism Problem	p. 149
Realization Problem	p. 150
Homology and cohomology invariants	p. 151
Asymptotic cycles	p. 152
Fundamental class of A. Katok	p. 155
Zorich's cycles	p. 160
Rotation sets of surface flows	p. 163
Smooth classification of flows	p. 164
Torus and Klein bottle	p. 165
Closed orientable surfaces of genus ≥ 2	p. 167
C*-algebras of surface flows	p. 175
Irrational rotation algebra	p. 175
Dimension groups	p. 176
Continued fractions	p. 178
Effros-Shen's Theorem	p. 179
Embedding of A_¿	p. 180
Projections of A_¿	p. 181
Morita Equivalence	p. 183
Artin's rotation algebra	p. 184
Myrberg's Approximationssatz	p. 184
Artin's numbers	p. 186
K-theory	p. 193
Torus with Reeb's components	p. 195
Baum-Connes Conjecture	p. 195
C*-algebras of Morse-Smale flows	p. 197
Semi-local theory	p. 201
Denjoy's and Schwarz's theorems	p. 202
Cherry's problem	p. 203
Local structure preventing quasiminimality	p. 207
Anosov-Weil problem	p. 209
Theorems of Weil and Anosov	p. 210
Asymptotic directions	p. 210
Weil's theorem and Weil's conjecture	p. 211
Anosov's theorem	p. 213
Proof of Weil's conjecture and Weil's theorem	p. 216
Asymptotic direction of individual curves	p. 217
Non-trivial recurrent semi-trajectories	p. 217
Trajectories of analytic flows	p. 219
Leaves of foliation	p. 219
Curves with restriction on the geodesic curvature	p. 220
Approximation of curve by trajectories of a flow	p. 221
Limit sets of curves and trajectories at the absolute	p. 222
Deviation of curves from the geodesies	p. 224
The deviation property of trajectories	p. 224
Deviation from the geodesic frameworks	p. 227
Branched coverings	p. 228
Swing of trajectories near hyperbolic lines	p. 229
Examples of unbounded deviation	p. 233
Surfaces of genus ≥ 2	p. 233
Irrational direction on torus	p. 235
Rational direction on torus	p. 236
Non-compact surfaces	p. 239
Kaplan's classification	p. 239
Level curves of harmonic functions	p. 242
Markus's classification	p. 245
Structural stability	p. 246
Neumann's example	p. 248
Inaba's example and Beniere-Meigniez's theorem	p. 250
Beniere-Hector's theorem	p. 252
Aranson-Zhuzhoma's example	p. 253
Triptych	p. 257
Geodesic frameworks revisited	p. 257
On continuity and collapse of geodesic frameworks	p. 260
C^r-closing lemma	p. 263
Table of Contents provided by Publisher. All Rights Reserved.

Lecture Notes in Mathematics

An Overview

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Paperback

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