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

Non-wandering Fatou components for strongly attracting polynomial skew products