Explosion Points and Topology of Julia Sets of Zorich Maps

Abstract

AbstractZorich maps are higher dimensional analogues of the complex exponential map. For the exponential family $$\lambda e^z$$ λ e z , $$\lambda >0$$ λ > 0 , it is known that for small values of $$\lambda $$ λ 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 $$\mathbb {R}^3$$ R 3 , unlike the corresponding two dimensional objects which are all ambiently homeomorphic.