Bounded Fatou and Julia components of meromorphic functions

Martí-Pete, David ORCID: 0000-0002-0541-8364, Rempe, Lasse ORCID: 0000-0001-8032-8580 and Waterman, James (2022) Bounded Fatou and Julia components of meromorphic functions. Mathematische Annalen.

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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.

Item Type:	Article
Additional Information:	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
Uncontrolled Keywords:	math.DS, math.DS, math.CV, 37F10 (primary), 30D05, 37B45, 54F15 (secondary)
Divisions:	Faculty of Science and Engineering > School of Physical Sciences
Depositing User:	Symplectic Admin
Date Deposited:	13 Sep 2023 08:20
Last Modified:	21 Dec 2023 08:07
Related URLs:	Author
URI:	https://livrepository.liverpool.ac.uk/id/eprint/3172703