Boundary behaviour of universal covering maps

Let $\mathrm{\Omega }\subset \widehat{\mathbb{C}}$ be a multiply connected domain, and let $\pi \text{\hspace{0.17em}⁣}:\mathbb{D}\to \mathrm{\Omega }$ 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 $\mathrm{\Omega }$. As an application, we describe accesses to the boundary of $\mathrm{\Omega }$ 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 Carathéodory–Torhorst Theorem to multiply connected domains.