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Quasi-conformal mapping theorem and bifurcations. (English) Zbl 0930.30019
Let $$H$$ be a germ of holomorphic diffeomorphism at $$0\in\mathbb{C}$$, such that $$H(0)= 0$$. Applying the quasi-conformal mapping theorem of Ahlfors-Bers, the author gives a direct construction of a germ of analytic multivalued mapping $$S$$, with $$S(0)= 0$$, such that $$S(z)$$ obtained after one turn around the origin is equal to $$H\circ S(z)$$, i.e. he solves the equation $S(ze^{2\pi i})= H\circ S(z),\quad S(0)= 0.$ With the aid of another method this problem was solved by [R. Pérez-Marco and J.-C. Yoccoz in: Complex analytic methods in dynamical systems, IMPA, January 1992, Astérisque 222, 345-371 (1994; Zbl 0809.32008)]. An application to the bifurcation theory of vector fields of the plane is given.

##### MSC:
 30C62 Quasiconformal mappings in the complex plane
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##### References:
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