The wandering domain problem for attracting polynomial skew products

Wandering Fatou components were recently constructed by Astorg et al. (2016) 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.