Smooth linearization of commuting circle diffeomorphisms

Fayad, Bassam; Khanin, Konstantin

doi:10.4007/annals.2009.170.961

Cited by 34 publications

(18 citation statements)

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“…In this paper, we determine a Baire-dense set of rotation numbers α such that if f and g are commuting C ∞ -diffeomorphisms, with f of rotation number α, then f and g are accumulated in the C ∞ norm by commuting C ∞ -diffeomorphisms that are C ∞ -conjugated to a rotation. This result is related to a theorem of Fayad and Khanin [6]. They showed that if (α, α ) are simultaneously Diophantine (i.e.…”

Section: Introductionmentioning

confidence: 69%

Circle diffeomorphisms: quasi-reducibility and commuting diffeomorphisms

Benhenda¹

2012

Nonlinearity

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In this article, we show two related results on circle diffeomorphisms. The first result is on quasi-reducibility: for a Baire-dense set of α, for any diffeomorphism f of rotation number α, it is possible to accumulate R α with a sequence h n f h −1 n , h n being a diffeomorphism. The second result is: for a Baire-dense set of α, given two commuting diffeomorphisms f and g, such that f has α for rotation number, it is possible to approach each of them by commuting diffeomorphisms f n and g n that are differentiably conjugated to rotations.In particular, it implies that if α is in this Baire-dense set, and if β is an irrational number such that (α, β) are not simultaneously Diophantine, then the set of commuting diffeomorphisms ( f, g) with singular conjugacy, and with rotation numbers (α, β) respectively, is C ∞ -dense in the set of commuting diffeomorphisms with rotation numbers (α, β).

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Section: Introductionmentioning

confidence: 69%

Circle diffeomorphisms: quasi-reducibility and commuting diffeomorphisms

Benhenda¹

2012

Nonlinearity

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“…In the C ∞ case, J. C. Yoccoz [14] extended this result to all Diophantine rotation numbers. Results in analytic class and in finite differentiability class subsequently enriched the global theory of circle diffeomorphisms [11,9,8,13,7,15,4,10]. In the perturbative theory, KAM theorems usually provide a bound on the norm of the conjugacy that involves the norm of the perturbation and the Diophantine constants of the number α (see [5,12,3] for example).…”

Section: Introductionmentioning

confidence: 99%

Estimates of the linearization of circle diffeomorphisms

Benhenda¹

2014

Bul. Soc. Math. France

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A celebrated theorem by Herman and Yoccoz asserts that if the rotation number α of a C ∞ -diffeomorphism of the circle f satisfies a Diophantine condition, then f is C ∞ -conjugated to a rotation. In this paper, we establish explicit relationships between the C k norms of this conjugacy and the Diophantine condition on α. To obtain these estimates, we follow a suitably modified version of Yoccoz's proof.

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“…Recently, Fayad and Khanin [6] proved that the related global theory of linearization for commuting circle diffeomorphisms is also available. Recently, Fayad and Khanin [6] proved that the related global theory of linearization for commuting circle diffeomorphisms is also available.…”

Section: Introductionmentioning

confidence: 99%