Connectivity of Fatou Components of Meromorphic Functions

TL;DR

This paper demonstrates the existence of transcendental meromorphic functions with various types of Fatou components, including doubly connected periodic components and wandering domains with specific connectivity properties, advancing understanding of complex dynamics.

Contribution

It establishes the existence of transcendental meromorphic functions with doubly connected periodic Fatou components and wandering domains with periodic connectivity, resolving longstanding open problems.

Findings

Existence of doubly connected periodic Fatou components that are attracting, parabolic, or Baker domains.

Existence of wandering domains with no eventual connectivity.

Periodic connectivity sequence of wandering domains with period two.

Abstract

In this paper, we show that there exist transcendental meromorphic functions with a cycle of 2-periodic Fatou components, where one is simply connected while the other is doubly connected. In particular, the doubly connected Fatou component can be an attracting, parabolic, or Baker domain. Thus, this settles the problem of whether a doubly connected periodic Fatou component must be a Herman ring. We also prove that there exists a transcendental meromorphic function with a wandering domain that has no eventual connectivity. In addition, we show that the connectivity sequence of this wandering domain is periodic with period two. This solves a problem about the nonexistence of the eventual connectivity of wandering domains and gives a different example constructed by Ferreira [J. London Math. Soc. (2022), DOI:10.1112/jlms.12613].