Nonuniform Hyperbolicity: Dynamics of Systems with Nonzero Lyapunov Exponents

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ISBN: 9780521832588 | 0521832586
Cover: Hardcover
Copyright: 9/3/2007

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Designed to work as a reference and as a supplement to an advanced course on dynamical systems, this book presents a self-contained and comprehensive account of modern smooth ergodic theory. Among other things, this provides a rigorous mathematical foundation for the phenomenon known as deterministic chaos - the appearance of 'chaotic' motions in pure deterministic dynamical systems. A sufficiently complete description of topological and ergodic properties of systems exhibiting deterministic chaos can be deduced from relatively weak requirements on their local behavior known as nonuniform hyperbolicity conditions. Nonuniform hyperbolicity theory is an important part of the general theory of dynamical systems. Its core is the study of dynamical systems with nonzero Lyapunov exponents both conservative and dissipative, in addition to cocycles and group actions. The results of this theory are widely used in geometry (e.g., geodesic flows and Teichmüller flows), in rigidity theory, in the study of some partial differential equations (e.g., the Schrödinger equation), in the theory of billiards, as well as in applications to physics, biology, engineering, and other fields.

Preface	p. xiii
Introduction	p. 1
Linear Theory
The Concept of Nonuniform Hyperbolicity	p. 9
Motivation	p. 9
Basic Setting	p. 13
Exponential Splitting and Nonuniform Hyperbolicity	p. 13
Tempered Equivalence	p. 14
The Continuous-Time Case	p. 15
Lyapunov Exponents Associated to Sequences of Matrices	p. 16
Definition of the Lyapunov Exponent	p. 16
Forward and Backward Regularity	p. 18
A Criterion of Forward Regularity for Triangular Matrices	p. 25
The Lyapunov-Perron Regularity	p. 31
Notes	p. 33
Lyapunov Exponents for Linear Extensions	p. 35
Cocycles over Dynamical Systems	p. 35
Cocycles and Linear Extensions	p. 35
Cohomology and Tempered Equivalence	p. 37
Examples and Basic Constructions	p. 39
Hyperbolicity of Cocycles	p. 41
Hyperbolic Cocycles	p. 41
Regular Sets of Hyperbolic Cocycles	p. 43
Cocycles over Topological Spaces	p. 46
Lyapunov Exponents for Cocycles	p. 46
Spaces of Cocycles	p. 50
Regularity of Cocycles	p. 53
The Lyapunov-Perron regularity	p. 53
Lyapunov Exponents and Basic Constructions	p. 57
Lyapunov Exponents and Hyperbolicity	p. 59
The Multiplicative Ergodic Theorem	p. 64
One-Dimensional Cocycles and Birkhoff's Ergodic Theorem	p. 64
Oseledets' Proof of the Multiplicative Ergodic Theorem	p. 65
Lyapunov Exponents and Subadditive Ergodic Theorem	p. 69
Raghunathan's Proof of the Multiplicative Ergodic Theorem	p. 70
Tempering Kernels and the Reduction Theorems	p. 75
Lyapunov Inner Products	p. 76
The Oseledets-Pesin Reduction Theorem	p. 77
A Tempering Kernel	p. 80
Zimmer's Amenable Reduction	p. 82
The Case of Noninvertible Cocycles	p. 82
More Results on Lyapunov-Perron Regularity	p. 83
Higher-Rank Abelian Actions	p. 83
The Case of Flows	p. 88
Nonpositively Curved Spaces	p. 91
Notes	p. 94
Methods for Estimating Exponents	p. 95
Cone and Lyapunov Function Techniques	p. 95
Lyapunov Functions	p. 96
A Criterion for Nonvanishing Lyapunov Exponents	p. 98
Invariant Cone Families	p. 101
Cocycles with Values in the Symplectic Group	p. 102
Monotone Operators and Lyapunov Exponents	p. 106
The Algebra of Potapov	p. 106
Lyapunov Exponents for J-Separated Cocycles	p. 108
The Lyapunov Spectrum for Conformally Hamiltonian Systems	p. 112
A Remark on Applications of Cone Techniques	p. 116
Notes	p. 117
The Derivative Cocycle	p. 118
Smooth Dynamical Systems and the Derivative Cocycle	p. 118
Nonuniformly Hyperbolic Diffeomorphisms	p. 119
Holder Continuity of Invariant Distributions	p. 122
Lyapunov Exponent and Regularity of the Derivative Cocycle	p. 125
On the Notion of Dynamical Systems with Nonzero Lyapunov Exponents	p. 129
Regular Neighborhoods	p. 130
Cocycles over Smooth Flows	p. 133
Semicontinuity of Lyapunov Exponents	p. 134
Examples and Foundations of the Nonlinear Theory
Examples of Systems with Hyperbolic Behavior	p. 139
Uniformly Hyperbolic Sets	p. 139
Hyperbolic Sets for Maps	p. 139
Hyperbolic Sets for Flows	p. 143
Linear Horseshoes	p. 144
Nonlinear Horseshoes	p. 147
Nonuniformly Hyperbolic Perturbations of Horseshoes	p. 152
Slow Expansion Near a Fixed Point	p. 152
Further Modifications	p. 154
Diffeomorphisms with Nonzero Lyapunov Exponents on Surfaces	p. 158
Analytic Nonuniformly Hyperbolic Diffeomorphisms	p. 159
A Nonuniformly Hyperbolic Diffeomorphism on the Sphere	p. 163
Nonuniformly Hyperbolic Diffeomorphisms on Compact Surfaces	p. 164
A Nonuniformly Hyperbolic Diffeomorphism of the Torus	p. 166
Pseudo-Anosov Maps	p. 167
Definitions and Basic Properties	p. 168
Smooth Models of Pseudo-Anosov Maps	p. 171
Nonuniformly Hyperbolic Flows	p. 182
Some Other Examples	p. 185
Notes	p. 187
Stable Manifold Theory	p. 188
The Stable Manifold Theorem	p. 188
Nonuniformly Hyperbolic Sequences of Diffeomorphisms	p. 191
The Hadamard-Perron Theorem: Hadamard's Method	p. 192
Invariant Cone Families	p. 193
Admissible Manifolds	p. 196
Existence of (s, [gamma])- and (u, [gamma])-Manifolds	p. 200
Invariant Families of Local Manifolds	p. 203
Higher Differentiability of Invariant Manifolds	p. 205
The Graph Transform Property	p. 206
The Hadamard-Perron Theorem: Perron's Method	p. 207
An Abstract Version of the Stable Manifold Theorem	p. 207
Smoothness of Local Manifolds	p. 215
Local Unstable Manifolds	p. 220
The Stable Manifold Theorem for Flows	p. 221
C[superscript 1] Pathological Behavior: Pugh's Example	p. 221
Notes	p. 225
Basic Properties of Stable and Unstable Manifolds	p. 226
Characterization and Sizes of Local Stable Manifolds	p. 226
Global Stable and Unstable Manifolds	p. 229
Foliations with Smooth Leaves	p. 231
Filtrations of Intermediate Local and Global Manifolds	p. 232
The Lipschitz Property of Intermediate Foliations	p. 236
The Absolute Continuity Property	p. 240
Absolute Continuity of Holonomy Maps	p. 242
Absolute Continuity of Local Stable Manifolds	p. 251
Foliation That Is Not Absolutely Continuous	p. 254
The Jacobian of the Holonomy Map	p. 255
Notes	p. 257
Ergodic Theory of Smooth and SRB Measures
Smooth Measures	p. 261
Ergodic Components	p. 261
Local Ergodicity	p. 266
The s- and u-Measures	p. 282
The Leaf-Subordinated Partition and the K-Property	p. 285
The Bernoulli Property	p. 290
The Continuous-Time Case	p. 297
Notes	p. 303
Measure-Theoretic Entropy and Lyapunov Exponents	p. 304
Entropy of Measurable Transformations	p. 304
The Margulis-Ruelle Inequality	p. 305
The Topological Entropy and Lyapunov Exponents	p. 308
The Entropy Formula	p. 311
Mane's Proof of the Entropy Formula	p. 315
Notes	p. 324
Stable Ergodicity and Lyapunov Exponents. More Examples of Systems with Nonzero Exponents	p. 326
Uniform Partial Hyperbolicity and Stable Ergodicity	p. 326
Partially Hyperbolic Systems with Nonzero Lyapunov Exponents	p. 329
Hyperbolic Diffeomorphisms with Countably Many Ergodic Components	p. 337
Existence of Hyperbolic Diffeomorphisms on Compact Manifolds	p. 349
Existence of Hyperbolic Flows on Compact Manifolds	p. 369
Foliations That Are Not Absolutely Continuous	p. 378
An Open Set of Diffeomorphisms with Nonzero Lyapunov Exponents on Tori	p. 382
Notes	p. 383
Geodesic Flows	p. 385
Hyperbolicity of Geodesic Flows	p. 385
Ergodic Properties of Geodesic Flows	p. 394
Entropy of Geodesic Flows	p. 404
Topological Properties of Geodesic Flows	p. 407
The Teichmuller Geodesic Flow	p. 409
Notes	p. 415
SRB Measures	p. 417
Definition and Ergodic Properties of SRB Measures	p. 417
A Characterization of SRB Measures	p. 421
Constructions of SRB Measures	p. 423
Notes	p. 430
General Hyperbolic Measures
Hyperbolic Measures: Entropy and Dimension	p. 433
Pointwise Dimensions and the Ledrappier-Young Entropy Formula	p. 433
Local Entropies	p. 434
Leaf Pointwise Dimensions	p. 438
The Ledrappier-Young Entropy Formula	p. 449
Local Product Structure of Hyperbolic Measures	p. 450
Applications to Dimension Theory	p. 461
Notes	p. 461
Hyperbolic Measures: Topological Properties	p. 463
The Closing Lemma	p. 463
The Shadowing Lemma	p. 472
The Livshitz Theorem	p. 473
Hyperbolic Periodic Orbits	p. 474
Topological Transitivity and Spectral Decomposition	p. 482
Entropy, Horseshoes, and Periodic Points	p. 482
Continuity Properties of Entropy	p. 485
Bibliography	p. 491
Index	p. 501
Table of Contents provided by Ingram. All Rights Reserved.

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Nonuniform Hyperbolicity: Dynamics of Systems with Nonzero Lyapunov Exponents

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Nonuniform Hyperbolicity: Dynamics of Systems with Nonzero Lyapunov Exponents

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