The ergodic theory of geodesic flows over certain manifolds without conjugate points

Abstract

In this article, we announce several new results in the ergodic theory of geodesic flows over uniform visibility manifolds without conjugate points. The topics include the ergodicity with respect to the Liouville measure, uniqueness and Bernoulli properties of the measure of maximal entropy and equilibrium states, counting closed geodesics and volume growth asymptotics, Margulis functions and related rigidity phenomenon, and the Hopf-Tsuji-Sullivan dichotomy with respect to the Bowen-Margulis measure. The detailed proof is given in [37].