Spider's webs of doughnuts

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 $g\ge 1$, 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 $n\ge 2$ and $J(f)$ is a Cantor set, then every periodic point is in $J(f)$ and is repelling.