Bounded Fatou and Julia components of meromorphic functions

TL;DR

This paper characterizes which bounded sets can be components of the Fatou and Julia sets of meromorphic functions, providing necessary and sufficient conditions and constructing examples using approximation theory.

Contribution

It offers a complete characterization of bounded Fatou and Julia components for meromorphic functions, linking geometric properties to dynamical components.

Findings

A bounded domain is a Fatou component iff it is regular.

A planar continuum is a Julia component iff it has empty interior.

Constructs meromorphic functions with wandering continua using approximation theory.

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.