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 $q{\phi }^u$, where ${\phi }^u$ is the geometric potential. We show that as $q\to 1-$, the weak${}^{\ast }$ limit of ${\mu }_q$ is the restriction of the Liouville measure to the regular set.