Small divisors and large multipliers

Boele Braaksma[1]; Laurent Stolovitch[2]

  • [1] University of Groningen Department of Mathematics P.O. Box 800 9700 AV Groningen (The Netherlands)
  • [2] CNRS UMR 5580 Université Paul Sabatier MIG, Laboratoire de Mathématiques Emile Picard 31062 Toulouse cedex 9 (France)

Annales de l’institut Fourier (2007)

  • Volume: 57, Issue: 2, page 603-628
  • ISSN: 0373-0956

Abstract

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We study germs of singular holomorphic vector fields at the origin of ℂ n of which the linear part is 1 -resonant and which have a polynomial normal form. The formal normalizing diffeomorphism is usually divergent at the origin but there exists holomorphic diffeomorphisms in some “sectorial domains” which transform these vector fields into their normal form. In this article, we study the interplay between the small divisors phenomenon and the Gevrey character of the sectorial normalizing diffeomorphisms. We show that the Gevrey order of the latter is linked to the diophantine type of the small divisors.

How to cite

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Braaksma, Boele, and Stolovitch, Laurent. "Small divisors and large multipliers." Annales de l’institut Fourier 57.2 (2007): 603-628. <http://eudml.org/doc/10233>.

@article{Braaksma2007,
abstract = {We study germs of singular holomorphic vector fields at the origin of $\mathbb\{C\}^n$ of which the linear part is $1$-resonant and which have a polynomial normal form. The formal normalizing diffeomorphism is usually divergent at the origin but there exists holomorphic diffeomorphisms in some “sectorial domains” which transform these vector fields into their normal form. In this article, we study the interplay between the small divisors phenomenon and the Gevrey character of the sectorial normalizing diffeomorphisms. We show that the Gevrey order of the latter is linked to the diophantine type of the small divisors.},
affiliation = {University of Groningen Department of Mathematics P.O. Box 800 9700 AV Groningen (The Netherlands); CNRS UMR 5580 Université Paul Sabatier MIG, Laboratoire de Mathématiques Emile Picard 31062 Toulouse cedex 9 (France)},
author = {Braaksma, Boele, Stolovitch, Laurent},
journal = {Annales de l’institut Fourier},
keywords = {Holomorphic dynamics; small divisors; normal forms; Gevrey functions; divergent series; holomorphic dynamics},
language = {eng},
number = {2},
pages = {603-628},
publisher = {Association des Annales de l’institut Fourier},
title = {Small divisors and large multipliers},
url = {http://eudml.org/doc/10233},
volume = {57},
year = {2007},
}

TY - JOUR
AU - Braaksma, Boele
AU - Stolovitch, Laurent
TI - Small divisors and large multipliers
JO - Annales de l’institut Fourier
PY - 2007
PB - Association des Annales de l’institut Fourier
VL - 57
IS - 2
SP - 603
EP - 628
AB - We study germs of singular holomorphic vector fields at the origin of $\mathbb{C}^n$ of which the linear part is $1$-resonant and which have a polynomial normal form. The formal normalizing diffeomorphism is usually divergent at the origin but there exists holomorphic diffeomorphisms in some “sectorial domains” which transform these vector fields into their normal form. In this article, we study the interplay between the small divisors phenomenon and the Gevrey character of the sectorial normalizing diffeomorphisms. We show that the Gevrey order of the latter is linked to the diophantine type of the small divisors.
LA - eng
KW - Holomorphic dynamics; small divisors; normal forms; Gevrey functions; divergent series; holomorphic dynamics
UR - http://eudml.org/doc/10233
ER -

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