The fast escaping set for quasiregular mappings

TL;DR

This paper extends the concept of the fast escaping set from transcendental entire functions to quasiregular mappings in higher dimensions, demonstrating equivalences and structural properties such as spider's web formations.

Contribution

It introduces the fast escaping set for quasiregular mappings, proves the equivalence of various definitions, and constructs examples with a spider's web structure.

Findings

Various definitions of the fast escaping set coincide for quasiregular mappings.

The fast escaping set can have a spider's web structure in higher dimensions.

The study generalizes known planar results to higher-dimensional quasiregular mappings.

Abstract

The fast escaping set of a transcendental entire function is the set of all points which tend to infinity under iteration as fast as compatible with the growth of the function. We study the analogous set for quasiregular mappings in higher dimensions and show, among other things, that various equivalent definitions of the fast escaping set for transcendental entire functions in the plane also coincide for quasiregular mappings. We also exhibit a class of quasiregular mappings for which the fast escaping set has the structure of a spider's web.