  We show a partial generalization of Sullivan's non-wandering domain theorem
in complex dimension two. More precisely, we show the non-existence of
wandering Fatou components for polynomial skew products of $ \mathbb{C}^2$ with
an invariant attracting fiber, under the assumption that the multiplier $
\lambda $ is small. We actually show a stronger result, namely that every
forward orbit of any vertical Fatou disk intersects a bulging Fatou component.
