We describe the moduli space of germs of generic families of analytic diffeomorphisms which unfold a parabolic fixed point of codimension 1. A complete modulus is given by unfolding the Écalle–Voronin modulus over a sector of opening greater than 2π in the canonical parameter ϵ. In the region of overlap (Glutsyuk sector of parameter space) where the two fixed points are connected by orbits, we identify the necessary compatibility between the two representatives of the modulus. The compatibility condition implies the existence of a normalization for which the modulus is -summable in ϵ, non-summability occurring in the direction of real multipliers of the fixed points. We show that the compatibility condition together with the summability is sufficient for realization of the modulus.
On donne l'espace des modules des germes de familles génériques de difféomorphismes analytiques déployant un point fixe parabolique de codimension 1. Un module complet est donné par le déploiement du module d'Écalle–Voronin sur un secteur d'ouverture plus grande que 2π du paramètre canonique. Dans le sous-secteur recouvert deux fois (sous-secteur Glutsyuk), là où les deux points fixes sont connectés par des orbites, on identifie une condition de compatibilité nécessaire satisfaite par les deux représentants du module. Cette condition implique l'existence d'une normalisation sous laquelle le module est -sommable en ϵ, la non-sommabilité se produisant dans la direction des multiplicateurs réels aux points fixes. On montre que la condition de compatibilité, jointe à cette propriété de sommabilité, est suffisante pour réaliser le module.
Accepted:
Published online:
Colin Christopher 1; Christiane Rousseau 2
@article{CRMATH_2007__345_12_695_0,
author = {Colin Christopher and Christiane Rousseau},
title = {The moduli space of germs of generic families of analytic diffeomorphisms unfolding a parabolic fixed point},
journal = {Comptes Rendus. Math\'ematique},
pages = {695--698},
publisher = {Elsevier},
volume = {345},
number = {12},
year = {2007},
doi = {10.1016/j.crma.2007.10.033},
language = {en},
}
TY - JOUR AU - Colin Christopher AU - Christiane Rousseau TI - The moduli space of germs of generic families of analytic diffeomorphisms unfolding a parabolic fixed point JO - Comptes Rendus. Mathématique PY - 2007 SP - 695 EP - 698 VL - 345 IS - 12 PB - Elsevier DO - 10.1016/j.crma.2007.10.033 LA - en ID - CRMATH_2007__345_12_695_0 ER -
%0 Journal Article %A Colin Christopher %A Christiane Rousseau %T The moduli space of germs of generic families of analytic diffeomorphisms unfolding a parabolic fixed point %J Comptes Rendus. Mathématique %D 2007 %P 695-698 %V 345 %N 12 %I Elsevier %R 10.1016/j.crma.2007.10.033 %G en %F CRMATH_2007__345_12_695_0
Colin Christopher; Christiane Rousseau. The moduli space of germs of generic families of analytic diffeomorphisms unfolding a parabolic fixed point. Comptes Rendus. Mathématique, Volume 345 (2007) no. 12, pp. 695-698. doi : 10.1016/j.crma.2007.10.033. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2007.10.033/
[1] Confluence of singular points and nonlinear Stokes phenomenon, Trans. Moscow Math. Soc., Volume 62 (2001), pp. 49-95
[2] Modulus of analytic classification for unfoldings of generic parabolic diffeomorphisms, Moscow Math. J., Volume 4 (2004), pp. 455-502
[3] Les séries k-sommables et leurs applications, Complex Analysis, Microlocal Calculus and Relativistic Quantum Theory, Proc. Internat. Colloq. Centre Phys., Les Houches, 1979 (Lecture Notes in Phys.), Volume vol. 126, Springer, Berlin, New York (1980), pp. 178-199 (in French)
[4] Hukuhara's domains and fundamental existence and uniqueness theorems for asymptotic solutions of Gevrey type, Asymptotic Anal., Volume 2 (1989), pp. 39-94
[5] X. Ribon, Modulus of analytic classification for unfoldings of resonant diffeomorphisms, Moscow Math. J., in press
Cited by Sources:
⁎ This work is supported by NSERC in Canada.
Comments - Policy