Ergodic Properties of Nonuniformly Hyperbolic Systems

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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.