## 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

## 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 âDynamics of polynomial skew products on $\mathbf {C}^2$â [*Math. Ann.*, 314(3): 403â447, 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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## Bibliographic Information

**Roland K. W. Roeder**- Affiliation: IUPUI Department of Mathematical Sciences, LD Building, Room 270, 402 North Blackford Street, Indianapolis, Indiana 46202-3267
- MR Author ID: 718580
- Email: rroeder@math.iupui.edu
- Received by editor(s): May 12, 2010
- Received by editor(s) in revised form: January 1, 2011, and January 2, 2011
- Published electronically: February 3, 2011
- Additional Notes: Research was supported in part by startup funds from the Department of Mathematics at IUPUI
- © Copyright 2011 Roland K. W. Roeder
- Journal: Conform. Geom. Dyn.
**15**(2011), 7-19 - MSC (2010): Primary 32H50; Secondary 37F20, 57R19
- DOI: https://doi.org/10.1090/S1088-4173-2011-00223-2
- MathSciNet review: 2769221