[Linéarisation explicite de germes unidimensionnels par développement en arbres]
We explain Écalle’s “arbomould formalism” in its simplest instance, showing how it allows one to give explicit formulas for the operators naturally attached to a germ of holomorphic map in one dimension. When applied to the classical linearization problem of non-resonant germs, which contains the well-known difficulties due to the so-called small divisor phenomenon, this elegant and concise tree formalism yields compact formulas, from which one easily recovers the classical analytical results of convergence of the solution under suitable arithmetical conditions on the multiplier. We rediscover this way Yoccoz’s lower bound for the radius of convergence of the linearization and can even reach a global regularity result with respect to the multiplier (-holomorphy) which improves on Carminati-Marmi’s result.
Nous expliquons le formalisme des « arbomoules » d’Écalle dans le cas le plus simple et montrons comment il permet d’obtenir des formules explicites pour les opérateurs naturellement associés à un germe d’application holomorphe en une dimension. Dans le cadre du problème classique de la linéarisation des germes non résonants, qui contient la difficulté bien connue due au phénomène des petits diviseurs, ce formalisme élégant et concis reposant sur des arbres fournit des formules compactes, dont on déduit aisément les résultats analytiques classiques de convergence de la solution moyennant des conditions arithmétiques appropriées sur le multiplicateur. Nous retrouvons de cette façon la borne inférieure due à Yoccoz pour le rayon de convergence de la linéarisation et obtenons même un résultat de régularité globale par rapport au multiplicateur (-holomorphie) qui améliore un résultat de Carminati et Marmi.
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DOI : 10.24033/bsmf.2757
Keywords: Dynamical systems, linearization, small divisors, moulds, trees, arborification, monogenic functions.
Mots-clés : Systèmes dynamiques, linéarisation, petits diviseurs, moules, arbres, arborification, fonctions monogènes.
Fauvet, Frédéric 1 ; Menous, Frédéric 2 ; Sauzin, David 3
@article{BSMF_2018__146_2_241_0,
author = {Fauvet, Fr\'ed\'eric and Menous, Fr\'ed\'eric and Sauzin, David},
title = {Explicit linearization of one-dimensional germs through tree-expansions},
journal = {Bulletin de la Soci\'et\'e Math\'ematique de France},
pages = {241--285},
year = {2018},
publisher = {Soci\'et\'e math\'ematique de France},
volume = {146},
number = {2},
doi = {10.24033/bsmf.2757},
mrnumber = {3933876},
zbl = {1406.37042},
language = {en},
url = {https://www.numdam.org/articles/10.24033/bsmf.2757/}
}
TY - JOUR AU - Fauvet, Frédéric AU - Menous, Frédéric AU - Sauzin, David TI - Explicit linearization of one-dimensional germs through tree-expansions JO - Bulletin de la Société Mathématique de France PY - 2018 SP - 241 EP - 285 VL - 146 IS - 2 PB - Société mathématique de France UR - https://www.numdam.org/articles/10.24033/bsmf.2757/ DO - 10.24033/bsmf.2757 LA - en ID - BSMF_2018__146_2_241_0 ER -
%0 Journal Article %A Fauvet, Frédéric %A Menous, Frédéric %A Sauzin, David %T Explicit linearization of one-dimensional germs through tree-expansions %J Bulletin de la Société Mathématique de France %D 2018 %P 241-285 %V 146 %N 2 %I Société mathématique de France %U https://www.numdam.org/articles/10.24033/bsmf.2757/ %R 10.24033/bsmf.2757 %G en %F BSMF_2018__146_2_241_0
Fauvet, Frédéric; Menous, Frédéric; Sauzin, David. Explicit linearization of one-dimensional germs through tree-expansions. Bulletin de la Société Mathématique de France, Tome 146 (2018) no. 2, pp. 241-285. doi: 10.24033/bsmf.2757
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