Unique equilibrium states for geodesic flows over surfaces without focal points

Abstract

Abstract In this paper, we study dynamics of geodesic flows over closed surfaces of genus greater than or equal to 2 without focal points. Especially, we prove that there is a large class of potentials having unique equilibrium states, including scalar multiples of the geometric potential, provided the scalar is less than 1. Moreover, we discuss ergodic properties of these unique equilibrium states, including the Bernoulli property and the fact that weighted regular periodic orbits are equidistributed relative to these unique equilibrium states.