本项目主要研究黎曼流形上测地流以及非平坦曲面上弹球系统的动力学性态。在测地流理论方面我们将主要研究以下一些问题：秩一无焦点流形上测地流的最大熵测度的统计性态；高维无共轭点流形上测地流的熵可扩性；广义秩一无共轭点流形上最大熵测度的存在唯一性及相关问题。流形上的弹球系统是测地流在带边流形上的推广，它可以视为弹球和测地流的复合。本项目中我们将研究2维负曲率球台上弹球系统的动力学行为及统计理论，并希望将其推广到更一般的非正曲率及无焦点球台上去。

In this project, we consider the dynamical behavior of the geodesic flows on Riemannian manifolds, and billiards on non-flat surfaces. On the theory of geodesic flows, we will carry out researches on the following topics: the statistical properties of invariant measures of maximal entropy for geodesic flows on rank 1 manifolds without focal points; the entropy-expansiveness for geodesic flows on manifolds without conjugate points; the existence, uniqueness and other important properties of invariant measures of maximal entropy for geodesic flows on generalized rank 1 manifolds without conjugate points. The theory of billiards on manifolds is a generalization of the classic theory of geodesic flows to manifolds with smooth boundaries. It is a composition of the classic billiards and geodesic flows. In this project, we plan to study the dynamics and the statistical properties of billiards on 2-dimensional billiard tables with negative curvatures. We expect to generalize this research to billiard tables with non-positive curvatures and tables without focal points.