There is only one KAM curve

Carminati, Carlo; Marmi, Stefano; Sauzin, David

doi:10.1088/0951-7715/27/9/2035

Cited by 8 publications

(21 citation statements)

References 22 publications

(49 reference statements)

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“…Like [MS03], [CM08], [MS11] or [CMS14], we will follow Herman to construct compact subsets of C on which regularity will be investigated. The strategy consists in defining, for each M > 0, a compact K M of C which intersects the unit circle along { q ∈ B | B(q) ≤ M } and a complex Banach space B M such that q ∈ K M → h q ∈ B M can be proved to be tame enough.…”

Section: -Holomorphic and Monogenic Functionsmentioning

confidence: 99%

“…This is a Banach space norm equivalent to the one indicated in [He85] or [MS03] (or to the one indicated in [CMS14], which is designed to be a Banach algebra norm whenever B is a Banach algebra 9 ). As usual, we simply denote by O(K) and C 1 hol (K) the spaces obtained when B = C. Here are the direct estimates of the norms of the functions q → S T (q) we have alluded to earlier: Proposition 8.2.…”

Section: Proof Of Theorem Cmentioning

confidence: 99%

“…" in[MS11] (p. 61) or the sets "K M " of[CMS14]; see the pictures there. Observe that the Haar measure of {|q| = 1} \ K M in the unit circle tends to 0 as M → ∞, and that the two-dimensional Lebesgue measure of C \ K M also tends to 0.The corresponding target spaces B M will be of the form H ∞ (D r ) for some real r > 0, with the notationsD r := { z ∈ C | |z| < r } and, for any open subset D of C, H ∞ (D) := {bounded holomorphic functions of D} (Banach space for the sup norm).…”

mentioning

confidence: 99%

“…Moreover, for any interior point q0 of C, the complex derivative of ψ at q0 exists and coincides with ψ (q0). 9 One gets a slighly simpler Banach algebra norm than in[CMS14] by taking a sum instead of a max in (56).…”

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confidence: 99%

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Explicit linearization of one-dimensional germs through tree-expansions

Fauvet¹,

Menous²,

Sauzin³

2018

Bul. Soc. Math. France

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We explainÉcalle's "arbomould formalism" in its simplest instance, showing how it allows one to give explicit formulas for the operators naturally attached to a germ of holomorphic map in one dimension. When applied to the classical linearization problem of non-resonant germs, which contains the well-known difficulties due to the so-called small divisor phenomenon, this elegant and concise tree formalism yields compact formulas, from which one easily recovers the classical analytical results of convergence of the solution under suitable arithmetical conditions on the multiplier. We rediscover this way Yoccoz's lower bound for the radius of convergence of the linearization and can even reach a global regularity result with respect to the multiplier (C 1-holomorphy) which improves on Carminati-Marmi's result.

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Section: -Holomorphic and Monogenic Functionsmentioning

confidence: 99%

Section: Proof Of Theorem Cmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Explicit linearization of one-dimensional germs through tree-expansions

Fauvet¹,

Menous²,

Sauzin³

2018

Bul. Soc. Math. France

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“…For a closed subset K of C, we denote by O (K, B) the space of all B-valued functions which are continuous on K and holomorphic in the interior of K; and we denote by C 1 hol (K, B) ⊂ O(K, B) the Banach space of all B-valued functions which are C 1 -holomorphic on K (i.e. Whitneydifferentiable in the complex sense-see [MS03] or [CMS14] for a precise definition).…”

Section: Poincaré Simple Pole Series and Monogenic Regularitymentioning

confidence: 99%

Generalised continuation by means of right limits

Sauzin

Tiozzo

2017

JAMA

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Several theories have been proposed to generalise the concept of analytic continuation to holomorphic functions of the disc for which the circle is a natural boundary. Elaborating on Breuer-Simon's work on right limits of power series, Baladi-Marmi-Sauzin recently introduced the notion of renascent right limit and rrl-continuation.We discuss a few examples and consider particularly the classical example of Poincaré simple pole series in this light. These functions are represented in the disc as series of infinitely many simple poles located on the circle; they appear for instance in small divisor problems in dynamics. We prove that any such function admits a unique rrl-continuation, which coincides with the function obtained outside the disc by summing the simple pole expansion. We also discuss the relation with monogenic regularity in the sense of Borel.

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On the divergence of Birkhoff Normal Forms

Krikorian

2022

Publ.math.IHES

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It is well known that a real analytic symplectic diffeomorphism of the $2d$ 2 d -dimensional disk ($d\geq 1$ d ≥ 1 ) admitting the origin as a non-resonant elliptic fixed point can be formally conjugated to its Birkhoff Normal Form, a formal power series defining a formal integrable symplectic diffeomorphism at the origin. We prove in this paper that this Birkhoff Normal Form is in general divergent. This solves, in any dimension, the question of determining which of the two alternatives of Pérez-Marco’s theorem (Ann. Math. (2) 157:557–574, 2003) is true and answers a question by H. Eliasson. Our result is a consequence of the fact that when $d=1$ d = 1 the convergence of the formal object that is the BNF has strong dynamical consequences on the Lebesgue measure of the set of invariant circles in arbitrarily small neighborhoods of the origin. Our proof, as well as our results, extend to the case of real analytic diffeomorphisms of the annulus admitting a Diophantine invariant torus.

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There is only one KAM curve

Cited by 8 publications

References 22 publications

Explicit linearization of one-dimensional germs through tree-expansions

Explicit linearization of one-dimensional germs through tree-expansions

Generalised continuation by means of right limits

On the divergence of Birkhoff Normal Forms

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