Wandering domains in quasiregular dynamics

TL;DR

This paper demonstrates the existence of wandering domains in the Fatou set of certain quasiregular mappings, contrasting with classical complex dynamics where such domains are absent for polynomials.

Contribution

It provides the first examples of wandering domains in quasiregular dynamics, including those with an essential singularity at infinity, expanding understanding beyond classical complex analysis.

Findings

Wandering domains can exist in polynomial-type quasiregular mappings.

Constructed examples with wandering domains in bounded regions.

Contrasts with classical complex dynamics where wandering domains are absent in polynomials.

Abstract

We show that wandering domains can exist in the Fatou set of a polynomial type quasiregular mapping of the plane. We also give an example of a quasiregular mapping of the plane, with an essential singularity at infinity, which has a sequence of wandering domains contained in a bounded part of the plane. This contrasts with the situation in the analytic case, where wandering domains are impossible for polynomials and, for transcendental entire functions, the existence of wandering domains in a bounded part of the plane has been an open problem for many years.