Flows on 2-dimensional Manifolds An Overview [Paperback]

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.1 Definitions and Examples: Preliminaries; Basic constructions; Basic examples.2 Poncare-Bendixon's theory: Existence of closed transversal; Absence of non-trivial recurrent trajectories on some surfaces; Hilmy's and Cherry's theorems on quasiminimal sets; Gutierrez's structure theorem; Limit set of individual trajectory3 Decomposition of flows: Decomposition theorems; Center of flow; Blowing-down of flows; Regular flows; Application: smoothing of flows.4 Local Theory: Topological normal forms; Analytical normal forms; Smooth normal forms; Finitely smooth normal forms, Degenerate critical points; C1 normal forms of degenerate singularities.5 Space of Flows and vector fields: Structural stability; Classification of Morse-Smale flows; Lyapunov's method, Connected components of Morse-Smale flows; Degrees of non-stability, Typical properties of non-stable flows.6 Ergodic theory: Liouville's theorem; Kolmogorov's theorem for flows on torus; Non-trivial invariant measures; Ergodicity; Mixing; Entropy.7 Invariants of surface flows: Topological classification of torus flows; Oriented surfaces of higher genus .z 2; Application of geodesic laminations; Transitive flows on non-orientable surfaces; Classification of exceptional minimal sets; Classification of regular flows; Classification of non-wandering flows; Cayley graph of a flow; Homology and cohomology invariants; Rotation sets of surface flows; Smooth classification oflÓ