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Dynamics of polynomials with disconnected Julia sets
The lagrange inversion formula on nonArchimedean fields, nonanalytical form of differential and finite difference equations
1.  Dipartimento di Matematica "U. Dini", viale Morgagni 67/A, 50134 Firenze, Italy 
We will be interested in linearization problems for germs of diffeomorphisms (Siegel center problem) and vector fields. In addition to analytic results, we give sufficient condition for the linearization to belong to some Classes of ultradifferentiable germs, closed under composition and derivation, including Gevrey Classes. We prove that Bruno's condition is sufficient for the linearization to belong to the same Class of the germ, whereas new conditions weaker than Bruno's one are introduced if one allows the linearization to be less regular than the germ. This generalizes to dimension $n> 1$ some results of [6]. Our formulation of the Lagrange inversion formula by mean of trees, allows us to point out the strong similarities existing between the two linearization problems, formulated (essentially) with the same functional equation. For analytic vector fields of $\mathbb C^2$ we prove a quantitative estimate of a previous qualitative result of [25] and we compare it with a result of [26].
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