Unique equilibrium states for geodesic flows in nonpositive curvature

TL;DR

This paper establishes the uniqueness of equilibrium states for geodesic flows on nonpositive curvature manifolds under certain regularity conditions, extending known results and providing new examples of potentials with unique equilibrium states.

Contribution

It proves the uniqueness of equilibrium states for a class of potential functions on rank 1 manifolds, including optimal results in dimension 2 and new conditions in higher dimensions.

Findings

Unique equilibrium states for regular potentials when the singular set has less than full pressure.

Optimal uniqueness results for scalar multiples of the geometric potential in dimension 2.

Examples of potentials with unique equilibrium states on a dense set of H"older potentials.

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.