Abstract:

Equilibrium states are important measures associated with dynamical systems, which carry physical and geometrical meanings. For example, the Liouville measure and the measure of the maximal entropy are equilibrium states for geodesics flows over compact negatively curved manifolds. In such cases, the equilibrium states are unique and possesses many ergodic properties. However, it is much less known for non-uniformly hyperbolic systems such as geodesic flows over manifolds with non-positive curvature, no focal point, no conjugate points, etc. In this work we study the equilibrium states for the geodesics flow over rank 1 manifolds over without focal points.  We show the equilibrium states are unique for Hölder potentials satisfying the pressure gap condition. In addition, we provide a criterion for a continuous potential to satisfy the pressure gap condition. Moreover, we derive several ergodic properties of the unique equilibrium states including the equidistribution and the K-property. This work is joint with Dong Chen, Kiho Park.