Boundary behaviour of universal covering maps

TL;DR

This work develops a boundary theory for the universal covering $\pi:\mathbb{D}\to\Omega$ of a hyperbolic multiply connected plane domain, linking radial limits, angular cluster sets, and deck-group limit sets to the boundary geometry. It constructs a prime-end framework compatible with $\pi$, using Ohtsuka exhaustions and ergodic-theoretic tools to generalize Carathéodory–Torhorst to multiply connected domains. The Main Theorem classifies boundary points into escaping, bounded, and bungee types via depth and membership in $\Lambda$ and $\Lambda_{NT}$, establishing a detailed correspondence between unit-circle data and boundary components of $\Omega$, and yielding consequences for accesses and prime ends. Applications include a correspondence between radial limits and boundary accesses, a robust prime-end theory for multiply connected domains, and explicit examples of domains with pathological limit sets, thereby extending classical simply connected theory to the infinitely connected setting. The results have potential implications for dynamics and potential theory on multiply connected domains and offer a framework for further exploration of boundary behavior under universal coverings.

Abstract

Let $Ω\subset\widehat{\mathbb{C}}$ be a multiply connected domain, and let $π\colon \mathbb{D}\toΩ$ be a universal covering map. In this paper, we analyze the boundary behaviour of $π$, describing the interplay between radial limits and angular cluster sets, the tangential and non-tangential limit sets of the deck transformation group, and the geometry and the topology of the boundary of $Ω$. As an application, we describe accesses to the boundary of $Ω$ in terms of radial limits of points in the unit circle, establishing a correspondence in the same spirit as in the simply connected case. We also develop a theory of prime ends for multiply connected domains which behaves properly under the universal covering, providing an extension of the Carathéodory--Torhorst Theorem to multiply connected domains.

Figures (17)

• Figure 2.1: Different sets related to $e^{i\theta}\in\partial\mathbb{D}$.
• Figure 2.2: Example of null-chain $\left\lbrace C_n\right\rbrace _n$ and its associated chain of crosscut neighbourhoods $\left\lbrace N_n\right\rbrace _n$ at the point $e^{i\theta}\in\partial\mathbb{D}$.
• Figure 2.3: The hyperbolic metric in the unit disk $\mathbb{D}$.
• Figure 2.4: Schematic representation of the action of a parabolic and a hyperbolic automorphism of $\mathbb{D}$, respectively.
• Figure 2.5: Hyperbolic geodesics passing through $e^{i\theta}\in\partial\mathbb{D}$, and a hyperbolic Stolz angle around one of them.

Theorems & Definitions (84)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Proposition 2.5
• Proposition 2.6
• Definition 2.7
• Lemma 2.8
• proof
• Definition 2.9