Linearization of germs: regular dependence on the multiplier

We prove that the linearization of a germ of holomorphic map of the type Fλ(z) = λ(z + O(z2)) has a C1–holomorphic dependence on the multiplier λ. C1–holomorphic functions are C1–Whitney smooth functions, defined on compact subsets and which belong to the kernel of the ¯∂ operator. The linearization is analytic for |λ| 6= 1 and the unit circle S1 appears as a natural boundary (because of resonances, i.e. roots of unity). However the linearization is still defined at most points of S1, 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 C1–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.