# Linearization of germs: regular dependence on the multiplier

Stefano Marmi; Carlo Carminati

Bulletin de la Société Mathématique de France (2008)

- Volume: 136, Issue: 4, page 533-564
- ISSN: 0037-9484

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topMarmi, Stefano, and Carminati, Carlo. "Linearization of germs: regular dependence on the multiplier." Bulletin de la Société Mathématique de France 136.4 (2008): 533-564. <http://eudml.org/doc/272353>.

@article{Marmi2008,

abstract = {We prove that the linearization of a germ of holomorphic map of the type $F_\lambda (z)=\lambda (z+O(z^2))$ has a $\{\mathcal \{C\}\}^1$-holomorphic dependence on the multiplier $\lambda $. $\{\mathcal \{C\}\}^1$-holomorphic functions are $\{\mathcal \{C\}\}^1$-Whitney smooth functions, defined on compact subsets and which belong to the kernel of the $\bar\{\partial \}$ operator.
The linearization is analytic for $|\lambda |\ne 1$ and the unit circle $\mathbb \{S\}^1$ appears as a natural boundary (because of resonances,i.e.roots of unity). However the linearization is still defined at most points of $\{\mathbb \{S\}\}^1$, 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 $\{\mathcal \{C\}\}^1$-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.},

author = {Marmi, Stefano, Carminati, Carlo},

journal = {Bulletin de la Société Mathématique de France},

keywords = {small divisors; linearization; monogenic functions; quasianalytic space; asymptotic expansion; diophantine condition},

language = {eng},

number = {4},

pages = {533-564},

publisher = {Société mathématique de France},

title = {Linearization of germs: regular dependence on the multiplier},

url = {http://eudml.org/doc/272353},

volume = {136},

year = {2008},

}

TY - JOUR

AU - Marmi, Stefano

AU - Carminati, Carlo

TI - Linearization of germs: regular dependence on the multiplier

JO - Bulletin de la Société Mathématique de France

PY - 2008

PB - Société mathématique de France

VL - 136

IS - 4

SP - 533

EP - 564

AB - We prove that the linearization of a germ of holomorphic map of the type $F_\lambda (z)=\lambda (z+O(z^2))$ has a ${\mathcal {C}}^1$-holomorphic dependence on the multiplier $\lambda $. ${\mathcal {C}}^1$-holomorphic functions are ${\mathcal {C}}^1$-Whitney smooth functions, defined on compact subsets and which belong to the kernel of the $\bar{\partial }$ operator.
The linearization is analytic for $|\lambda |\ne 1$ and the unit circle $\mathbb {S}^1$ appears as a natural boundary (because of resonances,i.e.roots of unity). However the linearization is still defined at most points of ${\mathbb {S}}^1$, 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 ${\mathcal {C}}^1$-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.

LA - eng

KW - small divisors; linearization; monogenic functions; quasianalytic space; asymptotic expansion; diophantine condition

UR - http://eudml.org/doc/272353

ER -

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