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Fixed points and normal families of quasiregular mappings. (English) Zbl 1133.30322

Summary: Let \(\mathcal F\) be a family of mappings \(K\)-quasiregular in some domain \(G\). We show that if for each \(f\in \mathcal F\), there exists \(k>1\) such that the \(k\)-th iterate \(f^k\) of \(f\) has no fixed point, then \(\mathcal F\) is normal. Moreover, we examine to what extent this result holds if we consider only repelling fixed points, rather than fixed points in general. We also prove that \(F\) is quasinormal, if \(F\) contains only quasiregular mappings that do not have periodic points of some period greater than one in \(G\). This implies that a quasiregular mapping \(f:\mathbb R^n \to \mathbb R^n\) with an essential singularity in \(\infty \) has infinitely many periodic points of any period greater than one. These results generalize results of M. Essén, S. Wu, D. Bargmann and W. Bergweiler for holomorphic functions.

MSC:

30C65 Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations
26B35 Special properties of functions of several variables, Hölder conditions, etc.
30D45 Normal functions of one complex variable, normal families
37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
39B12 Iteration theory, iterative and composite equations
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References:

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