Compact composition operators with symbol a universal covering map onto a multiply connected domain

We generalise previous results of the author concerning the compactness of composition operators on the Hardy spaces $H^p$, $1\leq p<\infty$, whose symbol is a universal covering map from the unit disk in the complex plane to general finitely connected domains. We demonstrate that the angular derivative criterion for univalent symbols extends to this more general case. We further show that compactness in this setting is equivalent to compactness of the composition operator induced by a univalent mapping onto the interior of the outer boundary component of the multiply connected domain