  We study toplogical properties of attracting sets for automorphisms of
$\mathbb{C}^k$. Our main result is that a generic volume preserving
automorphism has a hyperbolic fixed point with a dense stable manifold. We
prove the same result for volume preserving maps tangent to the identity. On
the other hand, we show that an attracting set can only contain a neighborhood
of the fixed point if the fixed point is an attracting fixed point. We will see
that the latter does not hold in the non-autonomous setting.
