Finiteness of gibbs measures on noncompact manifolds with pinched negative curvature

TL;DR

This paper characterizes when Gibbs measures are finite for geodesic flows on negatively curved noncompact manifolds, providing criteria that extend previous results and help identify new examples of finite measures.

Contribution

It introduces new criteria for the finiteness of Gibbs measures on noncompact manifolds, extending Sarig's symbolic dynamics approach to geometric settings.

Findings

Recovered Dal'bo-Otal-Peigné criterion for hyperbolic manifolds

Provided examples of geometrically infinite manifolds with finite Bowen-Margulis measure

Established criteria useful for identifying finite Gibbs measures in various geometries

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.