Phase transitions for the geodesic flow of a rank one surface with nonpositive curvature

We study the one parameter family of potential functions \(q\varphi^u\) associated with the geometric potential \(\varphi^u\) for the geodesic flow of a compact rank 1 surface of nonpositive curvature. For \(q 1\) it is known that an invariant measure is an equilibrium state if and only if it is supported on the singular set. We study the critical value \(q=1\) and show that the ergodic equilibrium states are either the restriction to the regular set of the Liouville measure, or measures supported on the singular set. In particular, when~\(q = 1\), there is a unique ergodic equilibrium state that gives positive measure to the regular set.