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Quasiregular semigroups. (English) Zbl 0860.30019
A quasiregular mapping is a quasiconformal mapping without the injectivity requirement. A quasiregular semigroup \(\Gamma\) is a family of mappings of \(\overline{\mathbb{R}}^n\) to itself closed under composition such that each element is \(K\)-quasiregular for some fixed \(K\). The authors construct such an example \(\Gamma\) for any \(n\geq 2\) and any \(K>2\); every \(f\in \Gamma\) has nonempty branch set. It is shown in [A. Hinkkanen: Ann. Acad. Sci. Fenn. Math. 21, 205-222 (1996)] that a plane quasiregular semigroup need not admit an invariant measurable conformal structure. With some additional hypothesis on \(\Gamma\), the authors show how to construct equivariant measurable conformal structures for \(\Gamma\). As an application it is shown the Julia set of a quasiregular mapping generating a quasiregular semigroup is a Cantor set.

MSC:
30C65 Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations
20M20 Semigroups of transformations, relations, partitions, etc.
30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
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