Unique equilibrium states for geodesic flows in nonpositive curvature

Abstract

We study geodesic flows over compact rank 1 manifolds and prove that sufficiently regular potential functions have unique equilibrium states if the singular set does not carry full pressure. In dimension 2, this proves uniqueness for scalar multiples of the geometric potential on the interval $(-\infty,1)$, which is optimal. In higher dimensions, we obtain the same result on a neighborhood of 0, and give examples where uniqueness holds on all of $\mathbb{R}$. For general potential functions $\varphi$, we prove that the pressure gap holds whenever $\varphi$ is locally constant on a neighborhood of the singular set, which allows us to give examples for which uniqueness holds on a $C^0$-open and dense set of H\"older potentials.