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Conformal Geometry and Dynamics

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A dichotomy for Fatou components of polynomial skew products

Author: Roland K. W. Roeder
Journal: Conform. Geom. Dyn. 15 (2011), 7-19
MSC (2010): Primary 32H50; Secondary 37F20, 57R19
Published electronically: February 3, 2011
MathSciNet review: 2769221
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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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Additional Information

Roland K. W. Roeder
Affiliation: IUPUI Department of Mathematical Sciences, LD Building, Room 270, 402 North Blackford Street, Indianapolis, Indiana 46202-3267

Keywords: Fatou components, linking numbers, closed currents, holomorphic motions
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
Article copyright: © Copyright 2011 Roland K. W. Roeder