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一致双曲型之外的动力学
  • 书号:9787030182906
    作者:(法)博纳蒂Bonatti, C.)等著
  • 外文书名:Dynamics Beyond Uniform Hyperbolicity A Global Geometric and Probabilistic Perspective
  • 丛书名:国外数学名著系列(影印版)
  • 装帧:圆脊精装
    开本:B5
  • 页数:408
    字数:473
    语种:英文
  • 出版社:科学出版社
    出版时间:2007-03-01
  • 所属分类:O31 理论力学(一般力学)
  • 定价: ¥68.00元
    售价: ¥54.40元
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  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。
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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
    3.2.1 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
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