Dynamics Beyond Uniform Hyperbolicity

by Bonatti, Christian; Diaz, Lorenzo J.; Viana, Marcelo

• ISBN13: 9783540220664
• ISBN10: 3540220666
• Format: Hardcover
• Copyright: 2004-12-16
• Publisher: Springer Verlag

Summary

The notion of uniform hyperbolicity, introduced by Steve Smale in the early sixties, unified important developments and led to a remarkably successful theory for a large class of systems: uniformly hyperbolic systems often exhibit complicated evolution which, nevertheless, is now rather well understood, both geometrically and statistically. Another revolution has been taking place in the last couple of decades, as one tries to build a global theory for "most" dynamical systems, recovering as much as possible of the conclusions of the uniformly hyperbolic case, in great generality. This book aims to put such recent developments in a unified perspective, and to point out open problems and likely directions for further progress. It is aimed at researchers, both young and senior, willing to get a quick, yet broad, view of this part of dynamics. Main ideas, methods, and results are discussed, at variable degrees of depth, with references to the original works for details and complementary information.

Table of Contents

1 Hyperbolicity and Beyond	1	(12)
1.1 Spectral decomposition	1	(2)
1.2 Structural stability	3	(1)
1.3 Sinai-Ruelle-Bowen theory	4	(2)
1.4 Heterodimensional cycles	6	(1)
1.5 Homoclinic tangencies	6	(1)
1.6 Attractors and physical measures	7	(2)
1.7 A conjecture on finitude of attractors	9	(4)
2 One-Dimensional Dynamics	13	(12)
2.1 Hyperbolicity	13	(3)
2.2 Non-critical behavior	16	(2)
2.3 Density of hyperbolicity	18	(1)
2.4 Chaotic behavior	18	(2)
2.5 The renormalization theorem	20	(1)
2.6 Statistical properties of unimodal maps	21	(4)
3 Homoclinic Tangencies	25	(30)
3.1 Homoclinic tangencies and Cantor sets	26	(1)
3.2 Persistent tangencies, coexistence of attractors	27	(7)
3.2.1 Open sets with persistent tangencies	32	(2)
3.3 Hyperbolicity and fractal dimensions	34	(4)
3.4 Stable intersections of regular Cantor sets	38	(6)
3.4.1 Renormalization and pattern recurrence	39	(2)
3.4.2 The scale recurrence lemma	41	(2)
3.4.3 The probabilistic argument	43	(1)
3.5 Homoclinic tangencies in higher dimensions	44	(6)
3.5.1 Intrinsic differentiability of foliations	45	(2)
3.5.2 Frequency of hyperbolicity	47	(3)
3.6 On the boundary of hyperbolic systems	50	(5)
4 Henon-like Dynamics	55	(42)
4.1 Henon-like families	56	(5)
4.1.1 Identifying the attractor	58	(1)
4.1.2 Hyperbolicity outside the critical regions	59	(2)
4.2 Abundance of strange attractors	61	(8)
4.2.1 The theorem of Benedicks-Carleson	61	(1)
4.2.2 Critical points of dissipative diffeomorphisms	62	(3)
4.2.3 Some conjectures and open questions	65	(4)
4.3 Sinai-Ruelle-Bowen measures	69	(10)
4.3.1 Existence and uniqueness	69	(5)
4.3.2 Solution of the basin problem	74	(5)
4.4 Decay of correlations and central limit theorem	79	(4)
4.5 Stochastic stability	83	(4)
4.6 Chaotic dynamics near homoclinic tangencies	87	(10)
4.6.1 Tangencies and strange attractors	87	(3)
4.6.2 Saddle-node cycles and strange attractors	90	(2)
4.6.3 Tangencies and non-uniform hyperbolicity	92	(5)
5 Non-Critical Dynamics and Hyperbolicity	97	(10)
5.1 Non-critical surface dynamics	97	(2)
5.2 Domination implies almost hyperbolicity	99	(1)
5.3 Homoclinic tangencies vs. Axiom A	100	(2)
5.4 Entropy and homoclinic points on surfaces	102	(2)
5.5 Non-critical behavior in higher dimensions	104	(3)
6 Heterodimensional Cycles and Blenders	107	(16)
6.1 Heterodimensional cycles	108	(6)
6.1.1 Explosion of homoclinic classes	108	(1)
6.1.2 A simplified example	109	(4)
6.1.3 Unfolding heterodimensional cycles	113	(1)
6.2 Blenders	114	(6)
6.2.1 A simplified model	115	(2)
6.2.2 Relaxing the construction	117	(3)
6.3 Partially hyperbolic cycles	120	(3)
7 Robust Transitivity	123	(24)
7.1 Examples of robust transitivity	124	(4)
7.1.1 An example of Shub	125	(1)
7.1.2 An example of Mañé	125	(1)
7.1.3 A local criterium for robust transitivity	126	(1)
7.1.4 Robust transitivity without hyperbolic directions	127	(1)
7.2 Consequences of robust transitivity	128	(10)
7.2.1 Lack of domination and creation of sinks or sources	130	(2)
7.2.2 Dominated splittings vs. homothetic transformations	132	(2)
7.2.3 On the dynamics of robustly transitive sets	134	(2)
7.2.4 Manifolds supporting robustly transitive maps	136	(2)
7.3 Invariant foliations	138	(9)
7.3.1 Pathological central foliations	138	(2)
7.3.2 Density of accessibility	140	(2)
7.3.3 Minimality of the strong invariant foliations	142	(2)
7.3.4 Compact central leaves	144	(3)
8 Stable Ergodicity	147	(10)
8.1 Examples of stably ergodic systems	148	(2)
8.1.1 Perturbations of time-1 maps of geodesic flows	148	(1)
8.1.2 Perturbations of skew-products	148	(1)
8.1.3 Stable ergodicity without partial hyperbolicity	149	(1)
8.2 Accessibility and ergodicity	150	(1)
8.3 The theorem of Pugh-Shub	151	(1)
8.4 Stable ergodicity of torus automorphisms	152	(1)
8.5 Stable ergodicity and robust transitivity	153	(1)
8.6 Lyapunov exponents and stable ergodicity	154	(3)
9 Robust Singular Dynamics	157	(32)
9.1 Singular invariant sets	158	(6)
9.1.1 Geometric Lorenz attractors	158	(3)
9.1.2 Singular horseshoes	161	(2)
9.1.3 Multidimensional Lorenz attractors	163	(1)
9.2 Singular cycles	164	(5)
9.2.1 Explosions of singular cycles	165	(1)
9.2.2 Expanding and contracting singular cycles	166	(2)
9.2.3 Singular attractors arising from singular cycles	168	(1)
9.3 Robust transitivity and singular hyperbolicity	169	(9)
9.3.1 Robust globally transitive flows	170	(3)
9.3.2 Robustness and singular hyperbolicity	173	(5)
9.4 Consequences of singular hyperbolicity	178	(5)
9.4.1 Singularities attached to regular orbits	178	(1)
9.4.2 Ergodic properties of singular hyperbolic attractors	179	(1)
9.4.3 From singular hyperbolicity back to robustness	180	(3)
9.5 Singular Axiom A flows	183	(3)
9.6 Persistent singular attractors	186	(3)
10 Generic Diffeomorphisms	189	(24)
10.1 A quick overview	189	(3)
10.2 Notions of recurrence	192	(1)
10.3 Decomposing the dynamics to elementary pieces	193	(6)
10.3.1 Chain recurrence classes and filtrations	195	(1)
10.3.2 Maximal weakly transitive sets	196	(1)
10.3.3 A generic dynamical decomposition theorem	197	(2)
10.4 Homoclinic classes and elementary pieces	199	(5)
10.4.1 Homoclinic classes and maximal transitive sets	199	(3)
10.4.2 Homoclinic classes and chain recurrence classes	202	(1)
10.4.3 Isolated homoclinic classes	202	(2)
10.5 Wild behavior vs. tame behavior	204	(3)
10.5.1 Finiteness of homoclinic classes	204	(1)
10.5.2 Dynamics of tame diffeomorphisms	205	(2)
10.6 A sample of wild dynamics	207	(7)
10.6.1 Coexistence of infinitely many periodic attractors	207	(1)
10.6.2 C¹ coexistence phenomenon in higher dimensions	208	(1)
10.6.3 Generic coexistence of aperiodic pieces	208	(5)
11 SRB Measures and Gibbs States	213	(40)
11.1 SRB measures for certain non-hyperbolic maps	214	(7)
11.1.1 Intermingled basins of attraction	214	(2)
11.1.2 A transitive map with two SRB measures	216	(1)
11.1.3 Robust multidimensional attractors	217	(2)
11.1.4 Open sets of non-uniformly hyperbolic maps	219	(2)
11.2 Gibbs u-states for Eu ECs systems	221	(12)
11.2.1 Existence of Gibbs u-states	221	(2)
11.2.2 Structure of Gibbs u-states	223	(2)
11.2.3 Every SRB measure is a Gibbs u-state	225	(6)
11.2.4 Mostly contracting central direction	231	(1)
11.2.5 Differentiability of Gibbs u-states	232	(1)
11.3 SRB measures for dominated dynamics	233	(7)
11.3.1 Non-uniformly expanding maps	234	(2)
11.3.2 Existence of Gibbs cu-states	236	(1)
11.3.3 Simultaneous hyperbolic times	237	(2)
11.3.4 Stability of cu-Gibbs states	239	(1)
11.4 Generic existence of SRB measures	240	(7)
11.4.1 A piecewise affine model	241	(2)
11.4.2 Transfer operators	243	(2)
11.4.3 Absolutely continuous invariant measure	245	(2)
11.5 Extensions and related results	247	(7)
11.5.1 Zero-noise limit and the entropy formula	247	(2)
11.5.2 Equilibrium states of non-hyperbolic maps	249	(4)
12 Lyapunov Exponents	253	(24)
12.1 Continuity of Lyapunov exponents	254	(4)
12.2 A dichotomy for conservative systems	258	(3)
12.3 Deterministic products of matrices	261	(3)
12.4 Abundance of non-zero exponents	264	(5)
12.4.1 Bundle-free cocycles	266	(1)
12.4.2 A geometric criterium for non-zero exponents	267	(1)
12.4.3 Conclusion and an application	267	(2)
12.5 Looking for non-zero Lyapunov exponents	269	(5)
12.5.1 Removing zero Lyapunov exponents	269	(1)
12.5.2 Lower bounds for Lyapunov exponents	270	(2)
12.5.3 Genericity of non-uniform hyperbolicity	272	(2)
12.6 Hyperbolic measures are exact dimensional	274	(3)
A Perturbation Lemmas	277	(10)
A.1 Closing lemmas	278	(1)
A.2 Ergodic closing lemma	279	(1)
A.3 Connecting lemmas	279	(2)
A.4 Some ideas of the proofs	281	(3)
A.5 A connecting lemma for pseudo-orbits	284	(1)
A.6 Realizing perturbations of the derivative	285	(2)
B Normal Hyperbolicity and Foliations	287	(12)
B.1 Dominated splittings	287	(6)
B.1.1 Definition and elementary properties	287	(3)
B.1.2 Proofs of the elementary properties:	290	(3)
B.2 Invariant foliations	293	(2)
B.3 Linear Poincare flows	295	(4)
C Non-Uniformly Hyperbolic Theory	299	(12)
C.1 The linear theory	299	(2)
C.2 Stable manifold theorem	301	(1)
C.3 Absolute continuity of foliations	302	(1)
C.4 Conditional measures along invariant foliations	303	(1)
C.5 Local product structure	304	(1)
C.6 The disintegration theorem	305	(6)
D Random Perturbations	311	(12)
D.1 Markov chain model	311	(2)
D.2 Iterations of random maps	313	(1)
D.3 Stochastic stability	314	(3)
D.4 Realizing Markov chains by random maps	317	(2)
D.5 Shadowing versus stochastic stability	319	(1)
D.6 Random perturbations of flows	320	(3)
E Decay of Correlations	323	(26)
E.1 Transfer operators: spectral gap property	324	(1)
E.2 Expanding and piecewise expanding maps	325	(1)
E.3 Invariant cones and projective metrics	326	(2)
E.4 Uniformly hyperbolic diffeomorphisms	328	(1)
E.5 Uniformly hyperbolic flows	329	(2)
E.6 Non-uniformly hyperbolic systems	331	(5)
E.7 Non-exponential convergence	336	(6)
E.8 Maps with neutral fixed points	342	(2)
E.9 Central limit theorem	344	(5)
Conclusion	349	(4)
References	353	(22)
Index	375