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Thermodynamical Formalism and Multifractal Analysis for Meromorphic Functions of Finite Order

Volker Mayer Université de Lille I, Villeneuve d’Ascq, France
Mariusz Urbański University of North Texas, Denton, TX
Available Formats:
Electronic ISBN: 978-1-4704-0568-7
Product Code: MEMO/203/954.E
107 pp
List Price: $72.00 MAA Member Price:$64.80
AMS Member Price: $43.20 Click above image for expanded view Thermodynamical Formalism and Multifractal Analysis for Meromorphic Functions of Finite Order Volker Mayer Université de Lille I, Villeneuve d’Ascq, France Mariusz Urbański University of North Texas, Denton, TX Available Formats:  Electronic ISBN: 978-1-4704-0568-7 Product Code: MEMO/203/954.E 107 pp  List Price:$72.00 MAA Member Price: $64.80 AMS Member Price:$43.20
• Book Details

Memoirs of the American Mathematical Society
Volume: 2032009
MSC: 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 $-t\log|f'|_\sigma$. 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.

• 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 Perron-Frobenius Operators and Topological Pressure
• 8. Multifractal analysis
• 9. Multifractal Analysis of Analytic Families of Dynamically Regular Functions
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Volume: 2032009
MSC: 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 $-t\log|f'|_\sigma$. 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.

• 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 Perron-Frobenius Operators and Topological Pressure
• 8. Multifractal analysis
• 9. Multifractal Analysis of Analytic Families of Dynamically Regular Functions
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