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Conformal Geometry and Dynamics

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Generic $2$-parameter perturbations of parabolic singular points of vector fields in $\mathbb {C}$

Authors: Martin Klimeš and Christiane Rousseau
Journal: Conform. Geom. Dyn. 22 (2018), 141-184
MSC (2010): Primary 37F75, 32M25, 32S65, 34M99
Published electronically: September 7, 2018
MathSciNet review: 3851392
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Abstract: We describe the equivalence classes of germs of generic $2$-parameter families of complex vector fields $\dot z = \omega _\epsilon (z)$ on $\mathbb {C}$ unfolding a singular parabolic point of multiplicity $k+1$: $\omega _0= z^{k+1} +o(z^{k+1})$. The equivalence is under conjugacy by holomorphic change of coordinate and parameter. As a preparatory step, we present the bifurcation diagram of the family of vector fields $\dot z = z^{k+1}+\epsilon _1z+\epsilon _0$ over $\mathbb {C}\mathbb {P}^1$. This presentation is done using the new tools of periodgon and star domain. We then provide a description of the modulus space and (almost) unique normal forms for the equivalence classes of germs.

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Additional Information

Martin Klimeš
Affiliation: Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria

Christiane Rousseau
Affiliation: Département de mathématiques et de statistique, Université de Montréal, C.P. 6128, Succursale Centre-ville, Montréal Quebec, H3C 3J7, Canada
MR Author ID: 192734

Received by editor(s): October 3, 2017
Received by editor(s) in revised form: July 24, 2018
Published electronically: September 7, 2018
Additional Notes: The first author thanks CRM where this research was first initiated.
The second author was supported by NSERC in Canada.
Article copyright: © Copyright 2018 American Mathematical Society