Statistical properties of finite and infinite physical measures

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This proposed research addresses statistic properties of dynamical systems with finite or infinite physical measures. These properties include rates of convergence to the equilibriums, rates of mixing, statistic stability, and stochastic stability. We focus on almost hyperbolic systems and some related systems. A smooth dynamical system is almost hyperbolic if it is hyperbolic everywhere except at a finite set of orbits. It is known from examples in one dimensional noninvertible maps that such systems have quite different statistic properties from uniformly hyperbolic systems. Moreover, the systems with finite physical measures and those with infinite physical measures have very different statistic behaviors. The latter ones are statistically deterministic, though they are topologically chaotic. The first goal of this project is to extend the results to more general cases such as invertible systems and higher dimensional systems and study some new properties such as stochastic stability. The second goal is to find similar properties in other nonuniformly hyperbolic systems.This project is devoted to the study of statistic properties of nonuniformly hyperbolic systems, in particular, almost hyperbolic systems. A smooth dynamical system is almost hyperbolic if it is hyperbolic everywhere except at a finite set of points. These systems lie on the boundary of the set of uniformly hyperbolic systems, and are the simplest but nontrivial nonuniformly hyperbolic systems. Statistic properties of such systems may be very different from those of uniformly hyperbolic systems. We will try to extend some known results to more general cases that include existence of SRB measures and infinite SRB measures for almost hyperbolic systems in higher dimensional spaces, rates of convergence to the equilibriums and rates of mixing for invertible systems. We will also explore some new properties such as statistic stability and stochastic stability for almost expanding maps on the interval and almost hyperbolic system on the torus. Further, we will try to find the properties, such as infinite physical measures, polynomial decay of correlations, in some other nonuniformly hyperbolic systems.