Branch sets of uniformly quasiregular maps

Martin, G.

doi:10.1090/S1088-4173-97-00016-7

Branch sets of uniformly quasiregular maps
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by G. J. Martin

Conform. Geom. Dyn. 1 (1997), 24-27

DOI: https://doi.org/10.1090/S1088-4173-97-00016-7

Published electronically: June 19, 1997

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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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