Examples of Uniformly Quasiregular Mappings with Fatou Set Having Infinitely Many Components

Gaven Martin; Kirsi Peltonen

Examples of Uniformly Quasiregular Mappings with Fatou Set Having Infinitely Many Components

Complex Variables 2026-06-26 v1

Authors: Gaven Martin , Kirsi Peltonen

Abstract

We construct examples of uniformly quasiregular mappings (uqr) acting on a sphere and having Fatou set consisting of infinitely many components. In particular we construct a uqr mapping providing a higher dimensional counterpart for the polynomial $z \mapsto z^2 - 1$ acting on the extended complex plane $\hat{\mathbb{C}}$ .

Keywords

uniformly quasiregular mappings Fatou set quasiregular dynamics complex dynamics

Cite

@article{arxiv.2606.27746,
  title  = {Examples of Uniformly Quasiregular Mappings with Fatou Set Having Infinitely Many Components},
  author = {Gaven Martin and Kirsi Peltonen},
  journal= {arXiv preprint arXiv:2606.27746},
  year   = {2026}
}