Ergodic Properties of Nonuniformly Hyperbolic Systems

非均匀双曲系统的遍历性质

• 批准号: 0202999
• 负责人: Nicolai Haydn
• 金额: $ 8.91万
• 依托单位: University of Southern California
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-07-01 至 2002-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0202999&HistoricalAwards=false
• 关键词: Ergodic; Properties; Nonuniformly; Hyperbolic; Systems

Proposal Number: DMS-0202999PI: Huyi Hu ABSTRACTThis project is devoted to the study of ergodic properties ofnonuniformly hyperbolic systems. We focus on almosthyperbolic systems and some related systems. An smooth dynamicalsystem is almost hyperbolic if it is hyperbolic everywhere exceptat a finite set of points. Such systems may have quite differentergodic behaviors from uniformly hyperbolic systems.In this project we will study existence of equilibrium states,including SRB measures and absolutely continuous invariantmeasures in multidimensional spaces; rates of convergence to theequilibrium states for both finite and infinite measure cases,and some related topics such as rates of decay of correlations ofthe systems and the central limit theorem; some other ergodicproperties of the systems such as stochastic stability, Gibbsproperties, topological conjugation. We are also interested inusing these or similar systems to construct varies examples ofsystems that have given properties, for instance, diffeomorphismsor flows on any manifolds that preserve the Riemannianvolume, have nonzero Lyapunov exponents almost everywhere, andhave countably many ergodic components.Ergodic theory concerns the statistic behavior of systems.Ergodic properties of uniformly hyperbolic systems were the mainresearch subject in smooth dynamical systems from 60's to 80's.The behaviors of such systems are regarded as chaotic.Now nonuniformly hyperbolic systems become a main research topicin the field. This project is devoted to the study of ergodicproperties of almost hyperbolic systems and some other relatedsystems. Almost hyperbolic systems are smooth dynamical systemsin which hyperbolic conditions are violated at only finitelynumber of points. These systems lie on the boundary of the setof uniformly hyperbolic systems, and are the simplest butnontrivial nonuniformly hyperbolic systems. Earlier studies onexamples of such systems, such as systems on the intervals andtorus, show that some ergodic properties may change dramatically.In this project we try to develop some theorems for generalalmost hyperbolic systems rather than individual examples.

提案编号：DMS-02029999PI：Huyi Hu Abstractthis项目专门研究诺隔均匀双曲线系统的ergodic特性。 我们专注于Almosthyperbolic系统和一些相关系统。 如果除了有限的点以外，那么平滑的动力系统几乎是双曲线。 这样的系统可能与均匀双曲线系统具有完全不同的行为。在这个项目中，我们将研究平衡状态的存在，包括多维空间中的SRB测量和绝对连续的不变量；有限和无限措施案例的收敛速率，以及某些相关主题，例如系统和中心极限定理的相关性衰减率； 这些系统的其他一些巨像，例如随机稳定性，gibbsproperties，拓扑结合。我们也有兴趣使用这些或类似的系统来构建各种系统的示例，例如，在保留riemannianvolume的任何流形上，差异性ss词几乎在任何地方都具有无数的lyapunov指数。从60到80年代的平滑动力系统。这种系统的行为被视为混乱。现在，非均匀的双曲系统成为该领域的主要研究主题。 该项目致力于研究几乎双曲线系统和其他一些相关系统的麦古迪杂质。几乎双曲线系统是平滑的动力学系统，仅在有限的点上违反双曲线条件。 这些系统位于固定双曲线系统的边界上，是最简单的肉体非均匀双曲线系统。 较早的研究系统的研究示象（例如在间隔和托鲁斯）的系统表明，某些千古特性可能会发生巨大变化。在该项目中，我们试图为一般的多纤维系统而不是单个示例开发一些定理。