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

Abstract

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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.

How to cite

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Marmi, 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 -

References

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