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Linearization of analytic and non-analytic germs of diffeomorphisms of (,0). (English) Zbl 0997.37017
The authors study Siegel’s center problem on the linearization of germs of diffeomorphisms in one complex variable. They first study the cases when the linearization operator is formal or analytic, and then give sufficient conditions for this operator to belong to a certain algebra of ultradifferentiable functions that includes the Gevrey functions. In the analytic case they give a direct proof (not using renormalization) of J.-C. Yoccoz’s result [Small divisors in dimension one (French), Astérisque 231, 3-88 (1995; Zbl 0836.30001)] on the optimality of the estimates obtained using the majorant series method. In the ultradifferentiable case they show that Bryuno’s generalization of the Diophantine condition is sufficient for the linearization to belong to the same class as the germ. If the linearization is less regular than the germ, the authors obtain new conditions weaker than the Bryuno condition.

MSC:
37F50Small divisors, rotation domains and linearization; Fatou and Julia sets
37F25Renormalization
37G05Normal forms
30C99Geometric function theory