Karpińska's paradox in dimension 3

Abstract

It was proved by Devaney and Krych, by McMullen, and by Karpińska that, for $0<\lambda <1/e$, the Julia set of $\lambda e^z$ is an uncountable union of pairwise disjoint simple curves tending to infinity, and the Hausdorff dimension of this set is $2$, but the set of curves without endpoints has Hausdorff dimension $1$. We show that these results have $3$-dimensional analogues when the exponential function is replaced by a quasi-regular self-map of ${\mathbb{R}}^3$ introduced by Zorich.