Gibbs measures for geodesic flow on CAT(-1) spaces

TL;DR

This paper constructs Gibbs measures for geodesic flows on CAT(-1) spaces with potentials satisfying the Bowen property, establishing their uniqueness as equilibrium states without previous restrictive conditions.

Contribution

It introduces a new method to build Gibbs measures on CAT(-1) spaces for a broad class of potentials, including bounded Hölder continuous functions, without shared segment restrictions.

Findings

Constructed weighted quasi-conformal Patterson densities.

Built Gibbs measures with local product structure.

Proved uniqueness of the equilibrium state when the measure is finite.

Abstract

For a proper geodesically complete CAT(-1) space equipped with a discrete non-elementary action, and a bounded continuous potential with the Bowen property, we construct weighted quasi-conformal Patterson densities and use them to build a Gibbs measure on the space of geodesic lines. Our construction yields a Gibbs measure with local product structure for any potential in this class, which includes bounded H\"older continuous potentials. Furthermore, if the Gibbs measure is finite, then we prove that it is the unique equilibrium state. In contrast to previous results in this direction, we do not require any condition that the potential must take the same value on two geodesic lines which share a common segment.