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 𝜇𝑞 for 𝑞⁢𝜑𝑢, where 𝜑𝑢 is the geometric potential. We show that as 𝑞→1−, the weak-* limit of 𝜇𝑞 is the restriction of the Liouville measure to the regular set.