We study the one parameter family of potential functions (Formula presented.) associated with the unstable Jacobian potential (or geometric potential) (Formula presented.) 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 (Formula presented.), 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.
- Dynamical systems
- equilibrium states
- geodesic flow
ASJC Scopus subject areas
- Computer Science Applications