Linearization of Analytic and Non-Analytic Germs of Diffeomorphisms of (C,0)

Timoteo Carletti, Stefano Marmi

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    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.
    Original languageEnglish
    Pages (from-to)69-85
    Number of pages17
    JournalBulletin SMF
    Volume128
    Publication statusPublished - 2000

    Keywords

    • Siegel's center problem
    • small divisors
    • Gevrey classes

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