  Wandering Fatou components were recently constructed by Astorg et al for
higher dimensional holomorphic maps on projective spaces. Their examples are
polynomial skew products with a parabolic invariant line. In this paper we
study this wandering domain problem for polynomial skew product $f$ with an
attracting invariant line $L$ (which is the more common case). We show that if
$f$ is unicritical (in the sense that the critical curve has a unique
transversal intersection with $L$), then every Fatou component of $f$ in the
basin of $L$ is an extension of a one-dimensional Fatou component of $f|_L$. As
a corollary there is no wandering Fatou component. We will also discuss the
multicritical case under additional assumptions.
