Analytic linearization of circle diffeomorphisms

Yoccoz, Jean-Christophe

doi:10.1007/978-3-540-47928-4_3

Cited by 68 publications

(90 citation statements)

References 14 publications

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“…Again it is not difficult to show that, if [·] denotes the integer part, one has B(ω) < ∞ if and only if B(ω − [ω]) < ∞; see for instance [43]. A somewhat similar Diophantine condition has been proposed by Rüssmann [36] in the context of skew-product systems and later on applied to the study of KAM tori [35,37,38].…”

Section: Some Commentsmentioning

confidence: 89%

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Invariant curves for exact symplectic twist maps of the cylinder with Bryuno rotation numbers

Gentile¹

2015

Nonlinearity

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Since Moser's seminal work it is well known that the invariant curves of smooth nearly integrable twist maps of the cylinder with Diophantine rotation number are preserved under perturbation. In this paper we show that, in the analytic class, the result extends to Bryuno rotation numbers. First, we will show that the series expansion for the invariant curves in powers of the perturbation parameter can be formally defined, then we shall prove that the series converges absolutely in a neighbourhood of the origin. This will be achieved using multiscale analysis and renormalisation group techniques to express the coefficients of the series as sums of values which are represented graphically as tree diagrams and then exploit cancellations between terms contributing to the same perturbation order. As a byproduct we shall see that, when perturbing linear maps, the series expansion for an analytic invariant curve converges for all perturbations if and only if the corresponding rotation number satisfies the Bryuno condition.

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Section: Some Commentsmentioning

confidence: 89%

“…The Bryuno condition is known to be optimal in the case of the Siegel problem [43] and the circle diffeomorphisms [42]. It has been conjectured to be optimal also in the case of twist maps by MacKay [27].…”

Section: Introductionmentioning

confidence: 99%

Invariant curves for exact symplectic twist maps of the cylinder with Bryuno rotation numbers

Gentile¹

2015

Nonlinearity

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“…When α = ρ( f ) is irrational, Denjoy showed that f is topologically conjugated to R α . However, this conjugacy is not always differentiable (see [1,5,7,15]). The regularity of this conjugacy depends on the Diophantine properties of the rotation number α (see Yoccoz's theorem 1.1).…”

Section: Preliminariesmentioning

confidence: 99%

“…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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“…A highlight result is that a diffeomorphism with a Diophantine rotation number is smoothly linearizable (without a local condition of closeness to a rotation). The proof of the first global smooth linearization theorem given by Herman [5], as well as all subsequent proofs and generalizations ( [12], [13], [9], [10], [8], [7]), extensively used the Gauss algorithm of continued fractions that yields the best rational approximations for a real number.…”

Section: Introductionmentioning

confidence: 99%