Dynamics of multi-resonant biholomorphisms
type: Item
instance of: scholarly article
zbMATH Open document ID: 1304.32010
author: Filippo Bracci
author: Jasmin Raissy
author: Dmitri Zaitsev
publication date: 2014-09-04T00:00:00Z
full work available at URL: https://arxiv.org/abs/1106.1962
Mathematics Subject Classification ID: 32H50
Mathematics Subject Classification ID: 32B10
Mathematics Subject Classification ID: 37F75
zbMATH DE Number: 6340080
zbMATH Keywords: germs of biholomorphisms of \(\mathbb C^n\)
zbMATH Keywords: basins of attraction
Wikidata QID: Q58623167
MaRDI profile type: Publication
OpenAlex ID: W2000388722
arXiv ID: 1106.1962
DOI: 10.1093/IMRN/RNS192
published in: IMRN. International Mathematics Research Notices
title: Dynamics of multi-resonant biholomorphisms
review text: The authors study the dynamics of \(F\), a germ of biholomorphism of \(\mathbb C^n\) at the origin, whose linear part is diagonalizable. They define \(F\) to be multi-resonant with respect to the first \(r\) eigenvalues of the linear part if the resonances between these eigenvalues are finitely generated, with \(m\) generators. Given such a biholomorphism \(F\) and a choice of the Poincaré-Dulac normal form, one can define a monomial map \(\pi: \mathbb C^n \to \mathbb C^m\) that semi-conjugates \(F\) to a germ \(f\) of biholomorphisms of \(\mathbb C^m\) tangent to the identity, called the parabolic shadow of \(F\). Using Hakim's theory on germs tangent to the identity, and under some natural conditions on \(f\), and on the dynamics of \(F\) on the fibers of \(\pi\), the authors are able to prove that the initial germ \(F\) admits a basin of attraction (and in fact at least as many as the weighted order of \(F\)).
reviewed by: Stéphane Lamy
description: scientific article; zbMATH DE number 6340080