Single and double toral band Fatou components in meromorphic dynamics

Abstract

We analyze the existence and types of unbounded Fatou components for elliptic functions and other meromorphic functions with doubly periodic Julia sets. We show that apart from Herman rings and Siegel disks, all types of dynamics can occur in these domains, which are called toral bands. We show that toral bands are not necessarily periodic, and we give results about the number of distinct residue classes of critical points in each toral band.