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Comptes Rendus Mathématique
Volume 352, n° 3
pages 255-261 (mars 2014)
Doi : 10.1016/j.crma.2013.12.005
Received : 29 August 2013 ;  accepted : 6 December 2013
The resurgent character of the Fatou coordinates of a simple parabolic germ
Résurgence des coordonnées de Fatou des germes paraboliques simples
 

Artem Dudko a, David Sauzin b
a Institute for Mathematical Sciences, University of Stony Brook, NY, USA 
b CNRS UMI 3483, Laboratorio Fibonacci, Centro di Ricerca Matematica Ennio De Giorgi, Scuola Normale Superiore di Pisa, Italy 

Abstract

Given a holomorphic germ at the origin of   with a simple parabolic fixed point, the local dynamics is classically described by means of pairs of attracting and repelling Fatou coordinates and the corresponding pairs of horn maps, of crucial importance for Écalle-Voronin's classification result and the definition of the parabolic renormalization operator. We revisit Écalle's approach to the construction of Fatou coordinates, which relies on Borel–Laplace summation, and give an original and self-contained proof of their resurgent character.

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Résumé

Pour un germe holomorphe à l'origine de   avec un point fixe parabolique simple, la dynamique locale se décrit classiquement à l'aide d'une paire de coordonnées de Fatou, qui permet de définir une paire d'applications de corne, cruciale pour le résultat de classification analytique d'Écalle–Voronin et pour la définition de l'opérateur de renormalisation parabolique. Nous revisitons l'approche d'Écalle fondée sur la sommation de Borel–Laplace pour construire les coordonnées de Fatou et donnons une démonstration originale complète de leur caractère résurgent.

The full text of this article is available in PDF format.
1  Notice that, for   large enough,   is close to 1, hence   where log is the principal branch of the logarithm.
2  The formal Borel transform   maps the Cauchy product of formal series to the convolution product:  , with termwise integration for formal series  , and with the obvious analytical meaning when   (then taking ζ close enough to 0).


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