Study of ergodic properties of partially hyperbolic dynamical systems

本文研究了部分双曲动力系统的遍历性。作为主要结果之一，我们证明了二维流形上的一般部分双曲自同态具有许多物理测度（具有吸引性质的不变测度），使得几乎所有初始点的轨道关于Lebesgue测度的分布渐近于其中一个测度.由于这类结果只对一致双曲型动力系统和一维动力系统是已知的，因此这是一种突破。作为一个密切相关的课题，我们还研究了泛函分析方法在动力系统中的应用。作为这方面的一个主要结果，我们构造了一个Ruelle转移算子自然作用于其上的Banach分布空间，并且证明了该作用的本质谱半径是小的。这导致了许多结果衰减的相关性和动态zeta函数的研究。

In this study, we investigate ergodic properties of partially hyperbolic dynamicals systems. As one of the main results, we showed that generic partially hyperbolic endomorphism one two dimensional manifolds has finitely many physical measures (invariant measure that has attracting nature) such that the distribution of the orbits for almost every initial points with respect to the Lebesgue measure is asymptotic to one of those measures. Since this kind of results has only known for uniformly hyperbolic dynamical systems and one dimensional dynamical systems, this result is a kind of break-through. As a closely related subject, we also studied applications of functional analytic method in dynamical sytems. As a main result in this direction, we contructed a Banach spaces of distributions on which the Ruelle transfer operators acts on naturally and the essential spectral radius of that action is shown to be small. This lead to many results on decay of correlations and study of dynamical zeta functions.