Gibbs measures for geodesic flow on CAT(-1) spaces
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.
Cite
@article{arxiv.2309.03297,
title = {Gibbs measures for geodesic flow on CAT(-1) spaces},
author = {Caleb Dilsavor and Daniel J. Thompson},
journal= {arXiv preprint arXiv:2309.03297},
year = {2025}
}
Comments
v4, 50 pages. Corrected typos. Final version. To appear in Advances in Mathematics