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

TL;DR

This paper investigates the behavior of equilibrium states for the geodesic flow on a rank one surface with nonpositive curvature, especially at the critical parameter value where phase transitions occur.

Contribution

It characterizes the equilibrium states at the critical potential parameter, revealing a unique ergodic state supported on the regular set and others on the singular set.

Findings

Unique ergodic equilibrium state at q=1 supported on the regular set

Existence of measures supported on the singular set as equilibrium states

Phase transition behavior at the critical parameter q=1

Abstract

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 there is a unique equilibrium state associated with $q\varphi^u$, 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.