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Recurrence and transience for suspension flows

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Abstract

We study the thermodynamic formalism for suspension flows over countable Markov shifts with roof functions not necessarily bounded away from zero. We establish conditions to ensure the existence and uniqueness of equilibrium measures for regular potentials. We define the notions of recurrence and transience of a potential in this setting. We define the renewal flow, which is a symbolic model for a class of flows with diverse recurrence features. We study the corresponding thermodynamic formalism, establishing conditions for the existence of equilibrium measures and phase transitions. Applications are given to suspension flows defined over interval maps having parabolic fixed points.

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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. The School of Mathematics, The University of Bristol, University Walk, Clifton, Bristol, BS8 1TW, UK

    Thomas Jordan

  3. Mathematical Institute, University of St Andrews, North Haugh, St Andrews, KY16 9SS, Scotland

    Mike Todd

Authors
  1. Godofredo Iommi
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  2. Thomas Jordan
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  3. Mike Todd
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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 1110040.

T. J. wishes to thank Proyecto Mecesup-0711 for funding his visit to PUC-Chile.

M. T. would like to thank I. Melbourne and D. Thompson for useful comments. He is also grateful for the support of Proyecto Fondecyt 1110040 for funding his visit to PUC-Chile and for partial support from NSF grant DMS 1109587.

All three authors thank the referees for their careful reading of the paper and useful suggestions.

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Iommi, G., Jordan, T. & Todd, M. Recurrence and transience for suspension flows. Isr. J. Math. 209, 547–592 (2015). https://doi.org/10.1007/s11856-015-1229-x

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  • DOI: https://doi.org/10.1007/s11856-015-1229-x

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