Geometric Thermodynamical Formalism and Real Analyticity for Meromorphic Functions of Finite Order
Abstract
Working with well chosen Riemannian metrics and employing Nevanlinna's theory, we make the thermodynamical formalism work for a wide class of hyperbolic meromorphic functions of finite order (including in particular exponential family, elliptic functions, cosine, tangent and the cosineroot 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.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 June 2006
 arXiv:
 arXiv:math/0606294
 Bibcode:
 2006math......6294M
 Keywords:

 Mathematics  Dynamical Systems;
 30D05
 EPrint:
 32 pages