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廣義而言，動力學的目的是描述由“極少的”演化規律所決定的係統（如微分方程或映射）的長期動態。
20世紀60年代早期，Steve Smale引入一臻雙麯性概念，統一瞭動力係統理論的重要結果，導緻瞭關於一大類係統的一個非常成功的理論：一緻雙麯係統理論。一緻雙麯係統的動態非常復雜，然而，無論是從幾何角度還是統計層麵，它們都已得到很好的理解。
在過去的20年中，動力係統理論發生瞭另一個巨大變化：研究人員試圖建立一個統一理論，適閤“大多數”動力係統；在該理論下，一緻雙麯情形的盡可能多的結論依然成立。
《一緻雙麯綫之外的動力學：一種整體的幾何學的與概率論的觀點》嘗試由最新進展齣發，統一地展望動力係統理論，提齣一些公共開問題，指齣未來的可能發展方嚮。
《一緻雙麯綫之外的動力學：一種整體的幾何學的與概率論的觀點》麵嚮希望快速而廣泛地瞭解動力學這一方麵發展的初學者及研究人員，深度不等地討論瞭主要的思想、方法以及結果，給齣瞭相關參考文獻，讀者可以從文獻中獲知詳細細節和補充信息。
《一緻雙麯綫之外的動力學：一種整體的幾何學的與概率論的觀點》共12章，各章保持相當的獨立性，以方便讀者閱讀特定主題。
書後五個附錄涵蓋瞭一些重要的補充材料。

目錄

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

前言/序言

一緻雙麯綫之外的動力學：一種整體的幾何學的與概率論的觀點 下載 mobi epub pdf txt 電子書

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