Oscillating simply connected wandering domains

Vasiliki Evdoridou,Philip J Rippon,Gwyneth M Stallard

doi:10.1017/etds.2021.169

Abstract

AbstractAlthough detailed descriptions of the possible types of behaviour inside periodic Fatou components have been known for over 100 years, a classification of wandering domains has only recently been given. Recently, simply connected wandering domains were classified into nine possible types and examples of escaping wandering domains of each of these types were constructed. Here we consider the case of oscillating wandering domains, for which only six of these types are possible. We use a new technique based on approximation theory to construct examples of all six types of oscillating simply connected wandering domains. This requires delicate arguments since oscillating wandering domains return infinitely often to a bounded part of the plane. Our technique is inspired by that used by Eremenko and Lyubich to construct the first example of an oscillating wandering domain, but with considerable refinements which enable us to show that the wandering domains are bounded, to specify the degree of the mappings between wandering domains and to give precise descriptions of the dynamical behaviour of these mappings.

Highlights

Let f be a transcendental entire function
We consider the iterates of f, which we denote by f n, n ≥ 1
The complex plane is divided into two sets: the Fatou set, F (f ), where the iterates (f n) form a normal family in a neighbourhood of every point, and its complement, the Julia set J (f )

Summary

Introduction

Let f be a transcendental entire function. We consider the iterates of f, which we denote by f n, n ≥ 1. In terms of convergence to the boundary, the orbits of all points stay away from the boundary, come arbitrarily close to the boundary but do not converge to it (bungee), or converge to the boundary (see Theorem 5.2) These two classifications give nine possible types of connected wandering domains. In order to obtain the transcendental entire function with the required properties, we consider an analytic function which is our model function and apply the following result which is an extension of the well-known Runge approximation theorem and was the Main Lemma in [EL87].

For n

We consider the function