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Conformal Geometry and Dynamics

Published by the American Mathematical Society, the Conformal Geometry and Dynamics (ECGD) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4173

The 2020 MCQ for Conformal Geometry and Dynamics is 0.5.

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A dichotomy for Fatou components of polynomial skew products
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by Roland K. W. Roeder PDF
Conform. Geom. Dyn. 15 (2011), 7-19

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.
References
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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
  • 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