  Let $g$ be a polynomial automorphism of $\C^2$. We study the Hausdorff
dimension and topological dimension of the Julia set of $g$. We show that when
$g$ is a hyperbolic mapping, then the Hausdorff dimension of the Julia set is
strictly greater than its topological dimension. Moreover, the Julia set cannot
be locally connected. We also provide estimates for the dimension of the Julia
sets in the general (not necessarily hyperbolic) case.
