Ergodic Theory and Dynamical Systems

Research Article

Geometric thermodynamic formalism and real analyticity for meromorphic functions of finite order

VOLKER MAYERa1 and MARIUSZ URBAŃSKIa2

a1 Université de Lille I, UFR de Mathématiques, UMR 8524 du CNRS, 59655 Villeneuve d’Ascq Cedex, France (email: volker.mayer@math.univ-lille1.fr)

a2 Department of Mathematics, University of North Texas, Denton, TX 76203-1430, USA (email: urbanski@unt.edu)

Abstract

Working with well chosen Riemannian metrics and employing Nevanlinna’s theory, we make the thermodynamic formalism work for a wide class of hyperbolic meromorphic functions of finite order (including in particular exponential family, elliptic functions, cosine, tangent and the cosine–root family and also compositions of these functions with arbitrary polynomials). In particular, the existence of conformal (Gibbs) measures is established and then the existence of probability invariant measures equivalent to conformal measures is proven. As a geometric consequence of the developed thermodynamic formalism, a version of Bowen’s formula expressing the Hausdorff dimension of the radial Julia set as the zero of the pressure function and, moreover, the real analyticity of this dimension, is proved.

(Received January 06 2007)

(Revised July 06 2007)