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If is a uniformly quasiregular mapping with Julia set , a genus Cantor set, for , then for any linearizer at any repelling periodic point of , the fast escaping set consists of a spiders' web structure containing embedded genus tori on any sufficiently large scale. In other words, 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 is a uniformly quasiregular mapping, for , and is a Cantor set, then every periodic point is in and is repelling.
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Alastair Fletcher, Daniel Stoertz, Spiders’ webs of doughnuts. Rev. Mat. Iberoam. 37 (2021), no. 1, pp. 161–176DOI 10.4171/RMI/1204