Simultaneous reduction to normal forms of commuting singular vector fields with linear parts having Jordan blocks

Masafumi Yoshino[1]; Todor Gramchev[2]

  • [1] Graduate School of Science Hiroshima University Higashi-Hiroshima, 739-8526 (Japan)
  • [2] Università di Cagliari Dipartimento di Matematica via Ospedale 72 09124 Cagliari (Italy)

Annales de l’institut Fourier (2008)

  • Volume: 58, Issue: 1, page 263-297
  • ISSN: 0373-0956

Abstract

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We study the simultaneous linearizability of d –actions (and the corresponding d -dimensional Lie algebras) defined by commuting singular vector fields in ℂ n fixing the origin with nontrivial Jordan blocks in the linear parts. We prove the analytic convergence of the formal linearizing transformations under a certain invariant geometric condition for the spectrum of d vector fields generating a Lie algebra. If the condition fails and if we consider the situation where small denominators occur, then we show the existence of divergent solutions of an overdetermined system of linearized homological equations. In the ℂ ∞ category, the situation is completely different. We show Sternberg’s theorem for a commuting system of ℂ ∞ vector fields with a Jordan block although they do not satisfy the condition.

How to cite

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Yoshino, Masafumi, and Gramchev, Todor. "Simultaneous reduction to normal forms of commuting singular vector fields with linear parts having Jordan blocks." Annales de l’institut Fourier 58.1 (2008): 263-297. <http://eudml.org/doc/10311>.

@article{Yoshino2008,
abstract = {We study the simultaneous linearizability of $d$–actions (and the corresponding $d$-dimensional Lie algebras) defined by commuting singular vector fields in $\{\mathbb\{C\}\}^n$ fixing the origin with nontrivial Jordan blocks in the linear parts. We prove the analytic convergence of the formal linearizing transformations under a certain invariant geometric condition for the spectrum of $d$ vector fields generating a Lie algebra. If the condition fails and if we consider the situation where small denominators occur, then we show the existence of divergent solutions of an overdetermined system of linearized homological equations. In the $\{\mathbb\{C\}\}^\infty $ category, the situation is completely different. We show Sternberg’s theorem for a commuting system of $\{\mathbb\{C\}\}^\infty $ vector fields with a Jordan block although they do not satisfy the condition.},
affiliation = {Graduate School of Science Hiroshima University Higashi-Hiroshima, 739-8526 (Japan); Università di Cagliari Dipartimento di Matematica via Ospedale 72 09124 Cagliari (Italy)},
author = {Yoshino, Masafumi, Gramchev, Todor},
journal = {Annales de l’institut Fourier},
keywords = {singular vector field; linearization; Jordan block; homological equation; Diophantine conditions; Gevrey spaces; decomposition},
language = {eng},
number = {1},
pages = {263-297},
publisher = {Association des Annales de l’institut Fourier},
title = {Simultaneous reduction to normal forms of commuting singular vector fields with linear parts having Jordan blocks},
url = {http://eudml.org/doc/10311},
volume = {58},
year = {2008},
}

TY - JOUR
AU - Yoshino, Masafumi
AU - Gramchev, Todor
TI - Simultaneous reduction to normal forms of commuting singular vector fields with linear parts having Jordan blocks
JO - Annales de l’institut Fourier
PY - 2008
PB - Association des Annales de l’institut Fourier
VL - 58
IS - 1
SP - 263
EP - 297
AB - We study the simultaneous linearizability of $d$–actions (and the corresponding $d$-dimensional Lie algebras) defined by commuting singular vector fields in ${\mathbb{C}}^n$ fixing the origin with nontrivial Jordan blocks in the linear parts. We prove the analytic convergence of the formal linearizing transformations under a certain invariant geometric condition for the spectrum of $d$ vector fields generating a Lie algebra. If the condition fails and if we consider the situation where small denominators occur, then we show the existence of divergent solutions of an overdetermined system of linearized homological equations. In the ${\mathbb{C}}^\infty $ category, the situation is completely different. We show Sternberg’s theorem for a commuting system of ${\mathbb{C}}^\infty $ vector fields with a Jordan block although they do not satisfy the condition.
LA - eng
KW - singular vector field; linearization; Jordan block; homological equation; Diophantine conditions; Gevrey spaces; decomposition
UR - http://eudml.org/doc/10311
ER -

References

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