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Gibbs and equilibrium measures for elliptic functions

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Abstract.

Because of its double periodicity, each elliptic function canonically induces a holomorphic dynamical system on a punctured torus. We introduce on this torus a class of summable potentials. With each such potential associated is the corresponding transfer (Perron-Frobenius-Ruelle) operator. The existence and uniquenss of “Gibbs states” and equilibrium states of these potentials are proved. This is done by a careful analysis of the transfer operator which requires a good control of all inverse branches. As an application a version of Bowen’s formula for expanding elliptic maps is obtained.

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Authors and Affiliations

  1. Université de Lille I, UFR de Mathématiques, UMR 8524 du CNRS, 59655, Villeneuve d’Ascq Cedex, France

    Volker Mayer

  2. Department of Mathematics, University of North Texas, P.O. Box 311430, Denton, TX, 76203-1430, USA

    Mariusz Urbański

Authors
  1. Volker Mayer
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  2. Mariusz Urbański
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Correspondence to Mariusz Urbański.

Additional information

The research of the second author was supported in part by the NSF Grant DMS 0400481 and INT 0306004.

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Mayer, V., Urbański, M. Gibbs and equilibrium measures for elliptic functions. Math. Z. 250, 657–683 (2005). https://doi.org/10.1007/s00209-005-0770-4

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  • DOI: https://doi.org/10.1007/s00209-005-0770-4

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