Memoirs of the American Mathematical Society 2009; 107 pp; softcover Volume: 203 ISBN10: 0821846590 ISBN13: 9780821846599 List Price: US$72 Individual Members: US$43.20 Institutional Members: US$57.60 Order Code: MEMO/203/954
 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  Introduction
 Balanced functions
 Transfer operator and Nevanlinna theory
 Preliminaries, Hyperbolicity and distortion properties
 PerronFrobenius operators and generalized conformal measures
 Finer properties of Gibbs states
 Regularity of PerronFrobenius operators and topological pressure
 Multifractal analysis
 Multifractal analysis of analytic families of dynamically regular functions
 Bibliography
 Index
