Thermodynamic formalism and the entropy at infinity of the geodesic flow

Abstract

In this paper we study the ergodic theory and thermodynamic formalism of the geodesic flow on non-compact pinched negatively curved manifolds. We consider two notions of entropy at infinity, the topological and the measure theoretic entropy at infinity. We prove that both notions coincide and we relate this quantity with the upper semicontinuity of the entropy map. This relationship was already studied in [RV] for geometrically finite manifolds; in this paper we extend such results to arbitrary pinched negatively curved manifolds. We obtain several applications to the existence and stability of equilibrium states. Our approach has the advantage to include certain uniformly continous potentials, a situation that can not be studied through the methods from [PPS]. Of particular importance are potentials that vanish at infinity and those that satisfy a critical gap condition, that is, strongly positive recurrent potentials. We also prove some equidistribution and counting results for closed geodesics (beyond the geometrically finite case). In particular we obtain a version of the prime geodesic theorem for strongly positive recurrent manifolds.