Spider's webs of doughnuts

TL;DR

This paper explores the complex structure of fast escaping sets in higher-dimensional quasiregular mappings, revealing a unique 'doughnut web' pattern that differs from planar cases and analyzing the nature of periodic points.

Contribution

It demonstrates that in higher dimensions, the fast escaping set forms a spiders' web of doughnuts and characterizes the nature of periodic points in such mappings.

Findings

Fast escaping sets form a spiders' web of doughnuts in higher dimensions.

Periodic points in these mappings are all in the Julia set and are repelling.

The structure is specific to higher dimensions and does not occur in planar transcendental functions.

Abstract

If $f:\mathbb{R}^3 \to \mathbb{R}^3$ is a uniformly quasiregular mapping with Julia set $J(f)$ a genus $g$ Cantor set, for $g\geq 1$, then for any linearizer $L$ at any repelling periodic point of $f$, the fast escaping set $A(L)$ consists of a spiders' web structure containing embedded genus $g$ tori on any sufficiently large scale. In other words, $A(L)$ contains a spiders' web of doughnuts. This type of structure is specific to higher dimensions, and cannot happen for the fast escaping set of a transcendental entire function in the plane. We also show that if $f:\mathbb{R}^n \to \mathbb{R}^n$ is uqr, for $n\geq 2$ and $J(f)$ is a Cantor set, then every periodic point is in $J(f)$ and is repelling.