The local range set of a meromorphic function

Abstract:

Let $f$ be a function meromorphic in a domain $G$ of the Riemann sphere. The global range set of $f$ is the set of values assumed infinitely often by $f$, and similarly the local range set of $f$ at a boundary point $p$ is the set of values assumed infinitely often in every neighborhood of $p$. Obviously any range set is a ${G_\delta }$ set. In this paper we show that every ${G_\delta }$ set is the local range set of some meromorphic function. This contrasts with the situation for the global range set. Our methods rely on prime end theory and Arakélian’s approximation theorems.