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Conformal Geometry and Dynamics

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Branch sets of uniformly quasiregular maps

Author: G. J. Martin
Journal: Conform. Geom. Dyn. 1 (1997), 24-27
MSC (1991): Primary 30C60
Published electronically: June 19, 1997
MathSciNet review: 1454921
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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

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.
Article copyright: © Copyright 1997 American Mathematical Society