Linéarisation des perturbations holomorphes des rotations et applications

Risler, Emmanuel

doi:10.24033/msmf.390

Cited by 26 publications

(40 citation statements)

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“…In this appendix we recall the definition of C 1 -holomorphic and C ∞ -holomorphic functions as they are given respectively in [7] and [13]. We follow quite closely Section 2 of [10] to which we refer for a more detailed discussion.…”

Section: Note Thatmentioning

confidence: 99%

“…To avoid this problem we use an idea of [13]: we will prove that h ∈ C ∞ hol (C * , H ∞ (Dρ)) for some ρ > 0, where this time C * will be somewhat smaller than the set CM considered in the previous section.…”

Section: Higher Regularitymentioning

confidence: 99%

“…It is Herman's point of view which was developed in [10] and which we will summarize in Appendix B, where we recall the formal definition of C 1 -holomorphic and C ∞ -holomorphic functions. Later Risler [13] extended considerably part of Herman's work proving various regularity results under less restrictive arithmetical conditions, namely using the Brjuno conditon as in [17] instead of a more classical diophantine condition. One should also mention that Whitney smooth dependence on parameters has been established also in the more general framework of KAM theory by Pöschel [12] who did not however consider neither complex frequencies nor Brjuno numbers.…”

Section: Introductionmentioning

confidence: 99%

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Linearization of germs: regular dependence on the multiplier

Marmi

Carminati²

2008

Bul. Soc. Math. France

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We prove that the linearization of a germ of holomorphic map of the type F λ (z) = λ(z + O(z 2 )) has a C 1 -holomorphic dependence on the multiplier λ. C 1 -holomorphic functions are C 1 -Whitney smooth functions, defined on compact subsets and which belong to the kernel of the∂ operator. The linearization is analytic for |λ| = 1 and the unit circle S 1 appears as a natural boundary (because of resonances, i.e. roots of unity). However the linearization is still defined at most points of S 1 , namely those points which lie "far enough from resonances", i.e. when the multiplier satisfies a suitable arithmetical condition. We construct an increasing sequence of compacts which avoid resonances and prove that the linearization belongs to the associated spaces of C 1 -holomorphic functions. This is a special case of Borel's theory of uniform monogenic functions [2], and the corresponding function space is arcwise-quasianalytic [11]. Among the consequences of these results, we can prove that the linearization admits an asymptotic expansion w.r.t. the multiplier at all points of the unit circle verifying the Brjuno condition: in fact the asymptotic expansion is of Gevrey type at diophantine points.

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Section: Note Thatmentioning

confidence: 99%

Section: Higher Regularitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Linearization of germs: regular dependence on the multiplier

Marmi

Carminati²

2008

Bul. Soc. Math. France

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“…Here, in contrast with Problem (S), we do not try to reach the optimal arithmetical condition, which is known to be the Bruno condition (11) as in the Siegel problem, by Risler's result based on renormalization-see [Ri99], [MY02]. We content ourselves with I (C) , which has still full measure in S.…”

Section: Linearizing the Conjugacy Equation (10) Written Asmentioning

confidence: 99%

“…Of course, to speak of complex rotation number, we need a generalization with respect to the classical theory of circle diffeomorphisms (in which g(θ) is assumed to be real for real values of θ and only real values of α and ε are considered)-see [Ri99] for a geometric insight on this generalization. Instead of solving first equation (18) and then the conjugacy equation (17), it is possible to obtain directly the pair (β, h) by rewriting the conjugacy equation as h(θ + α) − h(θ) − α + β = εg h(θ) and defining the operator…”

Section: Linearizing the Conjugacy Equation (10) Written Asmentioning

confidence: 99%

A quasianalyticity property for monogenic solutions of small divisor problems

Marmi

Sauzin

2011

Bull Braz Math Soc, New Series

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We discuss the quasianalytic properties of various spaces of functions suitable for one-dimensional small divisor problems. These spaces are formed of functions C 1 -holomorphic on certain compact sets K j of the Riemann sphere (in the Whitney sense), as is the solution of a linear or non-linear small divisor problem when viewed as a function of the multiplier (the intersection of K j with the unit circle is defined by a Diophantine-type condition, so as to avoid the divergence caused by roots of unity). It turns out that a kind of generalized analytic continuation through the unit circle is possible under suitable conditions on the K j 's. IntroductionFollowing V. Arnold and M. Herman, and in the same line of research as in [MS03], we consider "monogenic functions" in the sense ofÉmile Borel with a view to small divisor problems. In these problems of dynamical origin, there is a complex parameter q, called multiplier, which must be kept off the roots of unity in order to solve a functional equation; typically, q is the eigenvalue at a fixed point of a onedimensional complex map that one wants to linearize and one studies the equation (corresponding to the so-called Siegel problem)(where G(z) ∈ zC{z} is given, with G ′ (0) = 1, and h(q, . ) is sought in a Banach space of functions holomorphic in the variable z), or the linearized equation h(q, qz)− qh(q, z) = qg(z) (with g(z) ∈ z 2 C{z} given), or the more complicated non-linear equation corresponding to the conjugacy between a circle map and a rigid rotation with rotation number 1 2πi log q (see equation (2)). We are interested in the dependence of the solution on the multiplier q. Roots of unity act as resonances, because the coefficients of the solution of the problem are inductively defined by expressions which involve division by q k − 1, k ≥ 1. On the other hand the case where |q| = 1 is particularly interesting from the dynamical point of view. One is thus led to define compact sets K j of the Riemann sphere C by removing smaller and smaller neighbourhoods of the roots of unity. It is shown in [He85] and [CM08] for the above-mentioned non-linear problems and in [MS03] for the linear one, that the solution is Whitney smooth on the K j 's, which gives rise to an example of "monogenic" function (the definition is recalled in Section 2). In all the cases we consider, the union of the K j 's on which our monogenic functions are defined contains C \ S, where S denotes the unit circle, and also a subset of S defined

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Analytic linearization of circle diffeomorphisms

Yoccoz

2004

Lecture Notes in Mathematics

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Linéarisation des perturbations holomorphes des rotations et applications

Cited by 26 publications

References 8 publications

Linearization of germs: regular dependence on the multiplier

Linearization of germs: regular dependence on the multiplier

A quasianalyticity property for monogenic solutions of small divisor problems

Analytic linearization of circle diffeomorphisms

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