Study of ergodic properties of partially hyperbolic dynamical systems

在本研究中，我们研究了部分双曲动力系统的遍历性质。作为主要结果之一，我们证明了一维流形上的一般部分双曲自同态有有限多个物理测度(具有吸引性质的不变测度)，使得几乎每个初始点的轨道分布相对于勒贝格测度是渐近的。由于这种结果只对一致双曲动力系统和一维动力系统是已知的，这一结果是一种突破。作为一个密切相关的学科，我们还研究了泛函分析方法在动力系统中的应用。作为这一方向的主要结果，我们构造了Ruelle转移算子自然作用于其上的分布的Banach空间，并且该作用的本质谱半径是小的。这导致了许多关于关联衰变和动态Zeta函数研究的结果。

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.