Nonuniform Hyperbolicity : Dynamics of Systems with Nonzero Lyapunov Exponents.
Material type:
TextSeries: Encyclopedia of mathematics and its applicationsPublication details: Cambridge : Cambridge University Press, 2007.Description: 1 online resource (528 pages)Content type: - text
- computer
- online resource
- 9781107095915
- 1107095913
- 9781107089563
- 1107089565
- 9781107326026
- 1107326028
- 1107104009
- 9781107104006
- 531.11 531/.11
- QA871 .B365 2007
- 31.41
- 37-02 | 37D25 | 37A30
- SK 810
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Cover; Half Title; Series Page; Title; Copyright; Dedication; Contents; Preface; Introduction; Part I Linear Theory; 1 The Concept of Nonuniform Hyperbolicity; 1.1 Motivation; 1.2 Basic Setting; 1.2.1 Exponential Splitting and Nonuniform Hyperbolicity; 1.2.2 Tempered Equivalence; 1.2.3 The Continuous-Time Case; 1.3 Lyapunov Exponents Associated to Sequences of Matrices; 1.3.1 Definition of the Lyapunov Exponent; 1.3.2 Forward and Backward Regularity; 1.3.3 A Criterion of Forward Regularity for Triangular Matrices; 1.3.4 The Lyapunov-Perron Regularity; 1.4 Notes.
2 Lyapunov Exponents for Linear Extensions2.1 Cocycles over Dynamical Systems; 2.1.1 Cocycles and Linear Extensions; 2.1.2 Cohomology and Tempered Equivalence; 2.1.3 Examples and Basic Constructions; 2.2 Hyperbolicity of Cocycles; 2.2.1 Hyperbolic Cocycles; 2.2.2 Regular Sets of Hyperbolic Cocycles; 2.2.3 Cocycles over Topological Spaces; 2.3 Lyapunov Exponents for Cocycles; 2.4 Spaces of Cocycles; 3 Regularity of Cocycles; 3.1 The Lyapunov-Perron regularity; 3.2 Lyapunov Exponents and Basic Constructions; 3.3 Lyapunov Exponents and Hyperbolicity; 3.4 The Multiplicative Ergodic Theorem.
3.4.1 One-Dimensional Cocycles and Birkhoff's Ergodic Theorem3.4.2 Oseledets' Proof of the Multiplicative Ergodic Theorem; 3.4.3 Lyapunov Exponents and Subadditive Ergodic Theorem; 3.4.4 Raghunathan's Proof of the Multiplicative Ergodic Theorem; 3.5 Tempering Kernels and the Reduction Theorems; 3.5.1 Lyapunov Inner Products; 3.5.2 The Oseledets-Pesin Reduction Theorem; 3.5.3 A Tempering Kernel; 3.5.4 Zimmer's Amenable Reduction; 3.5.5 The Case of Noninvertible Cocycles; 3.6 More results on Lyapunov-Perron regularity; 3.6.1 Higher-Rank Abelian Actions; 3.6.2 The Case of Flows.
3.6.3 Nonpositively Curved Spaces3.7 Notes; 4 Methods for Estimating Exponents; 4.1 Cone and Lyapunov Function Techniques; 4.1.1 Lyapunov Functions; 4.1.2 A Criterion for Nonvanishing Lyapunov Exponents; 4.1.3 Invariant Cone Families; 4.2 Cocycles with Values in the Symplectic Group; 4.3 Monotone Operators and Lyapunov Exponents; 4.3.1 The Algebra of Potapov; 4.3.2 Lyapunov Exponents for J-Separated Cocycles; 4.3.3 The Lyapunov Spectrum for Conformally Hamiltonian Systems; 4.4 A Remark on Applications of Cone Techniques; 4.5 Notes; 5 The Derivative Cocycle.
5.1 Smooth Dynamical Systems and the Derivative Cocycle5.2 Nonuniformly Hyperbolic Diffeomorphisms; 5.3 Hölder Continuity of Invariant Distributions; 5.4 Lyapunov Exponent and Regularity of the Derivative Cocycle; 5.5 On the Notion of Dynamical Systems with Nonzero Lyapunov Exponents; 5.6 Regular Neighborhoods; 5.7 Cocycles over Smooth Flows; 5.8 Semicontinuity of Lyapunov Exponents; Part II Examples and Foundations of the Nonlinear Theory; 6 Examples of Systems with Hyperbolic Behavior; 6.1 Uniformly Hyperbolic Sets; 6.1.1 Hyperbolic Sets for Maps; 6.1.2 Hyperbolic Sets for Flows.
6.1.3 Linear Horseshoes.
A self-contained, comprehensive account of modern smooth ergodic theory, the mathematical foundation of deterministic chaos.
Print version record.
Includes bibliographical references (pages 491-500) and index.
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