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
3.3 Hyperbolicity and fractal dimensions
3.4 Stable intersections of regular Cantor sets
3.5 Homoclinic tangencies in higher dimensions
3.6 On the boundary of hyperbolic systems
4 Henon like Dynamics
4.1 Henon-like families
4.2 Abundance of strange attractors
4.3 Sinai-Ruelle-Bowen measures
4.4 Decay of correlations and central limit theorem
4.5 Stochastic stability
4.6 Chaotic dynamics near homoclinic tangencies
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
6.2 Blenders
6.3 Partially hyperbolic cycles
7 Robust Transitivity
7.1 Examples of robust transitivity
7.2 Consequences of robust transitivity
7.3 Invariant foliation
8 Stable Ergodieity
8.1 Examples of stably ergodic systems
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
9.2 Singular cycles
9.3 Robust transitivity and singular hyperbolicity
9.4 Consequences of singular hyperbolicity
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
10.4 Homoclinic classes and elementary pieces
10.5 Wild behavior vs. tame behavior
10.6 A sample of wild dynamics
11 SRB Measures and Gibbs States
11.1 SRB measures for certain non-hyperbolic maps
11.2 Gibbs u-states for EuEcs systems
11.3 SRB measures for dominated dynamics
11.4 Generic existence of SRB measures
11.5 Extensions and related results
12 Lyapunov Exponents
12.1 Continuity of Lyapunov exponents
12.2 A dichotomy for conservative systems
12.3 Deterministic products of matrices
12.4 Abundance of non-zero exponents
12.5 Looking for non-zero Lyapunov exponents
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.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
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