Boundary behaviour of universal covering maps

Let C be a multiply connected domain, and let D be a universal covering map. In this paper, we analyze the boundary behaviour of, 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. As an application, we describe accesses to the boundary of 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.