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Dynamics Beyond Uniform Hyperbolicity A Global Geometric and Probabilistic Perspective

:	Christian Bonatti, Lorenzo J. Díaz, Marcelo Viana
:	Dynamics Beyond Uniform Hyperbolicity A Global Geometric and Probabilistic Perspective
:	Springer-Verlag
:	9783540268444
:	1
:	CHF 161.20
:

:	Analysis
:	English

:	384
:	Wasserzeichen/DRM
:	PC/MAC/eReader/Tablet
:	PDF

What is Dynamics about? In broad terms, the goal of Dynamics is to describe the long term evolution of systems for which an 'infinitesimal' evolution rule is known. Examples and applications arise from all branches of science and technology, like physics, chemistry, economics, ecology, communications, biology, computer science, or meteorology, to mention just a few. These systems have in common the fact that each possible state may be described by a finite (or infinite) number of observable quantities, like position, velocity, temperature, concentration, population density, and the like. Thus, m the space of states (phase space) is a subset M of an Euclidean space M . Usually, there are some constraints between these quantities: for instance, for ideal gases pressure times volume must be proportional to temperature. Then the space M is often a manifold, an n-dimensional surface for some n< m. For continuous time systems, the evolution rule may be a differential eq- tion: to each state x G M one associates the speed and direction in which the system is going to evolve from that state. This corresponds to a vector field X(x) in the phase space. Assuming the vector field is sufficiently regular, for instance continuously differentiable, there exists a unique curve tangent to X at every point and passing through x: we call it the orbit of x.

	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