A dichotomy for Fatou components of polynomial skew products

Roeder, Roland

doi:10.1090/S1088-4173-2011-00223-2

A dichotomy for Fatou components of polynomial skew products
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by Roland K. W. Roeder

Conform. Geom. Dyn. 15 (2011), 7-19

DOI: https://doi.org/10.1090/S1088-4173-2011-00223-2

Published electronically: February 3, 2011

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Abstract:

We consider polynomial maps of the form $f(z,w) = (p(z),q(z,w))$ that extend as holomorphic maps of $\mathbb {CP}^2$. Mattias Jonsson introduces in “Dynamics of polynomial skew products on $\mathbf {C}^2$” [Math. Ann., 314(3): 403–447, 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.

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