Abstract
We study the one parameter family of potential functions associated with the unstable Jacobian potential (or geometric potential) 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 , 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.
Acknowledgments
This work was carried out in a workshop at the American Institute of Mathematics. We thank AIM for their support and hospitality. We also thank the referees for many useful suggestions.
Disclosure statement
No potential conflict of interest was reported by the authors.