A polynomial skew-product with a wandering Fatou-disk

TL;DR

This paper constructs examples of vertical disks in polynomial skew-products with attracting fibers that never intersect Fatou components and lie entirely in the Julia set, highlighting the complexity of wandering Fatou components in this setting.

Contribution

It demonstrates that Lilov's non-existence result for wandering Fatou components does not extend to attracting fibers, revealing new complexities in the dynamics of polynomial skew-products.

Findings

Existence of vertical disks accumulating at repelling fixed points

Vertical disks can lie entirely in the Julia set

Wandering Fatou components are more complex in attracting fibers

Abstract

Little is known about the existence of wandering Fatou components for rational maps in two complex variables. In 2003 Lilov proved the non-existence of wandering Fatou components for polynomial skew-products in the neighborhood of an invariant super-attracting fiber. In fact Lilov proved a stronger result, namely that the forward orbit of any vertical disk must intersect a fattened Fatou component of the invariant fiber. Naturally the next class of maps to study are polynomial skew-products with an invariant attracting (but not super-attracting) fiber. Here we show that Lilov's stronger result does not hold in this setting: for some skew-products there are vertical disks whose orbits accumulate at repelling fixed points in the invariant fiber, and that therefore never intersect the fattened Fatou components. These disks are necessarily Fatou disks, but we also prove that the vertical disks we construct lie entirely in the Julia set. Our results therefore do not answer the existence question of wandering Fatou components in the attracting setting, but show that the question is considerably more complicated than in the super-attracting setting.