In this paper we introduce measurable expanding random systems, develop the thermodynamical formalism and establish, in particular, exponential decay of correlations and analyticity of the expected pressure although the spectral gap property does not hold. This theory is then used to investigate fractal properties of conformal random systems. We prove a Bowen's formula and develop the multifractal formalism of the Gibbs states. Depending on the behavior of the Birkhoff sums of the pressure function we get a natural classifications of the systems into two classes: quasi-deterministic systems which share many properties of deterministic ones and essential random systems which are rather generic and never bilipschitz equivalent to deterministic systems. We show in the essential case that the Hausdorff measure vanishes which refutes a conjecture of Bogenschütz and Ochs. We finally give applications of our results to various specific conformal random systems and positively answer a question of Brück and Bürger concerning the Hausdorff dimension of random Julia sets.

The thermodynamical formalism has been developed in [MyU2] for a very general class of transcendental meromorphic functions. A function f : C →Ĉ of this class is called dynamically (semi-) regular. The key point in [MyU2] was that one worked with a well chosen Riemannian metric space (Ĉ, σ) and that the Nevanlinna theory was employed.In the present manuscript we first improve [MyU2] in providing a systematic account of the thermodynamical formalism for such a meromorphic function f and all potentials that are Hölder perturbations of −t log |f ′ |σ. In this general setting, we prove the variational principle, we show the existence and uniqueness of Gibbs states (with the definition appropriately adapted for the transcendental case) and equilibrium states of such potentials, and we demonstrate that they coincide. There is also given a detailed description of spectral and asymptotic properties (spectral gap, Ionescu-Tulcea and Marinescu Inequality) of Perron-Frobenius operators, and their stochastic consequences such as the Central Limit Theorem, K-mixing, and exponential decay of correlations.Then we provide various, mainly geometric, applications of this theory. Indeed, we examine the finer fractal structure of the radial (in fact nonescaping) Julia set by developing the multifractal analysis of Gibbs states. In particular, the Bowen's formula for the Hausdorff dimension of the radial Julia set from [MyU2] is reproved. Moreover, the multifractal spectrum function is proved to be convex, real-analytic and to be the Legendre transform conjugate to the temperature function. In the last chapter we went even further by showing that, for a analytic family satisfying a symmetric version of the growth condition (1.1) in a uniform way, the multifractal spectrum function is real-analytic also with respect to the parameter. Such a fact, up to our knowledge, has not been so far proved even for hyperbolic rational functions nor even for the quadratic family z → z 2 + c. As a by-product of our considerations we obtain real analyticity of the Hausdorff dimension function.

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...

We investigate local dynamics of uniformly quasiregular mappings, give new examples and show in particular that there is no quasiconformal analogue of the Leau-Fatou linearization of parabolic dynamics.