Dynamics Reported

This book contains four contributions in the field of dynamical systems. The topics treated are also related to topology, Hamiltonian systems and stochastics. All the authors give a careful readable presentation of recent research results, addressed to graduate students and researchers in these fields; the book is also suitable as a text for graduate level seminars in dynamical systems.

The "Spectral" Decomposition for One–Dimensional Maps.- 1. Introduction and Main Results.- 1.0 Preliminaries.- 1.1 Historical Remarks.- 1.2 A Short Description of the Approach Presented.- 1.3 Solenoidal Sets.- 1.4 Basic Sets.- 1.5 The Decomposition and Main Corollaries.- 1.6 The Limit Behavior and Generic Limit Sets for Maps Without Wandering Intervals.- 1.7 Topological Properties of Sets $$\overline {Per\,f}$$, w(f) and ?(f).- 1.8 Properties of Transitive and Mixing Maps.- 1.9 Corollaries Concerning Periods of Cycles for Interval Maps.- 1.10 Invariant Measures for Interval Maps.- 1.11 The Decomposition for Piecewise-Monotone Maps.- 1.12 Properties of Piecewise-Monotone Maps of Specific Kinds.- 1.13 Further Generalizations.- 2. Technical Lemmas.- 3. Solenoidal Sets.- 4. Basic Sets.- 5. The Decomposition.- 6. Limit Behavior for Maps Without Wandering Intervals.- 7. Topological Properties of the Sets Per f, ?(f) and ?(f).- 8. Transitive and Mixing Maps.- 9. Corollaries Concerning Periods of Cycles.- 10. Invariant Measures.- 11. Discussion of Some Recent Results of Block and Coven and Xiong Jincheng.- References.- A Constructive Theory of Lagrangian Tori and Computer-assisted Applications.- 1. Introduction.- 2. Quasi-Periodic Solutions and Invariant Tori for Lagrangian Systems: Algebraic Structure.- 2.1 Setup and Definitions.- 2.2 Approximate Solutions and Newton Scheme.- 2.3 The Linearized Equation.- 2.4 Solution of the Linearized Equation.- 3. Quasi-Periodic Solutions and Invariant Tori for Lagrangian Systems: Quantitative Analysis.- 3.1 Spaces of Analytic Functions and Norms.- 3.2 Analytic Tools.- 3.3 Norm-Parameters.- 3.4 Bounds on the Solution of the Linearized Equation.- 3.5 Bounds on the New Error Term.- 4. KAM Algorithm.- 4.1. A Self-Contained Description of the KAM Algorithm.- 5. A KAM Theorem.- 6. Application of the KAM Algorithm to Problems with Parameters.- 6.1 Convergent-Power-Series (Lindstedt-Poincaré-Moser Series).- 6.2 Improving the Lower Bound on the Radius of Convergence.- 7. Power Series Expansions and Estimate of the Error Term.- 7.1 Power Series Expansions.- 7.2 Truncated Series as Initial Approximations and the Majorant Method.- 7.3 Numerical Initial Approximations.- 8. Computer Assisted Methods.- 8.1 Representable Numbers and Intervals.- 8.2 Intervals on VAXes.- 8.3 Interval Operations.- 9. Applications: Three-Dimensional Phase Space Systems.- 9.1 A Forced Pendulum.- 9.2 Spin-Orbit Coupling in Celestial Mechanics.- 10. Applications: Symplectic Maps.- 10.1 Formalism.- 10.2 The Newton Scheme, the Linearized Equation, etc.- 10.3 Results.- Appendices.- References.- Ergodicity in Hamiltonian Systems.- 0. Introduction.- 1. A Model Problem.- 2. The Sinai Method.- 3. Proof of the Sinai Theorem.- 4. Sectors in a Linear Symplectic Space.- 5. The Space of Lagrangian Subspaces Contained in a Sector.- 6. Unbounded Sequences of Linear Monotone Maps.- 7. Properties of the System and the Formulation of the Results.- 8. Construction of the Neighborhood and the Coordinate System.- 9. Unstable Manifolds in the Neghborhood U.- 10. Local Ergodicity in the Smooth Case.- 11. Local Ergodicity in the Discontinous Case.- 12. Proof of Sinai Theorem.- 13. ‘Tail Bound’.- 14. Applications.- References.- Linearization of Random Dynamical Systems.- 1. Introduction.- 2. Random Difference Equations.- 2.1 Preliminaries.- 2.2 Quasiboundedness and Its Consequences.- 2.3 Random Invariant Fiber Bundles.- 2.4 Asymptotic Phases.- 2.5 Topological Decoupling.- 2.6 Topological Linearization.- 3. Random Dynamical Systems.- 3.1 Preliminaries and Hypotheses.- 3.2 Random Invariant Manifolds.- 3.3 Asymptotic Phases.- 3.4 The Hartman-Grobman Theorems.- 4. Local Results.- 4.1 The Discrete-Time Case.- 4.2 The Continuous-Time Case.- 5. Appendix.- References.