Gibbs and equilibrium measures for elliptic functions Authors Volker Mayer Université de Lille I, UFR de Mathématiques, UMR 8524 du CNRS Mariusz Urbański Department of Mathematics University of North Texas Article

First Online: 15 April 2005 Received: 09 March 2004 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
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
The research of the second author was supported in part by the NSF Grant DMS 0400481 and INT 0306004.

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