Linearization of germs: regular dependence on the multiplier

Stefano Marmi, Carlo Carminati
2008 Bulletin de la Société Mathématique de France  
We prove that the linearization of a germ of holomorphic map of the type F λ (z) = λ(z + O(z 2 )) has a C 1 -holomorphic dependence on the multiplier λ. C 1holomorphic functions are C 1 -Whitney smooth functions, defined on compact subsets and which belong to the kernel of the∂ operator. The linearization is analytic for |λ| = 1 and the unit circle S 1 appears as a natural boundary (because of resonances, i.e. roots of unity). However the linearization is still defined at most points of S 1 ,
more » ... t points of S 1 , namely those points which lie "far enough from resonances", i.e. when the multiplier satisfies a suitable arithmetical condition. We construct an increasing sequence of compacts which avoid resonances and prove that the linearization belongs to the associated spaces of C 1 -holomorphic functions. This is a special case of Borel's theory of uniform monogenic functions [2] , and the corresponding function space is arcwise-quasianalytic [11] . Among the consequences of these results, we can prove that the linearization admits an asymptotic expansion w.r.t. the multiplier at all points of the unit circle verifying the Brjuno condition: in fact the asymptotic expansion is of Gevrey type at diophantine points. Texte recu le 28 juin 2007, révisé et accepté le 27 février 2008
doi:10.24033/bsmf.2565 fatcat:ltovig672jhlnolslwciwf676y