The wandering domain problem for attracting polynomial skew products

Abstract

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.