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Conformal Geometry and Dynamics

Published by the American Mathematical Society, the Conformal Geometry and Dynamics (ECGD) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4173

The 2020 MCQ for Conformal Geometry and Dynamics is 0.5.

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Generic $2$-parameter perturbations of parabolic singular points of vector fields in $\mathbb {C}$
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by Martin Klimeš and Christiane Rousseau PDF
Conform. Geom. Dyn. 22 (2018), 141-184 Request permission

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
  • Email: martin.klimes@univie.ac.at
  • 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
  • Email: rousseac@dms.umontreal.ca
  • 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.
  • © Copyright 2018 American Mathematical Society
  • Journal: Conform. Geom. Dyn. 22 (2018), 141-184
  • MSC (2010): Primary 37F75, 32M25, 32S65, 34M99
  • DOI: https://doi.org/10.1090/ecgd/325
  • MathSciNet review: 3851392