Slow escaping points of quasiregular mappings

TL;DR

This paper demonstrates the existence of points in quasiregular mappings where iterates tend to infinity at a controlled, slow rate, extending complex dynamics results to higher dimensions and establishing new asymptotic escape rate results.

Contribution

It generalizes complex dynamics results to quasiregular maps in higher dimensions, showing slow escaping points and establishing new asymptotic escape rate theorems.

Findings

Existence of points with arbitrarily slow escape rates.

Asymptotic growth of iterates matches the iterated maximum modulus.

Extension of complex dynamics results to quasiregular mappings.

Abstract

This article concerns the iteration of quasiregular mappings on $\mathbb{R}^d$ and entire functions on $\mathbb{C}$. It is shown that there are always points at which the iterates of a quasiregular map tend to infinity at a controlled rate. Moreover, an asymptotic rate of escape result is proved that is new even for transcendental entire functions. Let $f:\mathbb{R}^d\to\mathbb{R}^d$ be quasiregular of transcendental type. Using novel methods of proof, we generalise results of Rippon and Stallard in complex dynamics to show that the Julia set of $f$ contains points at which the iterates $f^n$ tend to infinity arbitrarily slowly. We also prove that, for any large $R$, there is a point $x$ with modulus approximately $R$ such that the growth of $|f^n(x)|$ is asymptotic to the iterated maximum modulus $M^n(R,f)$.