Non-wandering Fatou components for strongly attracting polynomial skew products

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.