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Stoïlow factorization for quasiregular mappings in all dimensions. (English) Zbl 1190.30020
The classical Stoïlov theorem states that a discrete open map $$f$$ from a domain $$\Omega\subset\mathbb R^2$$ into $$\mathbb R^2$$ can be factorized as a composition of an analytic function $$\varphi$$ and a homeomorphisms $$h$$, $$f=\varphi\circ h$$. Discrete means here that, for any $$x\in\mathbb R^2$$, $$f^{-1}(x)$$ is a discrete subset of $$\Omega$$. It is known that if $$f$$ is quasiregular, then the $$h$$ in its factorization is quasiconformal.
In higher dimensions $$n\geq 3$$, the only analytic functions are restrictions of Möbius transformations. A natural substitution for analytic functions for the factorization problem is given by the so-called uniformly quasiregular (uqr) maps, i.e., quasiregular self-maps $$\varphi$$ of the $$n$$-dimensional sphere $$\mathbb S^n$$ having a uniform bound for the distortion of all iterates $$\varphi^{\circ n}$$. Uqr maps are always rational w.r.t. some measurable Riemannian structure on $$\mathbb S^n$$. Rickman’s version of Montel’s normality criterion makes it possible to develop an analogue of the Fatou – Julia theory for uqr maps, see [A. Hinkkanen, G. J. Martin and V. Mayer, Math. Scand. 95, No. 1, 80–100 (2004; Zbl 1067.30043)] and [G. Martin, V. Mayer and K. Peltonen, Proc. Am. Math. Soc. 134, No. 7, 2091–2097 (2006; Zbl 1094.30029)]. The Fatou set $$\mathcal F(\varphi)$$ is the open set where the iterates $$\varphi^{\circ n}$$ form a normal family (i.e., a family precompact w.r.t. locally uniform convergence). The Julia set is $$\mathcal J(\varphi):=\mathbb S^n\setminus \mathcal F(\varphi)$$.
The main result of the paper is the following analogue of the Stoïlow result for quasiregular mappings in higher dimensions:
Theorem 2.1. Suppose $$f:\mathbb S^n\to\mathbb S^n$$ is a non-constant quasiregular mapping, $$n\geq2$$. Then there exists a uniformly quasiregular mapping $$\varphi$$ and a quasiconformal homeomorphism $${h:\mathbb S^n\to\mathbb S^n}$$ such that $$f=\varphi\circ h$$, and $$\mathcal J(\varphi)$$ is a Cantor set.
The authors prove also the uniqueness of the above factorization up to a finite dimensional Lie group provided that the invariant conformal structure associated to $$\varphi$$ is fixed.

MSC:
 30C65 Quasiconformal mappings in $$\mathbb{R}^n$$, other generalizations 30G12 Finely holomorphic functions and topological function theory 30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable 37F30 Quasiconformal methods and Teichmüller theory, etc. (dynamical systems) (MSC2010)
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References:
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