Alastair N. Fletcher

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20182026

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math.DS2026

Cantor's onion: a uniformly quasiregular map whose Julia set is a Cantor set of spheres

Alastair N. Fletcher, Gaven J. Martin

For integers $d, m \geq 2$ with $1/ d + 1/ m < 1$ , we construct a uniformly quasiregular map of $S^{3}$ of degree $d^{2} + m^{2}$ whose Julia set is a Cantor set of concentric round spheres…

math.DS2026

Schwarz-Pick and Denjoy-Wolff for uniformly quasiregular maps

Alastair N. Fletcher

We prove a Denjoy-Wolff theorem for uniformly quasiregular self-maps of the unit ball in $R^{n}$ , $n \geq 2$ : either all orbits are relatively compact, or the iterates conver…

math.DS2026

Equipotentials and folds for quasiregular Mandelbrot sets

Alastair N. Fletcher

It is well-known that Douady and Hubbard proved that the Mandelbrot set is connected by constructing a uniformizing holomorphic map for its complement from the Böttcher coordinates…

math.DS2026

Sobolev mixing constants and compactness in rational moduli space

Alastair Fletcher, Ilya Krishtal

We study uniformity of Sobolev mixing estimates for rational maps and uniformly quasiregular mappings. For rational maps of fixed degree, the optimal centered Sobolev trace constan…

math.DS2026

On the ternary Cantor set and Julia sets

A. N. Fletcher

In this article, we show that the ternary Cantor set is not the Julia set of a rational map.