1 Hyperbolicity and Beyond
1.1 Spectral decomposition
1.2 Structural stability
1.3 Sinai-Ruelle-Bowen theory
1.4 Heterodimensional cycles
1.5 Homoclinic tangencies
1.6 Attractors and physical measures
1.7 A conjecture on finitude of attractors
2 One-Dimensional Dynamics
2.1 Hyperbolicity
2.2 Non-critical behavior
2.3 Density of hyperbolicity
2.4 Chaotic behavior
2.5 The renormalization theorem
2.6 Statistical properties of unimodal maps
3 Homoclinic Tangencies
3.1 Homoclinic tangencies and Cantor sets
3.2 Persistent tangencies, coexistence of attractors
3.2.1 Open sets with persistent tangencies
3.3 Hyperbolicity and fractal dimensions
3.4 Stable intersections of regular Cantor sets
3.4.1 Renormalization and pattern recurrence
3.4.2 The scale recurrence lemma
3.4.3 The probabilistic argument
3.5 Homoclinic tangencies in higher dimensions
3.5.1 Intrinsic differentiability of foliations
3.5.2 Frequency of hyperbolicity
3.6 On the boundary of hyperbolic systems
4 He(')non like Dynamics
4.1 He(')non-like families
4.1.1 Identifying the attractor
4.1.2 Hyperbolicity outside the critical regions
4.2 Abundance of strange attractors
4.2.1 The theorem of Benedicks-Carleson
4.2.2 Critical points of dissipativediffeomorphisms
4.2.3 Some conjectures and open questions
4.3 Sinai-Ruelle-Bowen measures
4.3.1 Existence and uniqueness
4.3.2 Solution of the basin problem
4.4 Decay of correlations and central limit theorem
4.5 Stochastic stability
4.6 Chaotic dynamics near homoclinic tangencies
4.6.1 Tangencies and strange attractors
4.6.2 Saddle-node cycles and strange attractors
4.6.3 Tangenciesand non-uniform hyperbolicity
5 Non-Critical Dynamics and Hyperbolicity
5.1 Non-critical surface dynamics
5.2 Domination implies almost hyperbolicity
5.3 Homoclinic tangencies vs. Axiom A
5.4 Entropy and homoclinic points on surfaces
5.5 Non-critical behavior in higher dimensions
6 Heterodimensional Cycles and Blenders
6.1 Heterodimensionalcycles
6.1.1 Explosion of homoclinic classes
6.1.2 A simplifiedexample
6.1.3 Unfolding heterodimensionalcycles
6.2 Blenders
6.2.1 Asimplified model
6.2.2 Relaxing the construction
6.3 Partially hyperbolic cycles
7 Robust Transitivity
7.1 Examples of robust transitivity
7.1.1 An example ofShub
7.1.2 An example ofMan(~)e(')
7.1.3 A local criterium for robust transitivity
7.1.4 Robust transitivity without hyperbolic directions
7.2 Consequences of robust transitivity
7.2.1 Lack of domination and creation of sinks or sources
7.2.2 Dominated splittings vs. homothetic transformations
7.2.3 On the dynamics of robustly transitive sets
7.2.4 Manifolds supporting robustly transitive maps
7.3 Invariant foliation
7.3.1 Pathological central foliations
7.3.2 Density of accessibility
7.3.3 Minimality of the strong invariant foliations
7.3.4 Compact central leaves
8 Stable Ergodieity
8.1 Examples of stably ergodic systems
8.1.i Perturbations of time-i maps of geodesic flows
8.1.2 Perturbations of skew-products
8.1.3 Stable ergodicity without partial hyperbolicity
8.2 Accessibility and ergodicity
8.3 The theorem of Pugh-Shub
8.4 Stable ergodicity of torus automorphisms
8.5 Stable ergodicity and robust transitivity
8.6 Lyapunov exponents and stable ergodicity
9 Robust Singular Dynamics
9.1 Singular invariant sets
9.1.1 Geometric Lorenz attractors
9.1.2 Singular horseshoes
9.1.3 Multidimensional Lorenz attractors
9.2 Singular cycles
9.2.1 Explosions of singular cycles
9.2.2 Expanding and contracting singular cycles
9.2.3 Singular attractors arising from singular cycles
9.3 Robust transitivity and singular hyperbolicity
9.3.1 Robust globally transitive flows
9.3.2 Robustness and singular hyperbolicity
9.4 Consequences of singular hyperbolicity
9.4.1 Singularities attached to regular orbits
9.4.2 Ergodic properties of singular hyperbolic attractors
9.4.3 From singular hyperbolicity back to robustness
9.5 Singular Axiom A flows
9.6 Persistent singular attractors
10 Generic Diffeomorphisms
10.1 A quick overview
10.2 Notions of recurrence
10.3 Decomposing the dynamics to elementary pieces
10.3.1 Chain recurrence classes and filtrations
10.3.2 Maximal weakly transitive sets
10.3.3 A generic dynamical decomposition theorem
10.4 Homoclinic classes and elementary pieces
10.4.1 Homoclinic classes and maximal transitive sets
10.4.2 Homoclinic classes and chain recurrence classes
10.4.3 Isolated homoclinic classes
10.5 Wild behavior vs. tame behavior
10.5.1 Finiteness of homoclinic classes
10.5.2 Dynamics of tame diffeomorphisms
10.6 A sample of wild dynamics
10.6.1 Coexistence of infinitely many periodic attractors
10.6.2 C1 coexistence phenomenon in higher dimensions
10.6.3 Generic coexistence of aperiodic pieces
11 SRB Measures and Gibbs States
11.1 SRB measures for certain non-hyperbolic maps
11.1.1 Intermingled basins of attraction
1.1.2 A transitive map with two SRB measures
11.1.3 Robust multidimensional attractors
11.1.4 Open sets of non-uniformly hyperbolic maps
11.2 Gibbs u-states for Eu(0+)Ecs systems
11.2.1 Existence of Gibbs u-states
11.2.2 Structure of Gibbs u-states
11.2.3 Every SRB measure is a Gibbs u-state
11.2.4 Mostly contracting central direction
11.2.5 Differentiability of Gibbs u-states
11.3 SRB measures for dominated dynamics
11.3.1 Non-uniformly expanding maps
11.3.2 Existence of Gibbs cu-states
11.3.3 Simultaneous hyperbolic times
11.3.4 Stability of cu-Gibbs states
11.4 Generic existence of SRB measures
11.4.1 A piecewise affine model
11.4.2 Transfer operators
11.4.3 Absolutely continuous invariant measure
11.5 Extensions and related results
11.5.1 Zero-noise limit and the entropy formula
11.5.2 Equilibrium states of non-hyperbolic maps
12 Lyapunov Exponents
2.1 Continuity of Lyapunov exponents
12.2 A dichotomy for conservative systems
12.3 Deterministic products of matrices
12.4 Abundance of non-zero exponents
12.4.1 Bundle-free cocycles
12.4.2 A geometric criterium for non-zero exponents
12.4.3 Conclusion and an application
12.5 Looking for non-zero Lyapunov exponents
12.5.1 Removing zero Lyapunov exponents
12.5.2 Lower bounds for Lyapunov exponents
12.5.3 Genericity of non-uniform hyperbolicity
12.6 Hyperbolic measures are exact dimensiona
A Perturbation Lemmas
A.1 Closing lemmas
A.2 Ergodic closing lemma
A.3 Connecting lemmas
A.4 Some ideas of the proofs
A.5 A connecting lemma for pseudo-orbits
A.6 Realizing perturbations of the derivative
B NormalHyperbolicity and Foliations
B.1 Dominated splittings
B.1.1 Definition and elementary properties
B.1.2 Proofs of the elementary properties:
B.2 Invariant foliations
B.3 Linear Poincare(') flows
C Non-Uniformly Hyperbolic Theory
C.1 The linear theory
C.2 Stable manifold theorem
C.3 Absolute continuity of foliations
C.4 Conditional measures along invariant foliations
C.5 Local product structure
C.6 The disintegration theorem
D Random Perturbations
D.1 Markov chain model
D.2 Iterations of random maps
D.3 Stochastic stability
D.4 Realizing Markov chains by random maps
D.5 Shadowing versus stochastic stability
D.6 Random perturbations of flows
E Decay of Correlations
E.1 Transfer operators: spectral gap property
E.2 Expanding and piecewise expanding maps
E.3 Invariant cones and projective metrics
E.4 Uniformly hyperbolic diffeomorphisms
E.5 Uniformly hyperbolic flows
E.6 Non-uniformly hyperbolic systems
E.7 Non-exponential convergence
E.8 Maps with neutral fixed points
E.9 Central limit theorem
Conclusion
References
Index |
