Branch sets of uniformly quasiregular maps

Abstract

Let n ≥ 2 n\geq 2 and f : S n → S n f: {\Bbb S}^n\to {\Bbb S}^n be a quasiregular mapping with branch set B f B_f , the set where f f fails to be locally injective. We show that there is a quasiregular mapping g : S n → S n g: {\Bbb S}^n\to {\Bbb S}^n with B g = B f B_g = B_f and such that g g can be chosen to be conformal (rational) with respect to some measurable Riemannian structure on S n {\Bbb S}^n . Hence g g is uniformly quasiregular. That is, g g and all its iterates are quasiregular with a uniform bound on the dilatation.