Wandering domains for entire functions of finite order in the Eremenko-Lyubich class

TL;DR

This paper constructs the first finite-order entire functions in the Eremenko-Lyubich class with wandering domains, using quasiregular interpolation to extend Bishop's infinite-order examples.

Contribution

It introduces a new construction method for finite-order functions with wandering domains in class , achieving the minimal possible order and allowing multiple wandering domain orbits.

Findings

Constructed finite-order entire functions with wandering domains.

Achieved the smallest possible order in class B.

Allowed for multiple or infinitely many wandering domain orbits.

Abstract

Recently Bishop constructed the first example of a bounded-type transcendental entire function with a wandering domain using a new technique called quasiconfomal folding. It is easy to check that his method produces an entire function of infinite order. We construct the first examples of entire functions of finite order in the class $\mathcal B$ with wandering domains. As in Bishop's example, these wandering domains are of oscillating type, that is, they have an unbounded non-escaping orbit. To construct such functions we use quasiregular interpolation instead of quasiconformal folding, which is much more straightforward. Our examples have order $p/2$ for any $p\in\mathbb{N}$ and, since the order of functions in the class $\mathcal B$ is at least $1/2$, we achieve the smallest possible order. Finally, we can modify the construction to obtain functions of finite order in the class $\mathcal B$ with any number of grand orbits of wandering domains, including infinitely many.