Simply Connected Wandering Domains of Small Order Entire Functions

TL;DR

This paper constructs entire functions of any order in (0,1) with bounded simply connected wandering domains, extending previous results and providing the first examples for each order in (0,1/2).

Contribution

It introduces a method to realize any bounded simply connected domain as a wandering domain of an entire function with a prescribed order in (0,1).

Findings

Constructed entire functions with wandering domains of any order in (0,1).

Provided the first examples of such functions with order in (0,1/2).

Extended previous results by Boc Thaler on wandering domains.

Abstract

We show that any bounded, simply connected domain with analytic boundary can be realised as a wandering domain of an entire function of any prescribed order in $(0, 1)$. Extending results of Boc Thaler, our construction simultaneously prescribes the domain and the exact order of the map. In particular, we produce the first examples of entire functions with bounded simply connected wandering domains of each order in $(0,1/2).$