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 C0-open and dense set of Hölder potentials.
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Acknowledgements
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.
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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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Burns, K., Climenhaga, V., Fisher, T. et al. Unique equilibrium states for geodesic flows in nonpositive curvature. Geom. Funct. Anal. 28, 1209–1259 (2018). https://doi.org/10.1007/s00039-018-0465-8
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DOI: https://doi.org/10.1007/s00039-018-0465-8