Unique equilibrium states for geodesic flows in nonpositive curvature

K. Burns; V. Climenhaga; T. Fisher; D. J. Thompson

doi:10.1007/s00039-018-0465-8

Unique equilibrium states for geodesic flows in nonpositive curvature

K. Burns, V. Climenhaga, T. Fisher, D. J. Thompson^*

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

15 Scopus citations

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 (- ∞, 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 R. For general potential functions φ, we prove that the pressure gap holds whenever 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.

Original language	English (US)
Pages (from-to)	1209-1259
Number of pages	51
Journal	Geometric and Functional Analysis
Volume	28
Issue number	5
DOIs	https://doi.org/10.1007/s00039-018-0465-8
State	Published - Oct 1 2018

Keywords

Equilibrium states
Geodesic flow
Topological pressure

ASJC Scopus subject areas

Analysis
Geometry and Topology

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title = "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 (- ∞, 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 R. For general potential functions φ, we prove that the pressure gap holds whenever 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 H{\"o}lder potentials.",

keywords = "Equilibrium states, Geodesic flow, Topological pressure",

author = "K. Burns and V. Climenhaga and T. Fisher and Thompson, {D. J.}",

note = "Funding Information: V.C. is supported by NSF Grants DMS-1362838 and DMS-1554794. T.F. is supported by Simons Foundation Grant # 239708. D.T. is supported by NSF Grant DMS-1461163. Keywords and phrases: Equilibrium states, Geodesic flow, Topological pressure Mathematics Subject Classification: 37D35 37D40 37C40 37D25 Funding Information: We would like to thank the anonymous referees for their helpful comments which have benefited this article. Much of this work was carried out in a SQuaRE program at the American Institute of Mathematics. We thank AIM for their support and hospitality. V.C. is supported by NSF Grants DMS-1362838 and DMS-1554794. T.F. is supported by Simons Foundation Grant # 239708. D.T. is supported by NSF Grant DMS-1461163.",

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doi = "10.1007/s00039-018-0465-8",

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T1 - Unique equilibrium states for geodesic flows in nonpositive curvature

AU - Burns, K.

AU - Climenhaga, V.

AU - Fisher, T.

AU - Thompson, D. J.

N1 - Funding Information: V.C. is supported by NSF Grants DMS-1362838 and DMS-1554794. T.F. is supported by Simons Foundation Grant # 239708. D.T. is supported by NSF Grant DMS-1461163. Keywords and phrases: Equilibrium states, Geodesic flow, Topological pressure Mathematics Subject Classification: 37D35 37D40 37C40 37D25 Funding Information: We would like to thank the anonymous referees for their helpful comments which have benefited this article. Much of this work was carried out in a SQuaRE program at the American Institute of Mathematics. We thank AIM for their support and hospitality. V.C. is supported by NSF Grants DMS-1362838 and DMS-1554794. T.F. is supported by Simons Foundation Grant # 239708. D.T. is supported by NSF Grant DMS-1461163.

PY - 2018/10/1

Y1 - 2018/10/1

N2 - 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 (- ∞, 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 R. For general potential functions φ, we prove that the pressure gap holds whenever 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.

AB - 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 (- ∞, 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 R. For general potential functions φ, we prove that the pressure gap holds whenever 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.

KW - Equilibrium states

KW - Geodesic flow

KW - Topological pressure

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Unique equilibrium states for geodesic flows in nonpositive curvature

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