Boundary behaviour of universal covering maps

Abstract

Let $\Omega \subset \hat{C}$ be a multiply connected domain, and let $\pi :D\to \Omega$ 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.