Un th\'eor\`eme sur les actions de groupes de dimension infinie

TL;DR

This paper presents an infinitesimal criterion in the analytic setting for a vector space to be locally homogeneous under a group action, utilizing the group structure and an iterative method inspired by Kolmogorov and Arnold.

Contribution

It introduces a novel infinitesimal criterion for local homogeneity that avoids inverse function theorems by leveraging the group structure and iterative techniques.

Findings

Provides an infinitesimal criterion for local homogeneity in the analytic setting.

Uses the iterative method of Kolmogorov and Arnold in the proof.

Replaces inverse map estimates with tangent vector estimates at the origin.

Abstract

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.