Thermodynamic formalism for non-compact systems with expansivity and specification

Abstract

We develop the theory of equilibrium states via specification properties for a wide class of continuous flows on complete separable metric spaces. We provide general dynamical criteria which guarantee that there is a unique equilibrium state. This measure is ergodic and satisfies a Gibbs property. Our framework applies to the geodesic flow over negatively curved manifolds beyond the pinched setting. These results also apply beyond the smooth setting to geodesic flows over locally CAT(-1) spaces. Since our phase space is non-compact, we need to establish all the basic definitions and results to make this theory work, including a suitable notion of topological pressure and the variational principle. We introduce the notion of a coherent family of metrics, which captures the properties of a natural family of metrics in our geodesic flow examples which are essential for dealing with cusps. We define Strong Positive Recurrence in this setting and establish it as a criterion to prove the existence and uniqueness of an equilibrium state.