Unique equilibrium states for geodesic flows in nonpositive curvature
By Keith Burns, Vaughn Climenhaga, Tom A. Fisher, Daniel J. Thompson. Geometric and Functional Analysis, 2017
66 citations recorded by the source corpus.
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 $${(-\infty,1)}$$(-∞,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 $${\mathbb{R}}$$R. For general potential functions $${\varphi}$$φ, we prove that the pressure gap holds whenever $${\varphi}$$φ is locally constant on a neighborhood of the singular set, which allows us to give examples for which uniqueness holds on a C0-open and dense set of Hölder potentials.
Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/b6c1421703051b40032c9712957d1dbaee4f4fd4
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Citations in the corpus
- Unique equilibrium states for flows and homeomorphisms with non-uniform structure (2015, unknown): “particular, the main theorems of this paper are applied to diffeomorphisms with weak forms of hyperbolicity in [CFT15] and to geodesic flows in nonpositive curvature in [BCFT16].”
- Measures of maximal entropy for surface diffeomorphisms (2018, background): “Flows of special interest have also been analyzed such as the Teichmüller flow [20], or the geodesic flow on compact surfaces with nonpositive curvature [25].”