Let $E_\lambda(z)=\lambda e^z$ be the exponential family for $\lambda>0$. It is well
known and proven by Misiurewicz that for $\lambda >1/e$ the Julia set is the
entire complex plane while for $0<\lambda \le1/e$ the Julia set is a 'Cantor
bouquet'. In this talk we will discuss the dynamics of a higher
dimensional generalization of the exponential map called the Zorich map,
denoted $Z$, in $R^3$. Bergweiler has shown that for small values of $\lambda >0$ the
Julia set of $\lambda Z$ is a disjoint collection of curves. We will show that
for large values of $\lambda$ the Julia set is the entire $R^3$. We will also
discuss how other well known theorems about the exponential hold for
Zorich maps as well.