Smooth Gevrey normal forms of vector fields near a fixed point

Stolovitch, Laurent

Annales de l'Institut Fourier, Tome 63 (2013) no. 1, pp. 241-267.

Nous étudions des germes lisses (i.e. $C^{\infty }$) de champs de vecteurs au voisinage d’un point fixe en lequel la partie linéaire est hyperbolique. Il est bien connu que les petits diviseurs sont «  invisibles  » dans les problèmes de linéarisation ou de mise sous forme normale lisse. Nous montrons qu’il en est tout autrement dans la catégorie Gevrey lisse. Nous montrons qu’un germe de champ de vecteurs $\alpha$-Gevrey lisse ayant une partie linéaire hyperbolique au point fixe admet une transformation $\beta$-Gevrey lisse vers une forme normale $\beta$-Gevrey lisse où l’indice $\beta$ dépend de la vitesse d’accumulation vers zéro des «  petits diviseurs  ». De plus, si le germe de champ de vecteurs est formellement linéarisable Gevrey lisse et admet une partie linéaire vérifiant la condition diophantienne de Brjuno alors il est linéarisable dans la même classe Gevrey.

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.

Classification : 34K17, 37J40, 37F50, 37F75, 37G05
Keywords: Hyperbolic dynamical systems, normal forms, linearization, small divisors, resonances, Gevrey classes.
Mot clés : Systèmes dynamique hyperbolique, formes normales, linéarisation, petits diviseurs, résonances, classes Gevrey.

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