zbMATH — the first resource for mathematics

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.

30C62 Quasiconformal mappings in the complex plane
Full Text: DOI
[1] [A] L. V. Ahlfors, ?Lectures on quasi conformal mappings?, The Wadsworth and
[2] [H] M. Herv?, ?Several Complex Variables?, Oxford University Press (1963).
[3] [I] Yu. Il’Yashenko, ?Limit cycles of polynomial vector fields with non-degenerate singular points on the real plane?, Funk. Anal. Ego. Pri.,18(3): (1984), 32-34, Func. Ana. and Appl.,18(3): (1985), 199-209. · Zbl 0549.34033
[4] [J] P. Joyal, ?The Generalized Homoclinic Bifurcation?, J.D.E.,107: (1994), 1-45. · Zbl 0801.34044 · doi:10.1006/jdeq.1994.1001
[5] [L] O. Lehto, ?Univalent Functions and Teichm?ller Spaces?, Graduate Texts in Mathematics,109: (1987), Springer-Verlag World Publishing Corp. · Zbl 0606.30001
[6] [M] P. Mardesic, ?Le d?ploiement versel du cusp d’ordre n?, th?se Universit? de Bourgogne (1992). ?Chebychev systems and the versal unfolding of the cusp of order n?. Travaux en Cours,57, (1998), 1-153.
[7] [P-Y] R. P?rez-Marco, J.-C. Yoccoz, ?Germes de feuilletages holomorphes ? holonomie prescrite?, in: Complex Analytic Methods in Dynamical Systems, IMPA january 1992, Ast?risque222: (1994), 345-371.
[8] [R 1] R. Roussarie, ?Cyclicit? finie des lacets et des points cuspidaux?, Nonlinearity, fasc.2: (1989), 73-117. · Zbl 0679.58037 · doi:10.1088/0951-7715/2/1/006
[9] [R 2] R. Roussarie, ?Bifurcations of Planar Vector Fields and Hilbert’s sixteenth problem?. Progress in Mathematics, Birkha?ser Ed.164, (1998), 1-204. · Zbl 0898.58039
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.