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动力系统IX:带有双曲性的动力系统
  • 书号:9787030234964
    作者:Anosov
  • 外文书名:Dynamical Systems Ⅸ:Dynamical Systems with Hyperbolic Behaviour
  • 丛书名:国外数学名著系列(影印版)
  • 装帧:精装
    开本:B5
  • 页数:248
    字数:296000
    语种:英文
  • 出版社:科学出版社
    出版时间:2009-01
  • 所属分类:
  • 定价: ¥108.00元
    售价: ¥108.00元
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The book deals with smooth dynamical systems with hyperbolic behaviour of trajectories filling out “large subsets” of the phase space. Such systems lead to complicated motion (so-called “chaos”). The book begins with a discussion of the topological manifestations of uniform and total hyperbolicity: hyperbolic sets, Smale’s Axiom A, structurally stable systems, Anosov systems, and hyperbolic attractors of dimension or codimension one. There are various modifications of hyperbolicity and in this connection the properties of Lorenz attractors, pseudo-analytic Thurston diffeomorphisms, and homogeneous flows with expanding and contracting foliations are investigated. These last two questions are discussed in the general context of the theory of homeomorphisms of surfaces and of homogeneous flows.
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目录

  • Preface (D. V. Anosov)
    References
    Chapter 1 Hyperbolic Sets (D. V. Anosov. V. V. Solodov)
    §1. Preliminary Notions
    1.1. Definition of a Hyperbolic Set
    1.2. Comments
    1.3. Stable and Unstable Manifolds
    1.4. Stable and Unstable Manifolds in Certain Special Cases
    §2. Some Examples
    2.1. Heteroclinic and Homoclinic Points
    2.2. The Smale Horseshoe
    2.3. Motions in a Neighbourhood of a Homoclinic Trajectory. ε-TRajectories
    2.4. Markov Partitioning for a Hyperbolic Automorphism of a Two-Dimensional Torus
    §3. Semilocal Theory
    3.1. ε-Trajectories
    3.2. Locally Maximal Hyperbolic Sets
    3.3. Remarks
    §4. Global Theory
    4.1. Smale's Axiom A and Structural Stability
    4.2. Anosov Systems
    4.3. Remarks
    4.4. Beyond the Limits of Hyperbolicity
    References
    Chapter 2 Strange Attractors (R.V. Plykin,E.A. Sataev,S. V. Shlyachkov)
    Introduction
    §1. Hyperbolic Attractors
    1.1. Quasi-attractors Related to the Horseshoe
    1.2. Attractors of Codimension 1
    1.3. One-dimensional Attractors
    §2. The Lorenz Attractor
    2.1. Physical Models Leading to a Lorenz System
    2.2. Bifurcation in a Lorenz System
    2.3. First-Return Map
    2.4. Invariant Stable Foliation
    2.5. Invariant Unstable Foliation
    2.6. Lacunae
    2.7. Classification Theorem
    2.8. Structurally Stable and Structurally Unstable Properties
    2.9. A Continuum of Non-homeomorphic Attractors
    2.10. An Alternative Treatment of the Attractor in a Lorenz System
    §3. Metric Properties of One-dimensional Attractors of Hyperbolic Maps with Singularities
    3.1. Objects of Investigation
    3.2. Invariant u-Gibbs Measures
    Application. ε-Trajectories and Stability Properties of Dynamical Systems
    References
    Chapter 3 Cascades on Surfaces (S.Kh. Aranson,V.Z. Grines)
    §1. Morse-Smale Diffeomorphisms
    §2. Cascades with a Countable Set of Periodic Points
    2.1. Topological Classification of Basic Sets
    2.2. Topological Classification of A-diffeomorphisrns with Non-trivial Basic Sets and Homeomorphisms with Two Invariant Transversal Foliations
    §3. Various Approaches to the Problem of the Realization of Homotopy Classes of Homeomorphisms with Given Topological Properties
    3.1. Geodesic Laminations and Their Role in the Construction of Representatives of Homotopy Classes of Homeomorphisms
    3.2. The Nielsen-Thurston Theory on the Homotopy Classification of Homeomorphisms of Surfaces
    References
    Chapter 4 Dynamical Systems with Transitive Symmetry Group. Geometric and Statistical Properties (A. V. Safonov. A. N. Starkov. A.M. Stepin)
    Introduction
    §1. Basic Concepts and Constructions. Examples
    §2. A Criterion for Ergodicity and the Ergodic Decomposition
    §3. Spectrum of Ergodic Flows on Homogeneous Spaces
    §4. Orbits of Homogeneous Flows
    §5. Statistical Properties of G-Induced Flows (and Actions)
    §6. Rigidity of Homogeneous Flows
    Appendix A. Structure of Spaces of Finite Volume
    Appendix B. Construction of Semisimple Splitting of Simply Connected Lie Groups
    Added to the English Translation
    References
    Author Index
    Subject Index
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