[Paper Review] Ergodic Theory, Geometric Measure Theory, Conformal Measures and the Dynamics of Elliptic Functions

Janina Kotus, Mariusz Urbański|arXiv (Cornell University)|Jul 26, 2020

Mathematical Dynamics and Fractals68 references4 citations

TL;DR

This paper establishes a comprehensive framework linking ergodic theory, geometric measure theory, and conformal measures to analyze the dynamics of elliptic functions. It proves that for elliptic functions in the Speiser class S, there are no wandering domains in the Julia set, resolving a long-standing conjecture via quasiconformal conjugacy and hyperbolic metric estimates.

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.

Motivation & Objective

• To establish a deep connection between ergodic theory, geometric measure theory, and conformal measures in the context of elliptic functions.
• To resolve the non-wandering conjecture for elliptic functions in the Speiser class S by proving the absence of wandering domains.
• To develop a general theory of conformal measures and invariant measures for meromorphic maps, particularly focusing on escaping sets and Julia sets.
• To extend thermodynamic formalism and pressure theory to infinite measure settings, enabling analysis of non-recurrent dynamics.
• To provide a complete topological and geometric description of Fatou and Julia sets for general meromorphic functions, especially elliptic functions.

Proposed method

• Utilizes a combination of ergodic theory, including Poincaré recurrence, Birkhoff's ergodic theorem, and the Darling–Kac theorem for infinite invariant measures.
• Applies geometric measure theory tools such as Hausdorff and packing dimensions, Frostmann's theorem, and covering theorems to analyze invariant sets.
• Employs quasiconformal conjugacies and the Riemann mapping theorem to construct conjugacies between dynamically equivalent maps.
• Uses hyperbolic (Poincaré) metrics and distortion estimates in simply connected domains to control the behavior of iterated maps near the boundary.
• Applies the theory of graph directed Markov systems (GDMS) and conformal GDMS to model the dynamics of elliptic functions.
• Employs the concept of conformal measures and their relation to Hausdorff dimension via Bowen’s formula and the variational principle.

Experimental results

Research questions

• RQ1 Do elliptic functions in the Speiser class S admit wandering domains in their Julia sets?
• RQ2 What is the Hausdorff dimension of the Julia set and escaping set of a general elliptic function?
• RQ3 How do conformal measures relate to the geometry and dynamics of the Julia set for elliptic functions?
• RQ4 What is the structure of Fatou components, especially Baker domains and Leau–Fatou petals, for meromorphic functions?
• RQ5 Can the dynamics of elliptic functions be fully described using thermodynamic formalism and conformal measures in infinite measure settings?

Key findings

• The paper proves that for elliptic functions in the Speiser class S, there are no wandering domains in the Julia set, confirming a key conjecture in complex dynamics.
• It establishes that the Julia set of a compactly non-recurrent elliptic function has Hausdorff dimension strictly greater than 1, and in some cases equals 2.
• The escaping set of a general elliptic function has Hausdorff dimension 2, indicating full geometric complexity.
• Conformal measures supported on the escaping set are shown to exist and are related to the pressure and dimension of the system via Bowen’s formula.
• The authors construct a quasiconformal conjugacy between dynamically equivalent maps, which extends continuously to the boundary and is the identity on the Julia set.
• The proof relies on a contradiction argument: assuming a wandering domain leads to a non-trivial quasiconformal map on the disk that fixes the boundary and is not identity, contradicting rigidity theorems in complex dynamics.