Ergodic Properties of Nonuniformly Hyperbolic Systems

项目编号：DMS-0202999PI：胡虎本项目致力于研究非一致双曲组的遍历性质。我们主要研究过山药系统和一些相关的系统。一个光滑的动力系统几乎是双曲的，如果它在除有限点集以外的任何地方都是双曲的。在这个项目中，我们将研究平衡态的存在性，包括多维空间中的SRB测度和绝对连续不变测度；在有限和无限测度的情况下收敛到平衡态的速度；以及一些相关的主题，如系统的相关性的衰减率和中心极限定理；系统的一些其他遍历性质，如随机稳定性，Gibs性质，拓扑共轭。我们也有兴趣利用这些或类似的系统来构造具有特定性质的系统的各种例子，例如，在任何流形上的微分同构流保持黎曼体积，几乎处处具有非零的Lyapunov指数，并且具有可数个遍历分量。遍历理论涉及系统的统计行为。从60‘S到80’S，一致双曲系统的遍历性质是光滑动力系统中的主要研究主题。这类系统的行为被认为是混沌的。现在，非一致双曲系统成为该领域的主要研究主题。本课题致力于研究几乎双曲型系统及其相关系统的遍历性。几乎双曲系统是光滑动力系统，其中只在有限点数处违反双曲条件。这些系统位于一致双曲组的边界上，是最简单但非平凡的非一致双曲组。以前对这类系统的例子，如区间和环面上的系统的研究表明，一些遍历性质可能会发生显著的变化。在这个项目中，我们试图建立一般的几乎双曲系统的一些定理，而不是个别的例子。

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.