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This book contains four contributions with topics in dynamical systems: The "spectral" decomposition for one-dimensional maps, a constructive theory of Lagrangian tori and computer-assisted applications, ergodicity in Hamiltonian systems, linearization of random dynamical systems. All the authors give a careful and readable presentation of recent research results, addressed not only to specialists but also to a broader range of readers including graduate students.

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CONTENTS
The "Spectral" Decomposition for One-Dimensional Maps / Alexander M. Blokh
  1. Introduction and Main Results = 1
    1.0 Preliminaries = 1
    1.1 Historical Remarks = 2
    1.2 A Short Description of the Approach Presented = 3
    1.3 Solenoidal Sets = 4
    1.4 Basic Sets = 5
    1.5 The Decomposition and Main Corollaries = 7
    1.6 The Limit Behavior and Generic Limit Sets for Maps Without Wandering Intervals = 8
    1.7 Topological Properties of Sets Perf ? , ω(f) and Ω(f) = 9
    1.8 Properties of Transitive and Mixing Maps = 10
    1.9 Corollaries Concerning Periods of Cycle for Interval Maps = 11
    1.10 Invariant Measures for Interval Maps = 12
    1.11 The Decomposition for Piecewise - Monotone Maps = 16
    1.12 Properties of Piecewise-Monotone Maps of Specific Kinds = 20
    1.13 Further Generalizations = 23
  2. Technical Lemmas = 25
  3. Solenoidal Sets = 27
  4. Basic Sets = 28
  5. The Decomposition = 33
  6. Limit Behavior for Maps Without Wandering Intervals = 36
  7. Topological Properties of the Sets Per f, ω(f) and Ω(f) = 37
  8. Transitive and Mixing Maps = 42
  9. Corollaries Concerning Periods of Cycles = 47
  10. Invariant Measures = 49
  11. Discussion of Some Recent Results of Block and Coven and Xiong Jincheng = 53
  References = 55
A Constructive Theory of Lagrangian Tori and Computer-assisted Applications / A. Celletti ; L. Chierchia
  1. Introduction = 60
  2. Quasi-Periodic Solutions add Invariant Tori for Lagrangian Systems : Algebraic Structure = 61
    2.1 Setup and Definitions = 61
    2.2 Approximate Solutions and Newton Scheme = 63
    2.3 The Linearized Equation = 65
    2.4 Solution of the Linearized Equation = 66
  3. Quasi-Periodic Solutions and Invariant Tori for Lagrangian Systems : Quantitative Analysis = 69
    3.1 Spaces of Analytic Functions and Norms = 69
    3.2 Analytic Tools = 71
    3.3 Norm-Parameters = 72
    3.4 Bounds on the Solution of the Linearized Equation = 74
    3.5 Bounds on the New Error Term = 76
  4. KAM Algorithm = 79
    4.1 A Self-Contained Description of the KAM Algorithm = 80
  5. A KAM Theorem = 81
  6. Application of the KAM Algorithm to Problems with Parameters = 87
    6.1 Convergent-Power-Series (Lindstedt-Poincar e' -Moser Series) = 87
    6.2 Improving the Lower Bound on the Radius of Convergence = 88
  7. Power Series Expansions and Estimate of the Error Term = 90
    7.1 Power Series Expansions = 90
    7.2 Truncated Series as Initial Approximations and the Majorant Method = 93
    7.3 Numerical Initial Approximations = 96
  8. Computer Assisted Methods = 96
    8.1 Representable Numbers and Intervals = 96
    8.2 Intervals on VAXes = 97
    8.3 Interval Operations = 98
  9. Applications : Three-Dimensional Phase Space Systems = 99
    9.1 A Forced Pendulum = 99
    9.2 Spin-Orbit Coupling in Celestial Mechanics = 101
  10. Applications : Symplectic Maps = 104
    10.1 Formalism = 104
    10.2 The Newton Scheme, the Linearized Equation, etc. = 105
    10.3 Results = 106
  Appendices = 107
  References = 127
Ergodicity in Hamiltonian Systems / C. Liverani ; M.P. Wojtkowski
  0. Introduction = 131
  1. A Model Problem = 132
  2. The Sinai Method = 137
  3. Proof of the Sinai Theorem = 141
  4. Sectors in a Linear Symplectic Space = 145
  5. The Space of Lagrangian Subspaces Contained in a Sector = 149
  6. Unbounded Sequences of Linear Monotone Maps = 153
  7. Properties of the System and the Formulation of the Results = 160
  8. Construction of the Neighborhood and the Coordinate System = 169
  9. Unstable Manifolds in the Neighborhood ??? = 172
  10. Local Ergodicity in the Smooth Case = 177
  11. Local Ergodicity in the Discontinuous Case = 180
  12. Proof of Sinai Theorem = 183
  13. 'Tail Bound' = 187
  14. Applications = 191
  References = 200
Linearization of Random Dynamical Systems / Thomas Wanner
  1. Introduction = 203
  2. Random Difference Equations = 208
    2.1 Preliminaries = 208
    2.2 Quasiboundedness and Its Consequences = 210
    2.3 Random Invariant Fiber Bundles = 221
    2.4 Asymptotic Phases = 227
    2.5 Topological Decoupling = 232
    2.6 Topological Linearization = 237
  3. Random Dynamical Systems = 242
    3.1 Preliminaries and Hypotheses = 242
    3.2 Random Invariant Manifolds = 246
    3.3 Asymptotic Phases = 250
    3.4 The Hartmen-Grobman Theorems = 253
  4. Local Results = 257
    4.1 The Discrete-Time Case = 257
    4.2 The Continuous-Time Case = 260
  5. Appendix = 266
  References = 268