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

Timoteo Carletti, Stefano Marmi

Research output: Contribution to journalArticle

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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

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Diffeomorphisms
Linearization
Optimality
Gevrey Classes
Center Problem
Sufficient
Closed
Algebra
Series
Sufficient Conditions
Estimate

Keywords

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

Cite this

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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.",
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Linearization of Analytic and Non-Analytic Germs of Diffeomorphisms of (C,0). / Carletti, Timoteo; Marmi, Stefano.

In: Bulletin SMF, Vol. 128, 2000, p. 69-85.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Linearization of Analytic and Non-Analytic Germs of Diffeomorphisms of (C,0)

AU - Carletti, Timoteo

AU - Marmi, Stefano

PY - 2000

Y1 - 2000

N2 - 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.

AB - 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.

KW - Siegel's center problem

KW - small divisors

KW - Gevrey classes

M3 - Article

VL - 128

SP - 69

EP - 85

JO - Bulletin SMF

JF - Bulletin SMF

ER -