Preface page ................................................. xiii
Introduction .................................................... 1
Part I Linear Theory
1 The Concept of Nonuniform Hyperbolicity ...................... 9
1.1 Motivation .............................................. 9
1.2 Basic Setting .......................................... 13
1.2.1 Exponential Splitting and Nonuniform
Hyperbolicity ................................... 13
1.2.2 Tempered Equivalence ............................ 14
1.2.3 The Continuous-Time Case ........................ 15
1.3 Lyapunov Exponents Associated to Sequences
of Matrices ............................................ 16
1.3.1 Definition of the Lyapunov Exponent ............. 16
1.3.2 Forward and Backward Regularity ................. 18
1.3.3 A Criterion of Forward Regularity for
Triangular Matrices ............................. 25
1.3.4 The Lyapunov-Perron Regularity .................. 31
1.4 Notes .................................................. 33
2 Lyapunov Exponents for Linear Extensions .................... 35
2.1 Cocycles over Dynamical Systems ........................ 35
2.1.1 Cocycles and Linear Extensions .................. 35
2.1.2 Cohomology and Tempered Equivalence ............. 37
2.1.3 Examples and Basic Constructions ................ 39
2.2 Hyperbolicity of Cocycles .............................. 41
2.2.1 Hyperbolic Cocycles ............................. 41
2.2.2 Regular Sets of Hyperbolic Cocycles ............. 43
2.2.3 Cocycles over Topological Spaces ................ 46
2.3 Lyapunov Exponents for Cocycles ........................ 46
2.4 Spaces of Cocycles ..................................... 50
3 Regularity of Cocycles ...................................... 53
3.1 The Lyapunov-Perron regularity ......................... 53
3.2 Lyapunov Exponents and Basic Constructions ............. 57
3.3 Lyapunov Exponents and Hyperbolicity ................... 59
3.4 The Multiplicative Ergodic Theorem ..................... 64
3.4.1 One-Dimensional Cocycles and Birkhoff's
Ergodic Theorem ................................. 64
3.4.2 Oseledets' Proof of the Multiplicative
Ergodic Theorem ................................. 65
3.4.3 Lyapunov Exponents and Subadditive Ergodic
Theorem ......................................... 69
3.4.4 Raghunathan's Proof of the Multiplicative
Ergodic Theorem ................................. 70
3.5 Tempering Kernels and the Reduction Theorems ........... 75
3.5.1 Lyapunov Inner Products ......................... 76
3.5.2 The Oseledets-Pesin Reduction Theorem ........... 77
3.5.3 A Tempering Kernel .............................. 80
3.5.4 Zimmer's Amenable Reduction ..................... 82
3.5.5 The Case of Noninvertible Cocycles .............. 82
3.6 More Results on Lyapunov-Perron Regularity ............. 83
3.6.1 Higher-Rank Abelian Actions ..................... 83
3.6.2 The Case of Flows ............................... 88
3.6.3 Nonpositively Curved Spaces ..................... 91
3.7 Notes .................................................. 94
4 Methods for Estimating Exponents ............................ 95
4.1 Cone and Lyapunov Function Techniques .................. 95
4.1.1 Lyapunov Functions .............................. 96
4.1.2 A Criterion for Nonvanishing Lyapunov
Exponents ....................................... 98
4.1.3 Invariant Cone Families ........................ 101
4.2 Cocycles with Values in the Symplectic Group .......... 102
4.3 Monotone Operators and Lyapunov Exponents ............. 106
4.3.1 The Algebra of Potapov ......................... 106
4.3.2 Lyapunov Exponents for J-Separated Cocycles .... 108
4.3.3 The Lyapunov Spectrum for Conformally
Hamiltonian Systems ............................ 112
4.4 A Remark on Applications of Cone Techniques ........... 116
4.5 Notes ................................................. 117
5 The Derivative Cocycle ..................................... 118
5.1 Smooth Dynamical Systems and the Derivative
Cocycle ............................................... 118
5.2 Nonuniformly Hyperbolic Diffeomorphisms ............... 119
5.3 Holder Continuity of Invariant Distributions .......... 122
5.4 Lyapunov Exponent and Regularity of the Derivative
Cocycle ............................................... 125
5.5 On the Notion of Dynamical Systems with Nonzero
Lyapunov Exponents .................................... 129
5.6 Regular Neighborhoods ................................. 130
5.7 Cocycles over Smooth Flows ............................ 133
5.8 Semicontinuity of Lyapunov Exponents .................. 134
Part II Examples and Foundations of the Nonlinear Theory
6 Examples of Systems with Hyperbolic Behavior ............... 139
6.1 Uniformly Hyperbolic Sets ............................. 139
6.1.1 Hyperbolic Sets for Maps ....................... 139
6.1.2 Hyperbolic Sets for Flows ...................... 143
6.1.3 Linear Horseshoes .............................. 144
6.1.4 Nonlinear Horseshoes ........................... 147
6.2 Nonuniformly Hyperbolic Perturbations of
Horseshoes ............................................ 152
6.2.1 Slow Expansion Near a Fixed Point .............. 152
6.2.2 Further Modifications .......................... 154
6.3 Diffeomorphisms with Nonzero Lyapunov Exponents on
Surfaces .............................................. 158
6.3.1 Analytic Nonuniformly Hyperbolic
Diffeomorphisms ................................ 159
6.3.2 A Nonuniformly Hyperbolic Diffeomorphism on
the Sphere ..................................... 163
6.3.3 Nonuniformly Hyperbolic Diffeomorphisms on
Compact Surfaces ............................... 164
6.3.4 A Nonuniformly Hyperbolic Diffeomorphism of
the Torus ...................................... 166
6.4 Pseudo-Anosov Maps .................................... 167
6.4.1 Definitions and Basic Properties ............... 168
6.4.2 Smooth Models of Pseudo-Anosov Maps ............ 171
6.5 Nonuniformly Hyperbolic Flows ......................... 182
6.6 Some Other Examples ................................... 185
6.7 Notes ................................................. 187
7 Stable Manifold Theory ..................................... 188
7.1 The Stable Manifold Theorem ........................... 188
7.2 Nonuniformly Hyperbolic Sequences of
Diffeomorphisms ....................................... 191
7.3 The Hadamard-Perron Theorem: Hadamard's Method ........ 192
7.3.1 Invariant Cone Families ........................ 193
7.3.2 Admissible Manifolds ........................... 196
7.3.3 Existence of (s, y)- and (u, y)-Manifolds ...... 200
7.3.4 Invariant Families of Local Manifolds .......... 203
7.3.5 Higher Differentiability of Invariant
Manifolds ...................................... 205
7.4 The Graph Transform Property .......................... 206
7.5 The Hadamard-Perron Theorem: Perron's Method .......... 207
7.5.1 An Abstract Version of the Stable Manifold
Theorem ........................................ 207
7.5.2 Smoothness of Local Manifolds .................. 215
7.6 Local Unstable Manifolds .............................. 220
7.7 The Stable Manifold Theorem for Flows ................. 221
7.8 Сl Pathological Behavior: Pugh's Example .............. 221
7.9 Notes ................................................. 225
8 Basic Properties of Stable and Unstable Manifolds .......... 226
8.1 Characterization and Sizes of Local Stable
Manifolds ............................................. 226
8.2 Global Stable and Unstable Manifolds .................. 229
8.3 Foliations with Smooth Leaves ......................... 231
8.4 Filtrations of Intermediate Local and Global
Manifolds ............................................. 232
8.5 The Lipschitz Property of Intermediate Foliations ..... 236
8.6 The Absolute Continuity Property ...................... 240
8.6.1 Absolute Continuity of Holonomy Maps ........... 242
8.6.2 Absolute Continuity of Local Stable
Manifolds ...................................... 251
8.6.3 Foliation That Is Not Absolutely Continuous .... 254
8.6.4 The Jacobian of the Holonomy Map ............... 255
8.7 Notes ................................................. 257
Part III Ergodic Theory of Smooth and SRB Measures
9 Smooth Measures ............................................ 261
9.1 Ergodic Components .................................... 261
9.2 Local Ergodicity ...................................... 266
9.3 The s- and u-Measures ................................. 282
9.4 The Leaf-Subordinated Partition and the
К -Property ........................................... 285
9.5 The Bernoulli Property ................................ 290
9.6 The Continuous-Time Case .............................. 297
9.7 Notes ................................................. 303
10 Measure-Theoretic Entropy and Lyapunov Exponents ........... 304
10.1 Entropy of Measurable Transformations ................. 304
10.2 The Margulis-Ruelle Inequality ........................ 305
10.3 The Topological Entropy and Lyapunov Exponents ........ 308
10.4 The Entropy Formula ................................... 311
10.5 Mane's Proof of the Entropy Formula ................... 315
10.6 Notes ................................................. 324
11 Stable Ergodicity and Lyapunov Exponents. More Examples
of Systems with Nonzero Exponents .......................... 326
11.1 Uniform Partial Hyperbolicity and Stable
Ergodicity ............................................ 326
11.2 Partially Hyperbolic Systems with Nonzero Lyapunov
Exponents ............................................. 329
11.3 Hyperbolic Diffeomorphisms with Countably Many
Ergodic Components .................................... 337
11.4 Existence of Hyperbolic Diffeomorphisms on Compact
Manifolds ............................................. 349
11.5 Existence of Hyperbolic Flows on Compact Manifolds .... 369
11.6 Foliations That Are Not Absolutely Continuous ......... 378
11.7 An Open Set of Diffeomorphisms with Nonzero
Lyapunov Exponents on Tori ............................ 382
11.8 Notes ................................................. 383
12 Geodesic Flows ............................................. 385
12.1 Hyperbolicity of Geodesic Flows ....................... 385
12.2 Ergodic Properties of Geodesic Flows .................. 394
12.3 Entropy of Geodesic Flows ............................. 404
12.4 Topological Properties of Geodesic Flows .............. 407
12.5 The Teichmuller Geodesic Flow ......................... 409
12.6 Notes ................................................. 415
13 SRB Measures ............................................... 417
13.1 Definition and Ergodic Properties of SRB Measures ..... 417
13.2 A Characterization of SRB Measures .................... 421
13.3 Constructions of SRB Measures ......................... 423
13.4 Notes ................................................. 430
Part IV General Hyperbolic Measures
14 Hyperbolic Measures: Entropy and Dimension ................. 433
14.1 Pointwise Dimensions and the Ledrappier-Young
Entropy Formula ....................................... 433
14.1.1 Local Entropies ................................ 434
14.1.2 Leaf Pointwise Dimensions ...................... 438
14.1.3 The Ledrappier-Young Entropy Formula ........... 449
14.2 Local Product Structure of Hyperbolic Measures ........ 450
14.3 Applications to Dimension Theory ...................... 461
14.4 Notes ................................................. 461
15 Hyperbolic Measures: Topological Properties ................ 463
15.1 The Closing Lemma ..................................... 463
15.2 The Shadowing Lemma ................................... 472
15.3 The Livshitz Theorem .................................. 473
15.4 Hyperbolic Periodic Orbits ............................ 474
15.5 Topological Transitivity and Spectral
Decomposition ......................................... 482
15.6 Entropy, Horseshoes, and Periodic Points .............. 482
15.7 Continuity Properties of Entropy ...................... 485
15.1 Bibliography .......................................... 491
Index ......................................................... 501
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