A remark on the phase transition for the geodesic flow of a rank one surface of nonpositive curvature

TL;DR

This paper investigates the behavior of equilibrium states for rank one nonpositively curved surfaces as a parameter approaches a critical value, revealing a phase transition where measures converge to Liouville measure on the regular set.

Contribution

It establishes the limiting behavior of equilibrium states for the geometric potential as the parameter approaches one from below, extending understanding of phase transitions in this setting.

Findings

Weak* limits of equilibrium states converge to Liouville measure on the regular set as q approaches 1 from below.

Unique equilibrium states exist for all q<1 for the geometric potential.

The result clarifies the nature of phase transitions in geodesic flows of nonpositively curved surfaces.

Abstract

For any rank 1 nonpositively curved surface $M$, it was proved by Burns-Climenhaga-Fisher-Thompson that for any $q<1$, there exists a unique equilibrium state $\mu_q$ for $q\varphi^u$, where $\varphi^u$ is the geometric potential. We show that as $q\to 1-$, the weak$^*$ limit of $\mu_q$ is the restriction of the Liouville measure to the regular set.