  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.

Dynamical Systems