Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Boundaries of multiply connected Fatou components. A unified approach

Gustavo R. Ferreira, Anna Jové

TL;DR

This work develops a unified framework to study boundaries of multiply connected Fatou components for transcendental maps by coupling universal covering maps with associated inner functions. It proves that every Fatou component admits a harmonic measure whose support is the whole boundary and uses radial limits to relate the limit sets of deck groups, boundary dynamics, and analytic properties of the inner function. The authors extend the approach to rational maps and maps in class $\\mathbb{K}$, including invariant basins and wandering domains, and they construct pathological boundary behaviors to illustrate the limits of the inner-function correspondence. The results yield precise ergodic and exactness conclusions for boundary maps in many cases, while also showing striking exceptions in the presence of essential singularities, thereby illuminating the boundary dynamics of complex dynamical systems with rich topologies.

Abstract

We analyze the boundaries of multiply connected Fatou components of transcendental maps by means of universal covering maps and associated inner functions. A unified approach is presented, which includes invariant Fatou components (of any type) as well as wandering domains. We prove that any Fatou component admits a harmonic measure on its boundary whose support is the whole boundary. Consequently, we relate, in a successful way, the geometric structure of such Fatou components (in terms of the limit sets of their universal covering maps), the dynamics induced on their boundary from an ergodic point of view, and analytic properties of the associated inner function.

Boundaries of multiply connected Fatou components. A unified approach

TL;DR

This work develops a unified framework to study boundaries of multiply connected Fatou components for transcendental maps by coupling universal covering maps with associated inner functions. It proves that every Fatou component admits a harmonic measure whose support is the whole boundary and uses radial limits to relate the limit sets of deck groups, boundary dynamics, and analytic properties of the inner function. The authors extend the approach to rational maps and maps in class , including invariant basins and wandering domains, and they construct pathological boundary behaviors to illustrate the limits of the inner-function correspondence. The results yield precise ergodic and exactness conclusions for boundary maps in many cases, while also showing striking exceptions in the presence of essential singularities, thereby illuminating the boundary dynamics of complex dynamical systems with rich topologies.

Abstract

We analyze the boundaries of multiply connected Fatou components of transcendental maps by means of universal covering maps and associated inner functions. A unified approach is presented, which includes invariant Fatou components (of any type) as well as wandering domains. We prove that any Fatou component admits a harmonic measure on its boundary whose support is the whole boundary. Consequently, we relate, in a successful way, the geometric structure of such Fatou components (in terms of the limit sets of their universal covering maps), the dynamics induced on their boundary from an ergodic point of view, and analytic properties of the associated inner function.

Paper Structure

This paper contains 19 sections, 27 theorems, 38 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Iteration in class $\mathbb{K}$ and associated inner functions
  3. Basic results on iteration
  4. Harmonic measure with respect to a Fatou component. Theorem \ref{['thmA']}
  5. The associated inner function and its radial extension
  6. The main construction: relating inner functions, limit sets and universal covering maps
  7. Background results. Boundary behaviour of universal covering maps
  8. Singular values
  9. Mapping properties of the associated inner function
  10. Invariance of limit sets in the sense of measure
  11. Multiply connected basins of rational maps. Theorem \ref{['thmB']}
  12. Inner functions associated with invariant Fatou components
  13. Proof of Theorem \ref{['thmB']}
  14. Transcendental functions with pathological boundaries
  15. Herman rings
  16. ...and 4 more sections

Key Result

Theorem 1

(Radial extension well-defined) Let $f\in\mathbb{K}$, and let $U$ be a Fatou component of $f$. Then, given a universal covering $\pi\colon \mathbb{D}\to U$, the radial extension is well-defined $\lambda$-almost everywhere, where $\lambda$ denotes the normalized Lebesgue measure on $\partial\mathbb{D}$. Equivalently, $U$ admits a harmonic measure $\omega_U$. Moreover, the support of $\omega_U$ is

Figures (5)

  • Figure 3.2: Sketch of the construction in the claim of Lemma \ref{['lemma-lifts-and-limit-sets-proper']}.
  • Figure 4.1: Sketch of the construction in Lemma \ref{['lem:fp']}. Following the notation of the Lemma, $z\in U$ represents the fixed point in $U$, and the blue disk around it, its absorbing domain. In the unit disk, the lift of such absorbing domain, accumulating at the Denjoy-Wolff point $p\in\partial\mathbb{D}$.
  • Figure 5.1: Sketch of a Herman ring and its universal covering map, giving rise to its associated inner function, for some $\theta\in \mathbb{R}\smallsetminus\mathbb{Q}$.
  • Figure 5.2: Skecht of a doubly connected wandering domain of an entire function, and its associated inner function.
  • Figure 5.3:

Theorems & Definitions (56)

  • Theorem 1
  • Remark
  • Theorem 2
  • Theorem 3
  • Remark
  • Theorem 4
  • Lemma 2.1
  • Theorem 2.2
  • Lemma 2.3
  • Theorem 2.4
  • ...and 46 more