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Teichmüller spaces and normal forms associated to wandering domains

Núria Fagella, Gustavo R. Ferreira, Leticia Pardo-Simón

TL;DR

This work analyzes the dynamical Teichmüller space associated to wandering domains of transcendental entire functions, establishing a sharp dichotomy: the associated Teichmüller space is infinite-dimensional precisely when the grand orbit relation is discrete. It develops a robust normal-form theory for internal wandering-domain dynamics, yielding global non-autonomous linearising coordinates in the discrete case and revealing power-map dynamics on annuli or punctured discs in the indiscrete case. The authors prove that discrete grand-orbit components contribute infinite-dimensional Teichmüller factors, while indiscrete components yield finite-dimensional (often one-dimensional) or trivial factors, and they characterize when the wandering-domain Teichmüller space is finite-dimensional. The methods combine geometric convergence of marked surfaces and deck groups, inner-function dynamics from Riemann maps, hyperbolic-geometric tools, and non-autonomous conjugacies, providing a comprehensive picture that answers longstanding questions about wandering domains and their moduli spaces.

Abstract

We study the dynamical Teichmüller space ${\mathcal T}(U,f)$ associated to a wandering domain $U$ of an entire function $f$. We show that a discrete grand orbit relation in $U$ forces ${\mathcal T}(U,f)$ to be infinite dimensional, thereby answering a question of Fagella--Henriksen. We further describe the geometry of these spaces by developing normal forms for the dynamics on wandering domains, yielding global linearising coordinates in the discrete case and power-type dynamics between annuli in the indiscrete case.

Teichmüller spaces and normal forms associated to wandering domains

TL;DR

This work analyzes the dynamical Teichmüller space associated to wandering domains of transcendental entire functions, establishing a sharp dichotomy: the associated Teichmüller space is infinite-dimensional precisely when the grand orbit relation is discrete. It develops a robust normal-form theory for internal wandering-domain dynamics, yielding global non-autonomous linearising coordinates in the discrete case and revealing power-map dynamics on annuli or punctured discs in the indiscrete case. The authors prove that discrete grand-orbit components contribute infinite-dimensional Teichmüller factors, while indiscrete components yield finite-dimensional (often one-dimensional) or trivial factors, and they characterize when the wandering-domain Teichmüller space is finite-dimensional. The methods combine geometric convergence of marked surfaces and deck groups, inner-function dynamics from Riemann maps, hyperbolic-geometric tools, and non-autonomous conjugacies, providing a comprehensive picture that answers longstanding questions about wandering domains and their moduli spaces.

Abstract

We study the dynamical Teichmüller space associated to a wandering domain of an entire function . We show that a discrete grand orbit relation in forces to be infinite dimensional, thereby answering a question of Fagella--Henriksen. We further describe the geometry of these spaces by developing normal forms for the dynamics on wandering domains, yielding global linearising coordinates in the discrete case and power-type dynamics between annuli in the indiscrete case.

Paper Structure

This paper contains 15 sections, 26 theorems, 62 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basic notation
  4. The Teichmüller space of an entire function
  5. Wandering domains: internal dynamics and orbit relations
  6. Dimension of Teichmüller spaces: Proof of Theorem \ref{['thm:main']}
  7. Proof of Theorem \ref{['thm:main']}
  8. Linearising coordinates: proof of Theorem \ref{['thm:discrete_intro']}
  9. Complex analytical tools: distortion estimates
  10. Proof of Theorem \ref{['thm:discrete_intro']}
  11. Teichmüller spaces: Proof of Theorem \ref{['thm:teich']}
  12. Proof of Theorem \ref{['thm:teich']}
  13. Indiscrete characterisation: Proof of Theorem \ref{['thm:indiscrete_intro']}
  14. Tools from hyperbolic geometry: collars
  15. Proof of Theorem \ref{['thm:indiscrete_intro']}

Key Result

Theorem 1.1

Let $f\colon {\mathbb{C}}\to{\mathbb{C}}$ be an entire function with a wandering domain $U$, let $V$ be a connected component of $\widehat{F}(f)\cap U$, and let $\widehat{V}$ denote the grand orbit of $V$. Then $\mathcal{T}(\widehat{V},f)$ is infinite-dimensional if and only if the grand orbit relat

Theorems & Definitions (46)

  • Theorem 1.1: Discreteness if and only if infinite-dimensional Teichmüller space
  • Remark
  • Theorem 1.2: Discrete orbit relation and linearising coordinates
  • Theorem 1.3
  • Theorem 1.4: Indiscrete orbit relation and power maps
  • Corollary 1.5
  • Remark
  • Definition 2.1
  • Theorem 2.2: FH09, MS98
  • Theorem 2.3: MS98, FH09
  • ...and 36 more