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

Abstract

Link to this article: http://journals.cambridge.org/abstract_S0143385707000648How to cite this article: VOLKER MAYER and MARIUSZ URBAŃSKI (2008). Geometric thermodynamic formalism and real analyticity for meromorphic functions of nite order. Ergodic Theory and Dynamical Systems, 28, 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. 916V. Mayer and M. Urbańskiis the derivative of f with respect to the metric dσ . With this tool in hand we obtain then geometric information about the Julia set J ( f ) and about the radial (or conical) Julia setWe now give a precise description of our results. 1.1. Thermodynamic formalism. Various versions of the thermodynamic formalism and finer fractal geometry of transcendental entire and meromorphic functions have been explored since the middle of the 1990s, and have speeded up since the year 2000 (see for example [Ba, CS1, CS2, KU1, KU2, KU3, MyU, UZ1, UZ2, UZ3], and especially the survey article [KU4]touching on most of the results obtained so far). Some interesting and important classes of functions, including exponential λe z and elliptic, are now fairly well understood. Essentially all of them were periodic; the methods used to deal with them broke down in the lack of periodicity, and required the dynamics to be projected down onto the appropriate quotient space, either torus or infinite cylinder. Workers have actually never completely gone back to the original phase space, the complex plane C. A nice exception is the case of critically non-recurrent elliptic functions treated in [KU2], where the special but most important potential −HD(J ( f )) log | f | was explored in detail. In this paper we propose an entirely different approach. We do not need periodicity and we work on the complex plane itself. The main idea, which among others allows us to abandon periodicity, is that we associate to a given meromorphic function f a Riemannian conformal metric dσ = γ |dz| with respect to which the Perron-Frobenius-Ruelle (or transfer) operatoris well defined and has all the required properties that make the thermodynamic formalism work. Such a good metric can be found for meromorphic functions f : C →Ĉ that are of finite order ρ and satisfy the following growth condition for the derivative.Rapid growth: There are α 2 > max{0, −α 1 } a...

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