Ergodic Theory, Geometric Measure Theory, Conformal Measures and the Dynamics of Elliptic Functions

Abstract

The ultimate goal of our book is to present a unified approach to the dynamics, ergodic theory, and geometry of elliptic functions from $\C$ to $\oc$. We consider elliptic functions as a most regular class of transcendental meromorphic functions. Poles form an essential feature of such functions but the set of critical values is finite and an elliptic function is "the same" on its of its fundamental regions. In a sense this is the class of transcendental meromorphic functions which resembles rational functions most. On the other hand, the differences are huge. We will touch on them in the course of this introduction. In order to comprehensively cover the dynamics and geometry of elliptic functions we make large preparations. This is done in the first two parts of the book: Part 1, "Ergodic Theory and Measures" and Part 2,"Geometry and Conformal Measures". We intend our book to be as self contained as possible and we use essentially all major results of Part~1 and Part~2 in Part~3 and Part~4 dealing with elliptic functions. This book can be thus treated as a fairly comprehensive account of dynamics, ergodic theory, and fractal geometry of elliptic functions but also as a reference book (with proofs) for many results of geometric measure theory, finite and infinite abstract ergodic theory, Young's towers, measure--theoretic Kolmogorov--Sinai entropy, thermodynamic formalism, geometric function theory (in particular Koebe's Distortion Theorems and Riemann--Hurwitz Formulas), various kinds of conformal measures, conformal graph Directed Markov systems and iterated function systems, classical general theory of elliptic functions, and topological dynamics of transcendental meromorphic functions.