Flows on 2-dimensional Manifolds: An Overview

ISBN-13: 9783540660804 / Angielski / Miękka / 1999 / 302 str.

Igor Nikolaev;Evgeny Zhuzhoma

Opis

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 Poincare-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.

Spis treści

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 trajectory 3 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 of flows. 8 C*-algebras of Surface Flows: Irrational rotation algebra; Artin's rotation algebra; K-theory; C*-algebra of Morse-Smale flows. 9 Semi-local Theory: Denjoy's and Schwarz's theorems; Cherry's problem; Local structure preventing quasiminimality. 10 Anosov-Weil Problem: Theorems of Weil and Anosov; Asymptotic direction of individual curves; Approximation of curve by trajectories of a flow; Limit sets of curves and trajectories at the absolute; Deviation of curves from the geodesics; Examples of unbounded deviation. 11 Non-compact Surfaces: Kaplan classification; Level curves of harmonic functions; Markus's classification; Structural stability; Neumann's example; Inaba example and Beniere-Meigniez theorem, Beniere-Hector theorem; Aranson-Zhuzhoma's example. 12 Triptych: Geodesic frameworks revisited; On continuity and collapse of geodesic frameworks; Cr-Closing lemma.