Flows on 2-dimensional manifolds : an overview
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書誌事項
Flows on 2-dimensional manifolds : an overview
（Lecture notes in mathematics, 1705）
Springer-Verlag, c1999
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Flows on two-dimensional manifolds
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注記
Includes bibliographical references (p. [269]-286) and index
内容説明・目次
内容説明
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.
目次
- 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.
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