A remark on the phase transition for the geodesic flow of a rank one surface of nonpositive curvature

Abstract

For any rank 1 nonpositively curved surface $M$, it was proved by Burns-Climenhaga-Fisher-Thompson that for any $q<1$, there exists a unique equilibrium state ${\mu }_q$ for $q{\phi }^u$, where ${\phi }^u$ is the geometric potential. We show that as $q\to 1-$, the weak-* limit of ${\mu }_q$ is the restriction of the Liouville measure to the regular set.

Keywords:

• Equilibrium states
• geodesic flow
• phase transition

Mathematics Subject Classifications::

• 37B40
• 37C40
• 37D40

Acknowledgments

The authors would like to thank Federico Rodriguez Hertz for bringing up this question and for suggesting improvements in the exposition. We are also grateful to the referee for many helpful comments and useful remarks.

Disclosure statement

No potential conflict of interest was reported by the author(s).