Boundary behaviour of universal covering maps

Abstract

Let Ω⊂Cˆ be a multiply connected domain, and let π:D→Ω 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.