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

We study the one parameter family of potential functions $qφ^u$ associated with the geometric potential $φ^u$ for the geodesic flow of a compact rank 1 surface of nonpositive curvature. For $q<1$ it is known that there is a unique equilibrium state associated with $qφ^u$, and it has full support. 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.

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