Bounded Fatou and Julia components of meromorphic functions
arXiv:2204.11781 · doi:10.1007/s00208-023-02725-4
Abstract
We completely characterise the bounded sets that arise as components of the Fatou and Julia sets of meromorphic functions. On the one hand, we prove that a bounded domain is a Fatou component of some meromorphic function if and only if it is regular. On the other hand, we prove that a planar continuum is a Julia component of some meromorphic function if and only if it has empty interior. We do so by constructing meromorphic functions with wandering continua using approximation theory.
15 pages, 4 figures. V2: We have revised the introduction, and introduced two new sections: Section 2 discusses and compare topological properties of Fatou components, while Section 3 establishes that certain bounded regular domains cannot arise as eventually periodic Fatou components of meromorphic functions