Oscillating simply connected wandering domains

TL;DR

This paper classifies and constructs examples of all six types of oscillating simply connected wandering domains in complex dynamics, using advanced approximation techniques to analyze their boundedness and dynamical properties.

Contribution

It introduces a new approximation-based method to construct and analyze all six types of oscillating wandering domains, refining previous techniques and providing detailed dynamical descriptions.

Findings

Constructed examples of all six types of oscillating wandering domains.

Proved that these wandering domains are bounded.

Analyzed the degree and behavior of mappings between wandering domains.

Abstract

Although 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.