Un théorème sur les actions de groupes de dimension infinie

Details Un théorème sur les actions de groupes de dimension infinie Un théorème sur les actions de groupes de dimension infinie

2009-04-16

arxiv.org/abs/0904.2455v2

Comptes Rendus de l Acad\'emie des Sciences - Series I - Mathematics 348, 7-8 (2010) 477-482

Mathematics

Dynamical Systems

Scientific paper

We give an infinitesimal criterion, in the analytic setting, for a vector space to be locally homogeneous under some group action. Our approach differs from those which resort to an inverse function theorem (e.g. those of Moser, Zehnder or Sergeraert), because we use the underlying group structure in an essential way. In particular, this allows to replace the estimate of the inverse map of the Lie algebra action at an arbitrary tangent plane, by an estimate of the vectors tangent at the origin. Our proof relies on the iterative method of Kolmogorov and Arnold in their proof of the invariant tori theorem. The theorem of this note will be used in subsequent works.

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