Finiteness of gibbs measures on noncompact manifolds with pinched negative curvature

Abstract

We characterize the finiteness of Gibbs measures for geodesic flows on negatively curved manifolds by several criteria, analogous to those proposed by Sarig for symbolic dynamical systems over an infinite alphabet. As an application, we recover Dal'bo-Otal-Peign{\'e} criterion of finiteness for the Bowen-Margulis measure on geometrically finite hyperbolic manifolds, as well as Peign{\'e}' examples of gemetrically infinite manifolds having a finite Bowen-Margulis measure. These criteria should be useful in the future to find more examples with finite Gibbs measures.