Mathematische Zeitschrift

, Volume 250, Issue 3, pp 657–683

Gibbs and equilibrium measures for elliptic functions

Article

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.

Mathematics Subject Classification (2000):

30D05 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  1. 1.Université de Lille I, UFR de Mathématiques, UMR 8524 du CNRSVilleneuve d’Ascq CedexFrance
  2. 2.Department of MathematicsUniversity of North TexasDentonUSA

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