The wandering domain problem for attracting polynomial skew products

TL;DR

This paper investigates the wandering domain problem for attracting polynomial skew products with an attracting invariant line, proving that in the unicritical case, all Fatou components are non-wandering extensions of one-dimensional components.

Contribution

It establishes that unicritical polynomial skew products with an attracting invariant line have no wandering Fatou components, extending understanding of Fatou components in higher dimensions.

Findings

No wandering Fatou components in the unicritical case

Fatou components are extensions of one-dimensional components

Discussion of multicritical cases under additional assumptions

Abstract

Wandering Fatou components were recently constructed by Astorg et al 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.