2018/07/18 by Alastair Fletcher, Fletcher, A., Daniel Stoertz +1 · 1 citation
Mathematics · #Analytic and geometric function theory #Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Meromorphic and Entire Functions
paper · pdf · doi:10.48550/arxiv.1807.07166
openalex publication_date 2018/07/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
If f:ℝ3 → ℝ3 is a uniformly quasiregular mapping with Julia set J(f) a genus g Cantor set, for g≥ 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:ℝn → ℝn is uqr, for n≥ 2 and J(f) is a Cantor set, then every periodic point is in J(f) and is repelling.