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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