非一致双曲系统关于最大熵测度的统计性质

Maximal entropy measure (MEM for short) is an important concept in the theory of dynamical systems and ergodic theory. The study of MEM for nonuniformly hyperbolic systems is now a popular topic. For C^r (r>1) diffeomorphisms on compact smooth two dimensional surfaces and C^r (r>1) flows on compact smooth three-dimensional manifolds, there are countable states symbolic dynamics on sets of full measure (w.r.t. an ergodic measure of positive entropy), and the existence of MEM. We will study statistical properties, especially the decay of correlations of MEM for such nonuniformly hyperbolic systems. We expect a upper bound for the decay of correlations, and hope it is exponentially small. Moreover, we will study almost sure rates of mixing for those systems under small perturbations. This research involves many areas, such as differentiable dynamical systems, topological dynamical systems, ergodic theory, probability, functional analysis, and etc. Part of the results will be a crucial step in the study of statistical properties for countable Markov shifts.

最大熵测度(简记为 MEM)是遍历论中的重要概念。非一致双曲系统中的 MEM 是近年来国际上研究的热点问题，其中已知带有正拓扑熵的 C^r (r>1) 二维曲面微分同胚和三维流存在可数符号空间和 MEM。本项目将通过 MEM 研究这两个非一致双曲系统的统计性质，尤其是相关衰减性；以及在随机小扰动作用下这两个系统的几乎确定相关衰减性。研究目标是估计出相关衰减的上界，希望其是至多指数小的。本项目研究涉及微分动力系统、拓扑动力系统、遍历论、概率论、泛函分析等多个领域。研究结果将有助于我们进一步探讨一般可数符号系统的相关衰减性。

最大熵测度是遍历论中的重要概念。非一致双曲系统中的最大熵测度是近年来国际上研究的热点问题，其中已知带有正拓扑熵的二维曲面微分同胚和三维流存在可数符号空间和 MEM。本项目将通过 MEM 研究这两个非一致双曲系统的统计性质，尤其是相关衰减性；以及在随机小扰动作用下这两个系统的几乎确定相关衰减性。研究目标是估计出相关衰减的上界，希望其是至多指数小的。本项目研究涉及微分动力系统、拓扑动力系统、遍历论、概率论、泛函分析等多个领域。研究结果将有助于我们进一步探讨一般可数符号系统的相关衰减性。