The size and topology of quasi-Fatou components of quasiregular maps

We consider the iteration of quasiregular maps of transcendental type from ${\mathbb{R}}^d$ to ${\mathbb{R}}^d$. In particular we study quasi-Fatou components, which are defined as the connected components of the complement of the Julia set. Many authors have studied the components of the Fatou set of a transcendental entire function, and our goal in this paper is to generalise some of these results to quasi-Fatou components. First, we study the number of complementary components of quasi-Fatou components, generalising, and slightly strengthening, a result of Kisaka and Shishikura. Second, we study the size of quasi-Fatou components that are bounded and have a bounded complementary component. We obtain results analogous to those of Zheng, and of Bergweiler, Rippon and Stallard. These are obtained using novel techniques, and may be of interest even in the case of transcendental entire functions.