# Thermodynamical Formalism and Multifractal Analysis for Meromorphic Functions of Finite Order

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*Volker Mayer; Mariusz Urbański*

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 the authors' 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 the authors first improve upon their
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, they prove the
variational principle, they show the existence and uniqueness of Gibbs
states (with the definition appropriately adapted for the
transcendental case) and equilibrium states of such potentials, and
they 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.

#### Table of Contents

# Table of Contents

## Thermodynamical Formalism and Multifractal Analysis for Meromorphic Functions of Finite Order

- Chapter 1. Introduction 18 free
- Chapter 2. Balanced functions 512 free
- Chapter 3. Transfer operator and Nevanlinna Theory 1320
- Chapter 4. Preliminaries, Hyperbolicity and Distortion Properties 1724
- Chapter 5. Perron--Frobenius Operators and Generalized Conformal Measures 2330
- Chapter 6. Finer properties of Gibbs States 3542
- Chapter 7. Regularity of Perron-Frobenius Operators and Topological Pressure 5562
- Chapter 8. Multifractal analysis 7986
- Chapter 9. Multifractal Analysis of Analytic Families of Dynamically Regular Functions 9198
- Bibliography 103110
- Index 107114 free