Small divisors and large multipliers
Annales de l'Institut Fourier, Volume 57 (2007) no. 2, p. 603-628

We study germs of singular holomorphic vector fields at the origin of n of which the linear part is 1-resonant and which have a polynomial normal form. The formal normalizing diffeomorphism is usually divergent at the origin but there exists holomorphic diffeomorphisms in some “sectorial domains” which transform these vector fields into their normal form. In this article, we study the interplay between the small divisors phenomenon and the Gevrey character of the sectorial normalizing diffeomorphisms. We show that the Gevrey order of the latter is linked to the diophantine type of the small divisors.

Nous étudions des germes de champs de vecteurs holomorphes singuliers à l’origine de n dont la partie linéaire est 1-résonante et qui admettent une forme normale polynomiale. En général, bien que le difféomorphisme formel normalisant soit divergent à l’origine, il existe néanmoins des difféomorphismes holomorphes dans des “domaines sectoriels” qui les transforment en leur forme normale. Dans cet article, nous étudions la relation qui existe entre le phénomène de petits diviseurs et le caractère Gevrey de ces difféomorphismes sectoriels normalisants. Nous montrons que l’ordre Gevrey de ce dernier est relié au type diophantien des petits diviseurs.

DOI : https://doi.org/10.5802/aif.2269
Classification:  34M30,  34M40,  32S65,  37F75,  37J40,  37J30,  70K45,  70K30
Keywords: Holomorphic dynamics, small divisors, normal forms, Gevrey functions, divergent series
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     author = {Braaksma, Boele and Stolovitch, Laurent},
     title = {Small divisors and large multipliers},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {57},
     number = {2},
     year = {2007},
     pages = {603-628},
     doi = {10.5802/aif.2269},
     mrnumber = {2310952},
     zbl = {1138.37028},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2007__57_2_603_0}
}
Braaksma, Boele; Stolovitch, Laurent. Small divisors and large multipliers. Annales de l'Institut Fourier, Volume 57 (2007) no. 2, pp. 603-628. doi : 10.5802/aif.2269. http://www.numdam.org/item/AIF_2007__57_2_603_0/

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