  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
$0<\kappa\leq1/e$ 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 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. We also
generalize a theorem of Ghys, Sullivan and Goldberg on the measurable dynamics
of the exponential.
