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

Abstract

We study the one parameter family of potential functions $q{\phi }^u$ associated with the geometric potential ${\phi }^u$ 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 $q{\phi }^u$, 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.

9 pages, 1 figure