Mathematische Zeitschrift

, Volume 250, Issue 3, pp 657–683

Gibbs and equilibrium measures for elliptic functions


DOI: 10.1007/s00209-005-0770-4

Cite this article as:
Mayer, V. & Urbański, M. Math. Z. (2005) 250: 657. doi:10.1007/s00209-005-0770-4


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):


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