If f\colon \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\colon \mathbb{R}^n \to \mathbb{R}^n is a uniformly quasiregular mapping, for n\geq 2 , and J(f) is a Cantor set, then every periodic point is in J(f) and is repelling.

We construct a geometrically self-similar Cantor set X of genus 2 in R 3 . This construction is the first for which the local genus is shown to be 2 at every point of X. As an application, we construct, also for the first time, a uniformly quasiregular mapping f : R 3 → R 3 for which the Julia set J(f ) is a genus 2 Cantor set.

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