Normal forms with exponentially small remainder and Gevrey normalization for vector fields with a nilpotent linear part
Annales de l'Institut Fourier, Volume 62 (2012) no. 6, p. 2211-2225
We explore the convergence/divergence of the normal form for a singularity of a vector field on n with nilpotent linear part. We show that a Gevrey-α vector field X with a nilpotent linear part can be reduced to a normal form of Gevrey-1+α type with the use of a Gevrey-1+α transformation. We also give a proof of the existence of an optimal order to stop the normal form procedure. If one stops the normal form procedure at this order, the remainder becomes exponentially small.
Nous investiguons la convergence/divergence de la forme normale d’une singularité d’un champ de vecteurs de n avec une partie linéaire nilpotente. Nous prouvons que chaque champ de vecteurs Gevrey-α avec une partie linéaire nilpotente peut être réduit à une forme normale Gevrey-1+α en utilisant une transformation Gevrey-1+α. Nous prouvons également que si on arrête la procédure de normalisation à un certain ordre optimal, le reste de la forme normale devient exponentiellement petit.
Classification:  37G05,  34C20,  37C10
Keywords: normal forms, nilpotent linear part, representation theory, Gevrey normalization
     author = {Bonckaert, Patrick and Verstringe, Freek},
     title = {Normal forms with exponentially small remainder and Gevrey normalization for vector fields with a nilpotent linear part},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {62},
     number = {6},
     year = {2012},
     pages = {2211-2225},
     doi = {10.5802/aif.2747},
     zbl = {1278.37044},
     mrnumber = {3060756},
     language = {en},
     url = {}
Normal forms with exponentially small remainder and Gevrey normalization for vector fields with a nilpotent linear part. Annales de l'Institut Fourier, Volume 62 (2012) no. 6, pp. 2211-2225. doi : 10.5802/aif.2747.

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