Bounded Fatou and Julia components of meromorphic functions
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.
Keywords
Cite
@article{arxiv.2204.11781,
title = {Bounded Fatou and Julia components of meromorphic functions},
author = {David Martí-Pete and Lasse Rempe and James Waterman},
journal= {arXiv preprint arXiv:2204.11781},
year = {2024}
}
Comments
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