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一致双曲性之外的动力学：一种整体的（影印版）
作者：（法）博纳蒂
出版社：科学出版社
出版时间：2007-01-01
ISBN：9787030182906
定价：¥68.00
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内容简介
广义而言，动力学的目的是描述由“极少的”演化规律所决定的系统（如微分方程或映射）的长期动态。20世纪60年代早期，Steve Smale引入一臻双曲性概念，统一了动力系统理论的重要结果，导致了关于一大类系统的一个非常成功的理论：一致双曲系统理论。一致双曲系统的动态非常复杂，然而，无论是从几何角度还是统计层面，它们都已得到很好的理解。在过去的20年中，动力系统理论发生了另一个巨大变化：研究人员试图建立一个统一理论，适合“大多数”动力系统；在该理论下，一致双曲情形的尽可能多的结论依然成立。本书尝试由最新进展出发，统一地展望动力系统理论，提出一些公共开问题，指出未来的可能发展方向。本书面向希望快速而广泛地了解动力学这一方面发展的初学者及研究人员，深度不等地讨论了主要的思想、方法以及结果，给出了相关参考文献，读者可以从文献中获知详细细节和补充信息。本书共12章，各章保持相当的独立性，以方便读者阅读特定主题。书后五个附录涵盖了一些重要的补充材料。
作者简介
暂缺《一致双曲性之外的动力学：一种整体的（影印版）》作者简介
目录
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
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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