A dichotomy for Fatou components of polynomial skew products

TL;DR

This paper establishes a dichotomy for Fatou components of polynomial skew products, showing they are either topologically simple or have infinitely generated homology, depending on certain dynamical properties.

Contribution

It proves a new dichotomy for Fatou components of polynomial skew products based on connectedness and Axiom-A conditions, extending understanding of their topological structure.

Findings

Connected skew products have Fatou components homeomorphic to open balls.

Non-connected skew products have at least one Fatou component with infinitely generated first homology.

The result links dynamical properties to topological complexity of Fatou components.

Abstract

We consider polynomial maps of the form f(z,w) = (p(z),q(z,w)) that extend as holomorphic maps of CP^2. Mattias Jonsson introduces in (Math. Ann., 1999) a notion of connectedness for such polynomial skew products that is analogous to connectivity for the Julia set of a polynomial map in one-variable. We prove the following dichotomy: if f is an Axiom-A polynomial skew product, and f is connected, then every Fatou component of f is homeomorphic to an open ball; otherwise, some Fatou component of f has infinitely generated first homology.