Boundary behaviour of universal covering maps

Gustavo R. Ferreira, Anna Jové
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Abstract

Let $\Omega \subset\widehat{\mathbb{C}}$ be a multiply connected domain, and let $\pi\colon \mathbb{D}\to\Omega$ be a universal covering map. In this paper, we analyze the boundary behaviour of $\pi$, 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 $\Omega$. As an application, we describe accesses to the boundary of $\Omega$ 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 Carath\'eodory--Torhorst Theorem to multiply connected domains.
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普遍覆盖图的边界行为
让 $\Omega \subset\widehat\{mathbb{C}}$ 是一个多连域,让 $\pi\colon \mathbb{D}\to\Omega$ 是一个普遍覆盖映射。在本文中,我们分析了 $\pi$ 的边界行为,描述了径向极限与角簇集、甲板变换组的切向与非切向极限集以及 $\Omega$ 边界的几何与拓扑之间的相互作用。作为应用,我们用单位圆中点的径向极限来描述对 $\Omega$ 边界的访问,建立了与简单连接情况相同的对应关系。我们还为多连通域发展了一种在普遍覆盖下表现适当的prime ends理论,提供了Carath\'eodory--TorhorstTheorem 对多连通域的扩展。
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