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