Karpi\'nska's paradox in dimension three

TL;DR

This paper explores a three-dimensional analogue of Karpińska's paradox, demonstrating that certain properties of Julia sets in complex dynamics extend to quasiregular maps in three-dimensional space.

Contribution

It introduces a three-dimensional version of Karpińska's paradox using a quasiregular map, extending known complex dynamics results to higher dimensions.

Findings

Julia set properties extend to 3D quasiregular maps

Hausdorff dimension results are analogous in 3D

Disjoint curves tend to infinity in the 3D setting

Abstract

For 0 < c < 1/e the Julia set of f(z) = c exp(z) is an uncountable union of pairwise disjoint simple curves tending to infinity [Devaney and Krych 1984], the Hausdorff dimension of this set is two [McMullen 1987], but the set of curves without endpoints has Hausdorff dimension one [Karpinska 1999]. We show that these results have three-dimensional analogues when the exponential function is replaced by a quasiregular self-map of three-space introduced by Zorich.