Statistical properties of finite and infinite physical measures

这项研究涉及有限或无限物理测量的动力系统的统计特性。这些性质包括收敛到平衡点的速率、混合速率、统计稳定性和随机稳定性。我们主要研究几乎双曲方程组和一些相关的方程组。一个光滑动力系统几乎是双曲的，如果它除了在有限轨道集之外处处是双曲的。由一维不可逆映射的例子可知，这类系统具有与一致双曲系统完全不同的统计性质。此外，有限物理量系统和无限物理量系统具有非常不同的统计行为。 后者是统计确定性的，尽管它们是拓扑混沌的。本项目的第一个目标是将结果推广到更一般的情况，如可逆系统和高维系统，并研究一些新的性质，如随机稳定性。 第二个目标是在其他非一致双曲型方程组中找到类似的性质，本项目致力于研究非一致双曲型方程组，特别是几乎双曲型方程组的统计性质。一个光滑动力系统是几乎双曲的，如果它除了在有限点集之外处处是双曲的。这些系统位于一致双曲系统集合的边界上，并且是最简单但非平凡的非一致双曲系统。这类系统的统计性质可能与一致双曲系统的统计性质有很大的不同。我们将尝试将一些已知的结果推广到更一般的情况，包括高维空间中几乎双曲系统的SRB测度和无穷SRB测度的存在性，可逆系统的平衡点的收敛速度和混合速度。 我们还将讨论一些新的性质，如区间上的几乎扩张映射和环面上的几乎双曲系统的统计稳定性和随机稳定性。此外，我们还将尝试在其他一些非一致双曲系统中找到一些性质，如无穷物理测度、相关的多项式衰减等。

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.