Explosion Points and Topology of Julia Sets of Zorich Maps

Abstract

Zorich maps are higher dimensional analogues of the complex exponential map. For the exponential family \(\lambda e^z\), \(\lambda >0\), it is known that for small values of \(\lambda \) the Julia set is an uncountable collection of disjoint curves. The same was shown to hold for Zorich maps by Bergweiler and Nicks. In this paper we introduce a topological model for the Julia sets of certain Zorich maps, similar to the so called straight brush of Aarts and Oversteegen. As a corollary we show that \(\infty \) is an explosion point for the set of endpoints of the Julia sets. Moreover we introduce an object called a hairy surface which is a compactified version of the Julia set of Zorich maps and we show that those objects are not uniquely embedded in \(\mathbb {R}^3\), unlike the corresponding two dimensional objects which are all ambiently homeomorphic.