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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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