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Linearization of germs: regular dependence on the multiplier

2008
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Bulletin de la Société Mathématique de France
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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 ,

doi:10.24033/bsmf.2565
fatcat:ltovig672jhlnolslwciwf676y