Flows on 2-dimensional Manifolds: An Overview

摘要: 1 Definitions and Examples: Preliminaries Basic constructions examples. 2 Poncare-Bendixon's theory: Existence of closed transversal Absence non-trivial recurrent trajectories on some surfaces Hilmy's Cherry's theorems quasiminimal sets Gutierrez's structure theorem Limit set individual trajectory 3 Decomposition flows: Center flow Blowing-down flows Regular Application: smoothing flows. 4 Local Theory: Topological normal forms Analytical Smooth Finitely smooth forms, Degenerate critical points C1 degenerate singularities. 5 Space Flows vector fields: Structural stability Classification Morse-Smale Lyapunov's method, Connected components Degrees non-stability, Typical properties non-stable 6 Ergodic Liouville's Kolmogorov's for torus Non-trivial invariant measures Ergodicity Mixing Entropy. 7 Invariants surface classification Oriented higher genus .z Application geodesic laminations Transitive non-orientable exceptional minimal regular non-wandering Cayley graph a Homology cohomology invariants Rotation 8 C*-algebras Surface Flows: Irrational rotation algebra Artin's K-theory C*-algebra 9 Semi-local Denjoy's Schwarz's problem preventing quasiminimality. 10 Anosov-Weil Problem: Theorems Weil Anosov Asymptotic direction curves Approximation curve by at the absolute Deviation from geodesics Examples unbounded deviation. 11 Non-compact Surfaces: Kaplan Level harmonic functions Markus's Neumann's example Inaba Beniere-Meigniez theorem, Beniere-Hector Aranson-Zhuzhoma's example. 12 Triptych: Geodesic frameworks revisited On continuity collapse Cr-Closing lemma.