JournalsrmiVol. 37, No. 1pp. 161–176

Spiders’ webs of doughnuts

  • Alastair Fletcher

    Northern Illinois University, DeKalb, USA
  • Daniel Stoertz

    Northern Illinois University, DeKalb, USA
Spiders’ webs of doughnuts cover
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Abstract

If f ⁣:R3R3f\colon \mathbb{R}^3 \to \mathbb{R}^3 is a uniformly quasiregular mapping with Julia set J(f)J(f), a genus gg Cantor set, for g1g\geq 1, then for any linearizer LL at any repelling periodic point of ff, the fast escaping set A(L)A(L) consists of a spiders' web structure containing embedded genus gg tori on any sufficiently large scale. In other words, A(L)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 ⁣:RnRnf\colon \mathbb{R}^n \to \mathbb{R}^n is a uniformly quasiregular mapping, for n2n\geq 2, and J(f)J(f) is a Cantor set, then every periodic point is in J(f)J(f) and is repelling.

Cite this article

Alastair Fletcher, Daniel Stoertz, Spiders’ webs of doughnuts. Rev. Mat. Iberoam. 37 (2021), no. 1, pp. 161–176

DOI 10.4171/RMI/1204