Fatou components of attracting skew products

TL;DR

This paper studies the existence of wandering Fatou components in polynomial skew-products in two complex variables, proving non-existence in certain attracting cases and analyzing critical orbit behavior.

Contribution

It extends previous work by proving non-existence of wandering domains for subhyperbolic attracting skew-products and analyzing critical orbit escape rates using linearization maps.

Findings

No wandering domains for subhyperbolic attracting skew-products.

Bounds on critical orbit escape rates in almost all fibers.

Use of linearization maps to describe unstable manifolds.

Abstract

We investigate the existence of wandering Fatou components for polynomial skew-products in two complex variables. In 2004 the non-existence of wandering domains near a super-attracting invariant fiber was shown in [8]. In 2014 it was shown in [1] that wandering domains can exist near a parabolic invariant fiber. In [9] the geometrically-attracting case was studied, and we continue this study here. We prove the non-existence of wandering domains for subhyperbolic attracting skew products; this class contains the maps studied in [9]. Using expansion properties on the Julia set in the invariant fiber, we prove bounds on the rate of escape of critical orbits in almost all fibers. Our main tool in describing these critical orbits is a possibly singular linearization map of unstable manifolds.