报告题目：The Bowen-Margulis Measures for the Geodesic Flows on Manifolds without Focal Points 

报告人：王方，首都师范大学

时间：2023年5月19日（周五）上午09:30-10:30

地点：东区管理科研楼1418教室

摘要：

In this talk, we introduce the so-called Bowen-Margulis (BM) measure which was first introduced by Bowen and Margulis in their works on the measures of maximal entropy (MME) for geodesic flows on compact manifolds of negative curvature. This measure was reformulated by Sullivan by using the Patterson-Sullivan measures on the ideal boundary of the universal covering space of hyperbolic manifold, and then extended by Knieper to construct the MME for geodesic flows on compact rank 1 manifolds with non-positive curvature. We extend Sullivan and Knieper’s works and construct the BM measure on manifolds without focal points. We show that the BM measure is the unique measure of maximal entropy for the geodesic flows on compact rank 1 manifolds without focal point, and furthermore, this measure is mixing. For non-compact regular manifolds without focal points, under the assumption of visibility, we show that the BS measure satisfies the Hopf-Tsujii-Sullivan dichotomy, which means that either the geodesic flow is conservative and ergodic with respect to the BS measure, or it is completely dissipative. 

This talk is based on a series works joint with Fei Liu, Xiaokai Liu and Weisheng Wu.