In this talk, we present our work on the dynamical theory of geodesic flows on manifolds without focal points. The Bowen-Margulis (BM) measure 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 later 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 and 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-Tsuji-Sullivan (HTS) dichotomy, which means that either the geodesic flow is conservative and ergodic with respect to the BS measure, or it is completely dissipative.
王方,首都师范大学数学科学学院副教授。博士毕业于美国西北大学。主要从事拉格朗日动力系统,测地流的动力学以及动力系统的统计性态等方面的研究。在中国科学,JDE,ETDS,Nonlinearity,DCDS-A等学术期刊发表论文十余篇,出版学术专著一部。