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Extending holomorphic motions. (English) Zbl 0619.30026
Let X be a subset of $${\mathbb{C}}$$, a holomorphic motion of X in $${\mathbb{C}}$$ is a map f:T$$\times X\to {\mathbb{C}}$$ defined for some connected open subset $$T\subset {\mathbb{C}}$$ containing 0 such that a) for any fixed $$x\in X$$, $$f_ t(x)=f(t,x)$$ is a holomorphic mapping of T to $${\mathbb{C}}$$, b) for any fixed $$t\in T$$, $$f_ t$$ is an embedding, and c) $$f_ 0$$ is the identity map of X.
The main result of this paper is the following theorem:
Theorem 1. There is a universal constant $$a>0$$ such that any holomorphic motion of any set $$X\subset {\mathbb{C}}$$ parametrized by the disk $$T=D_ 1$$ of radius 1 about 0 can be extended to a holomorphic motion of $${\mathbb{C}}$$ with time parameter in the disk $$D_ a$$ of radius a about 0.
Corollary. If f is a holomorphic motion of a set X, then for each time the map $$f_ t$$ extends to a quasiconformal map of $${\mathbb{C}}$$. Also the authors obtain some results about quasiconformal motions.
Reviewer: N.A.Gusevskij

##### MSC:
 30C62 Quasiconformal mappings in the complex plane
##### Keywords:
holomorphic motion; quasiconformal motions
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##### References:
 [1] Bers, L. &Royden, H. L., Holomorphic families of injections.Acta Math., 157 (1986), 259–286. · Zbl 0619.30027 · doi:10.1007/BF02392595 [2] Mañé, R., Sad, P. &Sullivan, D., On the dynamics of rational mapsAnn. Sci. Ecole Norm. Sup., 16 (1983), 193–217. · Zbl 0524.58025
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