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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)
Keywords:
semigroup; quasiregular mapping
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