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Entropy in the cusp and phase transitions for geodesic flows

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Abstract

In this paper we study the geodesic flow for a particular class of Riemannian non-compact manifolds with variable pinched negative sectional curvature. For a sequence of invariant measures we are able to prove results relating the loss of mass and bounds on the measure entropies. We compute the entropy contribution of the cusps. We develop and study the corresponding thermodynamic formalism. We obtain certain regularity results for the pressure of a class of potentials. We prove that the pressure is real analytic until it undergoes a phase transition, after which it becomes constant. Our techniques are based on the one hand on symbolic methods and Markov partitions, and on the other on geometric techniques and approximation properties at the level of groups.

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Authors and Affiliations

  1. Facultad de Matemáticas, Pontificia Universidad Católica de Chile (PUC), Avenida Vicuña Mackenna 4860, Santiago, Chile

    Godofredo Iommi

  2. Instituto de Matemáticas, Pontificia Universidad Católica de Valparaíso, Blanco Viel 596, Cerro Barón, Valparaíso, Chile

    Felipe Riquelme

  3. Princeton University, Princeton, NJ, 08544-1000, USA

    Anibal Velozo

Authors
  1. Godofredo Iommi
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  2. Felipe Riquelme
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  3. Anibal Velozo
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Corresponding author

Correspondence to Godofredo Iommi.

Additional information

G.I. was partially supported by the Center of Dynamical Systems and Related Fields código ACT1103 and by Proyecto Fondecyt 1150058.

F.R. was supported by Programa de Cooperación Cient´ıfica Internacional CONICYT-CNRS código PCCI 14009 and partially supported by Programa Postdoctorado FONDECYT 3170049.

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Iommi, G., Riquelme, F. & Velozo, A. Entropy in the cusp and phase transitions for geodesic flows. Isr. J. Math. 225, 609–659 (2018). https://doi.org/10.1007/s11856-018-1670-8

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  • DOI: https://doi.org/10.1007/s11856-018-1670-8

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