DYNAMICS REPORTED reports on recent developments in dynamical systems. Dynamical systems of course originated from ordinary differential equations. Today, dynamical systems cover a much larger area, including dynamical processes described by functional and integral equations, by partial and stochastic differential equations, etc. Dynamical systems have involved remarkably in recent years. A wealth of new phenomena, new ideas and new techniques are proving to be of considerable interest to scientists in rather different fields. It is not surprising that thousands of publications on the theory itself and on its various applications are appearing DYNAMICS REPORTED presents carefully written articles on major subjects in dy namical systems and their applications, addressed not only to specialists but also to a broader range of readers including graduate students. Topics are advanced, while detailed exposition of ideas, restriction to typical results - rather than the most general one- and, last but not least, lucid proofs help to gain the utmost degree of clarity. It is hoped, that DYNAMICS REPORTED will be useful for those entering the field and will stimulate an exchange of ideas among those working in dynamical systems Summer 1991 Christopher K. R. T Jones Drs Kirchgraber Hans-Otto Walther Managing Editors Table of Contents The "Spectral" Decomposition for One-Dimensional Maps Alexander M. Blokh Introduction and Main Results 1. 1 Preliminaries. 1 1. 0. 1. 1. Historical Remarks. 2 1. 2. A Short Description of the Approach Presented. 3 1. 3. Solenoidal Sets. 4 Basic Sets. 1. 4.

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InhaltsangabeThe "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.