Boundary behaviour of universal covering maps

Abstract

Let $\Omega \subset \hat{\mathbb{C}}$ be a multiply connected domain, and let $\pi :\mathbb{D}\to \Omega$ 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 Carathéodory--Torhorst Theorem to multiply connected domains.

49 pages, 17 figures

References in corpus (1)

• Boundary dynamics in unbounded Fatou components