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

2000/03/17 by Timotéo Carletti, T. Carletti, Carletti, T. +3
Mathematics · #05C38 #15A15 (Primary) 05A15 #15A18 (Secondary) #Advanced Algebra and Geometry #Advanced Differential Equations and Dynamical Systems #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #math.DS #msc:05A15 #msc:05C38 #msc:15A15 #msc:15A18

paper · pdf · doi:10.48550/arxiv.math/0003105

AMS-Latex2e, 11 pages, in press Bulletin Societe Mathematique de France

arxiv created 2000/03/17 · openalex publication_date 2000/03/17 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study Siegel's center problem on the linearization of germs of diffeomorphisms in one variable. In addition of the classical problems of formal and analytic linearization, we give sufficient conditions for the linearization to belong to some algebras of ultradifferentiable germs closed under composition and derivation, including Gevrey classes. In the analytic case we give a positive answer to a question of J.-C. Yoccoz on the optimality of the estimates obtained by the classical majorant series method. In the ultradifferentiable case we prove that the Brjuno condition is sufficient for the linearization to belong to the same class of the germ. If one allows the linearization to be less regular than the germ one finds new arithmetical conditions, weaker than the Brjuno condition. We briefly discuss the optimality of our results.

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