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Linearization of analytic and non-analytic germs of diffeomorphisms of \((\mathbb C,0)\). (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:

37F50 Small divisors, rotation domains and linearization in holomorphic dynamics
37F25 Renormalization of holomorphic dynamical systems
37G05 Normal forms for dynamical systems
30C99 Geometric function theory

Citations:

Zbl 0836.30001
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References:

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