  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 in \cite{Misiu}. Later in \cite{Devaney1984} Devaney
and Krych showed that for $0<\kappa\leq1/e$ the Julia set is an uncountable
union of pairwise disjoint simple curves tending to infinity. In \cite{bergk}
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 $\mathbb{R}^3$ 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 in \cite{REMPE2010}. We also generalize a theorem of Ghys, Sullivan and
Goldberg on the measurable dynamics of the exponential.
