A dichotomy for Fatou components of polynomial skew products
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.
Keywords
Cite
@article{arxiv.1005.2252,
title = {A dichotomy for Fatou components of polynomial skew products},
author = {Roland K. W. Roeder},
journal= {arXiv preprint arXiv:1005.2252},
year = {2011}
}
Comments
To appear in Conformal Geometry and Dynamics. 13 pages. Version 2 is updated with some additional details in the proof, a new example, and a discussion of how much of the Axiom-A hypothesis is needed. Comments welcome