Ecalle's arborification-coarborification transforms and Connes-Kreimer Hopf algebra
[Les transformations d'arborification-coarborification d'Ecalle et l'algèbre de Hopf de Connes-Kreimer]
Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 50 (2017) no. 1, pp. 39-83.

Nous donnons une description complète et naturelle du formalisme d'arborification/coarborification d'Ecalle en termes d'algèbres de Hopf. L'arborification apparaît alors comme une factorisation de caractères, impliquant les algèbres shuffle ou quasishuffle, en vertu d'une propriété universelle satisfaite par l'algèbre de Connes-Kreimer. Dans ce cadre, nous obtenons de façon directe le procédé fondamental de coarborification homogène, en utilisant la dualité explicite entre les algèbres de Hopf décorées de Connes-Kreimer et Grossman-Larson. Enfin, nous introduisons une nouvelle algèbre de Hopf qui est sous-jacente aux calculs de normalisation des systèmes dynamiques locaux.

We give a natural and complete description of Ecalle's mould-comould formalism within a Hopf-algebraic framework. The arborification transform thus appears as a factorization of characters, involving the shuffle or quasishuffle Hopf algebras, thanks to a universal property satisfied by Connes-Kreimer Hopf algebra. We give a straightforward characterization of the fundamental process of homogeneous coarborification, using the explicit duality between decorated Connes-Kreimer and Grossman-Larson Hopf algebras. Finally, we introduce a new Hopf algebra that systematically underlies the calculations for the normalization of local dynamical systems.

Publié le :
DOI : 10.24033/asens.2315
Classification : 05E05, 16T05, 34M35.
Keywords: Dynamical systems, normal forms, Hopf algebras, trees, Faà di Bruno, moulds, arborification, coarborification.
Mot clés : Systèmes dynamiques, formes normales, algèbres de Hopf, arbres, Faà di Bruno, moules, arborification, coarborification. 
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     author = {Fauvet, Fr\'ed\'eric and Menous, Fr\'ed\'eric},
     title = {Ecalle's arborification-coarborification transforms and {Connes-Kreimer}  {Hopf} algebra},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {39--83},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
     volume = {Ser. 4, 50},
     number = {1},
     year = {2017},
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Fauvet, Frédéric; Menous, Frédéric. Ecalle's arborification-coarborification transforms and Connes-Kreimer  Hopf algebra. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 50 (2017) no. 1, pp. 39-83. doi : 10.24033/asens.2315. http://www.numdam.org/articles/10.24033/asens.2315/

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