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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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Branch sets of uniformly quasiregular maps
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by G. J. Martin PDF
Conform. Geom. Dyn. 1 (1997), 24-27 Request permission

Abstract:

Let $n\geq 2$ and $f: {\Bbb S}^n\to {\Bbb S}^n$ be a quasiregular mapping with branch set $B_f$, the set where $f$ fails to be locally injective. We show that there is a quasiregular mapping $g: {\Bbb S}^n\to {\Bbb S}^n$ with $B_g = B_f$ and such that $g$ can be chosen to be conformal (rational) with respect to some measurable Riemannian structure on ${\Bbb S}^n$. Hence $g$ is uniformly quasiregular. That is, $g$ and all its iterates are quasiregular with a uniform bound on the dilatation.
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Additional Information
  • G. J. Martin
  • Affiliation: Department of Mathematics, University of Auckland, Auckland, New Zealand
  • MR Author ID: 120465
  • Email: martin@math.auckland.ac.nz
  • Received by editor(s): January 5, 1997
  • Received by editor(s) in revised form: April 16, 1997
  • Published electronically: June 19, 1997
  • Additional Notes: Research supported in part by a grant from the N.Z. Marsden Fund.
  • © Copyright 1997 American Mathematical Society
  • Journal: Conform. Geom. Dyn. 1 (1997), 24-27
  • MSC (1991): Primary 30C60
  • DOI: https://doi.org/10.1090/S1088-4173-97-00016-7
  • MathSciNet review: 1454921