The Julia set of the exponential family $E_{\kappa }:z\mapsto \kappa e^z$ , $\kappa>0$ was shown to be the entire complex plane when $\kappa>1/e$ essentially by Misiurewicz. Later, Devaney and Krych showed that for $0<\kappa \leq 1/e$ the Julia set is an uncountable union of pairwise disjoint simple curves tending to infinity. Bergweiler generalized the result of Devaney and Krych for a three-dimensional analogue of the exponential map called the Zorich map. We show that the Julia set of certain Zorich maps with symmetry is the whole of $\mathbb {R}^3$ , generalizing Misiurewicz’s result. Moreover, we show that the periodic points of the Zorich map are dense in $\mathbb {R}^3$ and that its escaping set is connected, generalizing a result of Rempe. We also generalize a theorem of Ghys, Sullivan and Goldberg on the measurable dynamics of the exponential.

Zorich maps are higher dimensional analogues of the complex exponential map. For the exponential family λe z , λ > 0, it is known that for small values of 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 ∞ 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 R 3 , unlike the corresponding two dimensional objects which are all ambiently homeomorphic.

The Julia set of the exponential family Eκ : z → κe z , κ > 0 was shown to be the entire complex plane when κ > 1/e essentially by Misiurewicz in [23]. Later in [13] Devaney and Krych showed that for 0 < κ ≤ 1/e the Julia set is an uncountable union of pairwise disjoint simple curves tending to infinity. In [3] Bergweiler generalized the result of Devaney and Krych for a three dimensional analogue of the exponential map called the Zorich map. We show that, under some mild assumptions on the construction of the Zorich map, it also has as its Julia set the entire R 3 generalizing Misiurewicz's result. Moreover, we show that the periodic points of the Zorich map are dense in R 3 and that its escaping set is connected, generalizing a result of Rempe in [27]. We also generalize a theorem of Ghys, Sullivan and Goldberg on the measurable dynamics of the exponential.

Zorich 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.