an:01241188
Zbl 0930.30019
Roussarie, Robert
Quasi-conformal mapping theorem and bifurcations
EN
Bol. Soc. Bras. Mat., Nova S??r. 29, No. 2, 229-251 (1998).
00048580
1998
j
30C62
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 [\textit{R. P??rez-Marco} and \textit{J.-C. Yoccoz} in: Complex analytic methods in dynamical systems, IMPA, January 1992, Ast??risque 222, 345-371 (1994; Zbl 0809.32008)]. An application to the bifurcation theory of vector fields of the plane is given.
J.Matkowski (Bielsko-Bia??a)
Zbl 0809.32008