Smooth Gevrey normal forms of vector fields near a fixed point

Laurent Stolovitch[1]

  • [1] CNRS, Laboratoire J.-A. Dieudonné U.M.R. 6621, Université de Nice - Sophia Antipolis, Parc Valrose 06108 Nice Cedex 02, France.

Annales de l’institut Fourier (2013)

  • Volume: 63, Issue: 1, page 241-267
  • ISSN: 0373-0956

Abstract

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We study germs of smooth vector fields in a neighborhood of a fixed point having an hyperbolic linear part at this point. It is well known that the “small divisors” are invisible either for the smooth linearization or normal form problem. We prove that this is completely different in the smooth Gevrey category. We prove that a germ of smooth α -Gevrey vector field with an hyperbolic linear part admits a smooth β -Gevrey transformation to a smooth β -Gevrey normal form. The Gevrey order β depends on the rate of accumulation to 0 of the small divisors. We show that a formally linearizable smooth Gevrey germ with the linear part satisfying Brjuno’s small divisors condition can be linearized in the same Gevrey class.

How to cite

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Stolovitch, Laurent. "Smooth Gevrey normal forms of vector fields near a fixed point." Annales de l’institut Fourier 63.1 (2013): 241-267. <http://eudml.org/doc/275549>.

@article{Stolovitch2013,
abstract = {We study germs of smooth vector fields in a neighborhood of a fixed point having an hyperbolic linear part at this point. It is well known that the “small divisors” are invisible either for the smooth linearization or normal form problem. We prove that this is completely different in the smooth Gevrey category. We prove that a germ of smooth $\alpha $-Gevrey vector field with an hyperbolic linear part admits a smooth $\beta $-Gevrey transformation to a smooth $\beta $-Gevrey normal form. The Gevrey order $\beta $ depends on the rate of accumulation to $0$ of the small divisors. We show that a formally linearizable smooth Gevrey germ with the linear part satisfying Brjuno’s small divisors condition can be linearized in the same Gevrey class.},
affiliation = {CNRS, Laboratoire J.-A. Dieudonné U.M.R. 6621, Université de Nice - Sophia Antipolis, Parc Valrose 06108 Nice Cedex 02, France.},
author = {Stolovitch, Laurent},
journal = {Annales de l’institut Fourier},
keywords = {Hyperbolic dynamical systems; normal forms; linearization; small divisors; resonances; Gevrey classes; hyperbolic dynamical systems},
language = {eng},
number = {1},
pages = {241-267},
publisher = {Association des Annales de l’institut Fourier},
title = {Smooth Gevrey normal forms of vector fields near a fixed point},
url = {http://eudml.org/doc/275549},
volume = {63},
year = {2013},
}

TY - JOUR
AU - Stolovitch, Laurent
TI - Smooth Gevrey normal forms of vector fields near a fixed point
JO - Annales de l’institut Fourier
PY - 2013
PB - Association des Annales de l’institut Fourier
VL - 63
IS - 1
SP - 241
EP - 267
AB - We study germs of smooth vector fields in a neighborhood of a fixed point having an hyperbolic linear part at this point. It is well known that the “small divisors” are invisible either for the smooth linearization or normal form problem. We prove that this is completely different in the smooth Gevrey category. We prove that a germ of smooth $\alpha $-Gevrey vector field with an hyperbolic linear part admits a smooth $\beta $-Gevrey transformation to a smooth $\beta $-Gevrey normal form. The Gevrey order $\beta $ depends on the rate of accumulation to $0$ of the small divisors. We show that a formally linearizable smooth Gevrey germ with the linear part satisfying Brjuno’s small divisors condition can be linearized in the same Gevrey class.
LA - eng
KW - Hyperbolic dynamical systems; normal forms; linearization; small divisors; resonances; Gevrey classes; hyperbolic dynamical systems
UR - http://eudml.org/doc/275549
ER -

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