Ergodic Properties of Nonuniformly Hyperbolic Systems

项目编号：DMS-0202999项目负责人：胡毅摘要本项目致力于研究非一致双曲系统的遍历性。 我们主要研究准抛物型方程组及其相关的方程组。 一个光滑动力系统几乎是双曲的，如果它在除有限点集之外的任何地方都是双曲的。 这类系统可能具有与一致双曲型系统完全不同的遍历性态，本项目将研究多维空间中平衡态的存在性，包括SRB测度和绝对连续不变测度，有限和无限测度情形下到平衡态的收敛速度，以及系统相关性的衰减速度和中心极限定理;系统的随机稳定性、Gibbs性质、拓扑共轭等遍历性。我们也有兴趣使用这些或类似的系统来构造各种各样的系统的例子，这些系统具有给定的性质，例如，在任何流形上保持黎曼体积，几乎处处具有非零的李雅普诺夫指数，一致双曲系统的遍历性是光滑动力系统的主要研究对象，60年代到80年代，这类系统的行为被认为是混沌的，现在非一致双曲系统已成为该领域的一个主要研究课题。 本项目致力于研究几乎双曲型方程组及其它相关方程组的遍历性。几乎双曲型系统是仅在有限个点处违反双曲条件的光滑动力系统。 这些方程组位于一致双曲方程组的边界上，是最简单但非平凡的非一致双曲方程组。 早期对这类系统的例子，如区间上的系统和环面上的系统的研究表明，某些遍历性质可能会发生显著的变化，在本项目中，我们试图建立一般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.