Ergodic Properties of Nonuniformly Hyperbolic Systems

摘要本课题主要研究非均匀双曲系统的遍历性质。我们主要研究近双曲型系统和一些相关的系统。如果一个光滑的动力系统除了有限的点集以外在任何地方都是双曲的，那么它几乎是双曲的。这样的系统可能具有与均匀双曲系统完全不同的行为。在这个项目中，我们将研究平衡态的存在性，包括SRB测度和多维空间中的绝对连续不变测度；收敛到平衡状态的速度为有限和无限测量的情况下，和一些相关的主题，如衰减速率的相关系统和中心极限定理；系统的一些其他遍历性质，如随机稳定性、吉布斯性质、拓扑共轭。我们也有兴趣使用这些或类似的系统来构造各种具有给定性质的系统的例子，例如，在任何流形上的微分模流，这些流形保持黎曼体积，几乎到处都有非零李雅普诺夫指数，并且有可数的遍历分量。遍历理论关注系统的统计行为。均匀双曲系统的遍历性是60 ~ 80年代光滑动力系统的主要研究课题。这种系统的行为被认为是混沌的。目前，非均匀双曲系统已成为该领域的一个主要研究课题。本课题致力于研究几乎双曲系统及其相关系统的遍历性。几乎双曲系统是只在有限个点上违反双曲条件的光滑动力系统。这些系统位于一致双曲系统集的边界上，是最简单但非平凡的非一致双曲系统。早期对这类系统的研究，例如区间和环面上的系统，表明一些遍历性质可能会发生巨大变化。在这个项目中，我们试图为一般的几乎双曲系统而不是个别的例子发展一些定理。

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.