Dynamics Beyond Uniform Hyperbolicity (eBook)

Preface 6
Contents 12
Hyperbolicity and Beyond 18
1.1 Spectral decomposition 18
1.2 Structural stability 20
1.3 Sinai-Ruelle-Bowen theory 21
1.4 Heterodimensional cycles 23
1.5 Homoclinic tangencies 23
1.6 Attract ors and physical measures 24
1.7 A conjecture on finitude of attractors 26
One-Dimensional Dynamics 29
2.1 Hyperbolicity 29
2.2 Non-critical behavior 32
2.3 Density of hyperbolicity 34
2.4 Chaotic behavior 34
2.5 The renormalization theorem 36
2.6 Statistical properties of unimodal maps 37
Homoclinic Tangencies 41
3.1 Homoclinic tangencies and Cantor sets 42
3.2 Persistent tangencies, coexistence of attractors 43
3.3 Hyperbolicity and fractal dimensions 50
3.4 Stable intersections of regular Cantor sets 54
3.5 Homoclinic tangencies in higher dimensions 60
3.6 On the boundary of hyperbolic systems 66
Henon-like Dynamics 71
4.1 Henon-like families 72
4.2 Abundance of strange attractors 77
4.3 Sinai-Ruelle-Bowen measures 85
4.4 Decay of correlations and central limit theorem 95
4.5 Stochastic stability 99
4.6 Chaotic dynamics near homoclinic tangencies 103
Non-Critical Dynamics and Hyperbolicity 112
5.1 Non-critical surface dynamics 112
5.2 Domination implies almost hyperbolicity 114
5.3 Homoclinic tangencies vs. Axiom A 115
5.4 Entropy and homoclinic points on surfaces 117
5.5 Non-critical behavior in higher dimensions 119
Heterodimensional Cycles and Blenders 122
6.1 Heterodimensional cycles 123
6.2 Blenders 129
6.3 Partially hyperbolic cycles 135
Robust Transitivity 137
7.1 Examples of robust transitivity 138
7.2 Consequences of robust transitivity 142
7.3 Invariant foliations 152
Stable Ergodicity 161
8.1 Examples of stably ergodic systems 162
8.2 Accessibility and ergodicity 164
8.3 The theorem of Pugh-Shub 165
8.4 Stable ergodicity of torus automorphisms 166
8.5 Stable ergodicity and robust transitivity 167
8.6 Lyapunov exponents and stable ergodicity 168
Robust Singular Dynamics 170
9.1 Singular invariant sets 171
9.2 Singular cycles 177
9.3 Robust transitivity and singular hyperbolicity 182
9.4 Consequences of singular hyperbolicity 191
9.5 Singular Axiom A flows 196
9.6 Persistent singular attractors 199
Generic Diffeomorphisms 202
10.1 A quick overview 202
10.2 Notions of recurrence 205
10.3 Decomposing the dynamics to elementary pieces 206
10.4 Homoclinic classes and elementary pieces 212
10.5 Wild behavior vs. tame behavior 217
10.6 A sample of wild dynamics 220
SRB Measures and Gibbs States 226
11.1 SRB measures for certain non-hyperbolic maps 227
11.2 Gibbs u-states for Eu © Ecs systems 234
11.3 SRB measures for dominated dynamics 246
11.4 Generic existence of SRB measures 253
11.5 Extensions and related results 260
Lyapunov Exponents 265
12.1 Continuity of Lyapunov exponents 266
12.2 A dichotomy for conservative systems 270
12.3 Deterministic products of matrices 273
12.4 Abundance of non-zero exponents 276
12.5 Looking for non-zero Lyapunov exponents 281
12.6 Hyperbolic measures are exact dimensional 286
A Perturbation Lemmas 288
B Normal Hyperbolicity and Foliations 298
C Non-Uniformly Hyperbolic Theory 309
D Random Perturbations 321
E Decay of Correlations 332
Conclusion 357
References 360
Index 381