ISBN: 3-540-66080-1
TITLE: Flows on 2-dimensional Manifolds
AUTHOR: Nikolaev, Igor; Zhuzhoma, Evgeny
TOC:

1 Definitions and examples 1
1.1 Preliminaries 1
1.2 Basic constructions 4
1.2.1 The projection method 5
1.2.2 The universal covering method 8
1.2.3 The suspension method 6
1.2.4 Whitney theorem 11
1.2.5 Connected sum of flows 11
1.2.6 The branch covering method 12
1.3 Basic examples 12
1.3.1 Gradient and Morse-Smale flows 12
1.3.2 Transitive flows 13
1.3.3 Flows with Cantor type limit sets 14
1.3.4 Area preserving and Hamiltonian flows 17
1.3.5 Harmonic and geodesic vector fields 19
2 Poincare-Bendixson's theory 23
2.1 Existence of closed transversal 23
2.2 Absence of non-trivial recurrent trajectories on some surfaces 27
2.3 Hilmy's and Cherry's theorems on quasiminimal sets 28
2.4 Maier's theorems on quasiminimal sets 30
2.5 Gutierrez's structure theorem 34
2.6 Limit set of individual trajectory 35
2.6.1 List of limit and minimal sets 35
2.6.2 Results of Solntzev and Vinograd 37
2.6.3 On the existence of minimal sets 38
3 Decomposition of flows 41
3.1 Decomposition theorems 41
3.1.1 Irreducible flows on torus 41
3.1.2 Canonical neighborhood 42
3.1.3 Gardiner  Levitt's decomposition 42
3.1.4 Pants decomposition 45
3.1.5 Decomposition of area preserving and Hamiltonian flows 47
3.2 Center of flow 50
3.3 Blowing-down of flows 53
3.4 Regular flows 55
3.4.1 Singular trajectories 55
3.4.2 Cells 56
3.5 Application: smoothing of flows 59
4 Local theory 63
4.1 Topological normal forms 63
4.2 Analytical normal forms 64
4.3 Smooth normal forms 66
4.4 Finitely smooth normal forms 66
4.5 Degenerate critical points 67
4.6 C^1 normal forms of degenerate singularities 70
5 Space of flows and vector fields 73
5.1 Structural stability 73
5.2 Peixoto's graphs. Classification of Morse-Smale flows 76
5.2.1 Rotation systems 76
5.2.2 Peixoto theorems 79
5.2.3 Peixoto's counterexample revisited 81
5.3 Lyapunov's method 83
5.3.1 Lyapunov functions 83
5.3.2 Lyapunov graphs 86
5.4 Connected components of Morse-Smale flows 87
5.5 Degrees of non-stability 89
5.6 Typical properties of non-stable flows 92
6 Ergodic theory 95
6.1 Liouville's theorem 96
6.2 Kolmogorov's theorem for flows on torus 98
6.3 Non-trivial invariant measures 100
6.4 Ergodicity 101
6.5 Mixing 106
6.6 Entropy 111
7 Invariants of surface flows 115
7.1 Topological classification of torus flows 116
7.1.1 Rotation numbers 116
7.1.2 Classification of minimal flows 119
7.1.3 Classification of the Denjoy flows 120
7.1.4 Classification of flows of the Cherry type 121
7.2 Oriented surfaces of genus >= 2 128
7.2.1 Aranson-Grines homotopy rotation class 125
7.2.2 Homotopy rotation orbit 126
7.2.3 Equivalence of irrational flows 127
7.2.4 Properties of the homotopy rotation classes 128
7.3 Application of geodesic laminations 129
7.4 Transitive flows on non-orientable surfaces 132
7.4.1 Torus with a cross-cup 132
7.4.2 Non-orientable surfaces of genus >= 4 134
7.5 Classification of exceptional minimal sets 134
7.6 Classification of the regular flows 135
7.6.1 Leontovich-Maier's theorem for sphere flows 136
7.6.2 Neumann-O'Brien's orbit complex 136
7.6.3 Bolsinov-Fomenko's classification of Hamiltonian flows 139
7.7 Classification of non-wandering flows 141
7.7.1 Elementary cells of non-wandering flows 142
7.7.2 Conley-Lyapunov-Peixoto graphs 144
7.7.3 Equivalence Problem 145
7.7.4 Realization Problem 146
7.8 Cayley graph of a flow 147
7.8.1 Finite groups and Cayley graphs 147
7.8.2 Isomorphism Problem 149
7.8.3 Realization Problem 150
7.9 Homology and cohomology invariants 151
7.9.1 Asymptotic cycles 152
7.9.2 Fundamental class of A. Katok 155
7.9.3 Zorich's cycles 160
7.10 Rotation sets of surface flows 163
7.11 Smooth classification of flows 164
7.11.1 Torus and Klein bottle 165
7.11.2 Closed orientable surfaces of genus >= 2 167
8 C*-algebras of surface flows 175
8.1 Irrational rotation algebra 175
8.1.1 Dimension groups 176
8.1.2 Continued fractions 178
8.1.3 Effros-Shen's Theorem 179
8.1.4 Embedding of A_alpha 180
8.1.5 Projections of A_alpha 181
8.1.6 Morita Equivalence 183
8.2 Artin's rotation algebra 184
8.2.1 Myrberg's Approximationssatz 184
8.2.2 Artin's numbers 186
8.3 K-theory 193
8.3.1 Torus with Reeb's components 195
8.3.2 Baum-Connes Conjecture 195
8.4 C*-algebras of Morse-Smale flows 197
9 Semi-local theory 201
9.1 Denjoy's and Schwarz's theorems 202
9.2 Cherry's problem 203
9.3 Local structure preventing quasiminimality 207
10 Anosov-Weil problem 209
10.1 Theorems of Weil and Anosov 210
10.1.1 Asymptotic directions 210
10.1.2 Weil's theorem and Weil's conjecture 211
10.1.3 Anosov's theorem 213
10.1.4 Proof of Weil's conjecture and Weil's theorem 216
10.2 Asymptotic direction of individual curves 217
10.2.1 Non-trivial recurrent semi-trajectories 217
10.2.2 Trajectories of analytic flows 219
10.2.3 Leaves of foliation 219
10.2.4 Curves with restriction on the geodesic curvature 220
10.3 Approximation of curve by trajectories of a flow 221
10.4 Limit sets of curves and trajectories at the absolute 222
10.5 Deviation of curves from the geodesies 224
10.5.1 The deviation property of trajectories 224
10.5.2 Deviation from the geodesic frameworks 227
10.5.3 Branched coverings 228
10.5.4 Swing of trajectories near hyperbolic lines 229
10.6 Examples of unbounded deviation 233
10.6.1 Surfaces of genus >= 2 233
10.6.2 Irrational direction on torus 235
10.6.3 Rational direction on torus 236
11 Non-compact surfaces 239
11.1 Kaplan's classification 239
11.2 Level curves of harmonic functions 242
11.3 Markus's classification 245
11.4 Structural stability 246
11.5 Neumann's example 248
11.6 Inaba's example and Beniere-Meigniez's theorem 250
11.7 Beniere-Hector's theorem 252
11.8 Aranson-Zhuzhoma's example 253
12 Triptych 257
12.1 Geodesic frameworks revisited 257
12.2 On continuity and collapse of geodesic frameworks 260
12.3 C^r-closing lemma 263
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