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Small divisors and large multipliers

Boele BraaksmaLaurent Stolovitch — 2007

Annales de l’institut Fourier

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....

Multisummability for some classes of difference equations

Boele L. J. BraaksmaBernard F. Faber — 1996

Annales de l'institut Fourier

This paper concerns difference equations y ( x + 1 ) = G ( x , y ) where G takes values in C n and G is meromorphic in x in a neighborhood of in C and holomorphic in a neighborhood of 0 in C n . It is shown that under certain conditions on the linear part of G , formal power series solutions in x - 1 / p , p N , are multisummable. Moreover, it is shown that formal solutions may always be lifted to holomorphic solutions in upper and lower halfplanes, but in general these solutions are not uniquely determined by the formal solutions.

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