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

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.