Patterson-Sullivan construction of equilibrium states and weighted counting in nonpositive curvature

Abstract

Consider the geodesic flow on a closed rank one manifold of nonpositive curvature. For certain H\"{o}lder continuous potential, there exists a unique equilibrium state by \cite{BCFT}. In this paper, we introduce the notions of core limit set, regular radial limit set and uniformly recurrent and regular vectors, and then construct a family of Patterson-Sullivan measures on the boundary at infinity in two separate settings. Then we give an explicit construction of the above unique equilibrium state using Patterson-Sullivan measures. This enables us to prove the Bernoulli property of the equilibrium states. Using the Patterson-Sullivan construction and mixing properties of equilibrium states, we count the number of free homotopy classes with weights in nonpositive curvature.