Non-wandering Fatou components for strongly attracting polynomial skew products

TL;DR

This paper extends Sullivan's theorem to certain complex two-dimensional polynomial skew products, proving the non-existence of wandering Fatou components when the fiber's multiplier is small, and showing all Fatou disks eventually intersect a bulging component.

Contribution

It generalizes Sullivan's non-wandering domain theorem to complex dimension two for skew products with small multipliers, establishing new intersection properties of Fatou disks.

Findings

No wandering Fatou components for small multiplier skew products

All Fatou disks intersect a bulging Fatou component

Extension of Sullivan's theorem to higher dimensions

Abstract

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.