eBook ISBN:  9781470405687 
Product Code:  MEMO/203/954.E 
List Price:  $72.00 
MAA Member Price:  $64.80 
AMS Member Price:  $43.20 
eBook ISBN:  9781470405687 
Product Code:  MEMO/203/954.E 
List Price:  $72.00 
MAA Member Price:  $64.80 
AMS Member Price:  $43.20 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 203; 2009; 107 ppMSC: Primary 30; 37
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 \(t\logf'_\sigma\). 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, IonescuTulcea and Marinescu Inequality) of PerronFrobenius operators, and their stochastic consequences such as the Central Limit Theorem, Kmixing, and exponential decay of correlations.

Table of Contents

Chapters

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 PerronFrobenius Operators and Topological Pressure

8. Multifractal analysis

9. Multifractal Analysis of Analytic Families of Dynamically Regular Functions


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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 \(t\logf'_\sigma\). 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, IonescuTulcea and Marinescu Inequality) of PerronFrobenius operators, and their stochastic consequences such as the Central Limit Theorem, Kmixing, and exponential decay of correlations.

Chapters

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 PerronFrobenius Operators and Topological Pressure

8. Multifractal analysis

9. Multifractal Analysis of Analytic Families of Dynamically Regular Functions