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

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.