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Thermodynamical formalism and multifractal analysis for meromorphic functions of finite order

About this Title

Volker Mayer, Université de Lille I, UFR de Mathématiques, UMR 8524 du CNRS, 59655 Villeneuve d’Ascq Cedex, France and Mariusz Urbański, Department of Mathematics, University of North Texas, Denton, Texas 76203-1430

Publication: Memoirs of the American Mathematical Society
Publication Year: 2010; Volume 203, Number 954
ISBNs: 978-0-8218-4659-9 (print); 978-1-4704-0568-7 (online)
Published electronically: August 26, 2009
Keywords: Holomorphic dynamics, thermodynamical formalism, transcendental functions, fractal geometry
MSC: Primary 30D05, 37F10

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Table of Contents


  • 1. Introduction
  • 2. Balanced functions
  • 3. Transfer operator and Nevanlinna Theory
  • 4. Preliminaries, Hyperbolicity and Distortion Properties
  • 5. Perron–Frobenius Operators and Generalized Conformal Measures
  • 6. Finer properties of Gibbs States
  • 7. Regularity of Perron-Frobenius Operators and Topological Pressure
  • 8. Multifractal analysis
  • 9. Multifractal Analysis of Analytic Families of Dynamically Regular Functions


The thermodynamical formalism has been developed by the authors for a very general class of transcendental meromorphic functions. A function $f:\mathbb {C}\to \hat {{\mathbb C}}$ of this class is called dynamically (semi-) regular. The key point in our earlier paper (2008) was that one worked with a well chosen Riemannian metric space $(\hat {{\mathbb C}} , \sigma )$ and that the Nevanlinna theory was employed.

In the present manuscript we first improve upon our earlier paper in providing a systematic account of the thermodynamical formalism for such a meromorphic function $f$ and all potentials that are Hölder perturbations of $-t\log |f’|_\sigma$. 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 non-escaping) 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 our earlier paper 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 $z\mapsto z^2+c$. As a by-product of our considerations we obtain real analyticity of the Hausdorff dimension function.

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