Normal families of covering maps

Abstract

Suppose that (fn)n∈N is a sequence of meromorphic covering maps which is uniformly convergent in a neighbourhood of a pointx∈Ĉ such that the limit function is non-constant. It is proved that the convergence extends to the largest domain where the sequence eventually is defined and that the limit function again is a covering map. As a consequence of this result, we obtain a rescaling lemma for holomorphic covering maps, a version of the Caratheodory Kernel Theorem for arbitrary domains in the sphere, and an elementary access to the Riemann Uniformization Theorem for arbitrary domains in the sphere. An application to complex dynamics of transcendental entire functions provides that the existence of an invariant Baker domain implies a certain frequency of singularities of the inverse function.