The moduli space of germs of generic families of analytic diffeomorphisms unfolding of a codimension one resonant diffeomorphism or resonant saddle
type: Item
instance of: scholarly article
title: The moduli space of germs of generic families of analytic diffeomorphisms unfolding of a codimension one resonant diffeomorphism or resonant saddle
zbMATH Open document ID: 1204.37048
author: Christiane Rousseau
published in: Journal of Differential Equations
publication date: 2010-03-29T00:00:00Z
review text: This interesting paper completes previous works by the author and C. Christopher giving a complete description of the moduli space under holomorphic conjugacy for germs of generic 1-parameter families unfolding a 1-dimensional resonant biholomorphism or a 2-dimensional resonant saddles. This is obtained in several stages. First of all, the author shows that any germ of a family unfolding a resonant biholomorphism can be realized as the holonomy map of a separatrix of a germ of family of vector fields having a second separatrix whose family of holonomies is again a germ of family of resonant biholomorphisms. Iterating this trick a finite number of times allows the author to reduce the problem to the study of the moduli space of families unfolding a germ of biholomorphism tangent to the identity and with a double fixed point at the origin. This requires relating the (Écalle, Martinet-Ramis, and/or Glutsyuk) invariants associated to the holonomies of the two separatrices in a germ of family unfolding a resonant saddle; the author, mostly using the Martinet-Ramis point of view, is able to give exact formulas for the relationships between the formal and the analytic part of the two invariants. The invariants of a family (in prepared form) unfolding a germ tangent to the identity with a double fixed point are: the formal invariant \(a(\varepsilon)\), depending analytically on the parameter \(\varepsilon\) of the unfolding; and the analytical invariant given by two families of germs of biholomorphisms \((\psi^0_{\varepsilon,\pm},\psi^\infty_{\varepsilon,\pm})|_{\varepsilon\in V_{\pm}}\), where \(V_{\pm}\) are two sectors whose union is a punctured neighborhood of the origin in the \(\varepsilon\)-space, and \(\psi^0_{\varepsilon,\pm}\) (respectively, \(\psi^\infty_{\varepsilon,\pm}\)) is a germ of biholomorphism at the origin (respectively, at infinity), satisfying a suitable compatibility condition implying that, possibly after a linear change of coordinates in \(\mathbb{…
reviewed by: Marco Abate
Mathematics Subject Classification ID: 37F45
Mathematics Subject Classification ID: 30D05
Mathematics Subject Classification ID: 34M40
zbMATH DE Number: 5687483
zbMATH Keywords: resonant biholomorphism
zbMATH Keywords: resonant saddle
zbMATH Keywords: moduli space
zbMATH Keywords: unfoldings
MaRDI profile type: Publication
full work available at URL: j.jde.2009.12.003
OpenAlex ID: W2006798731
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DOI: 10.1016/J.JDE.2009.12.003
description: scientific article; zbMATH DE number 5687483