Ecalle's arborification-coarborification transforms and Connes-Kreimer Hopf algebra

2012/12/19 by Fauvet, Frédéric, Menous, Frederic · 5 citations
#Combinatorics (math.CO) #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.1212.4740

Abstract

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 algebras. Finally, we introduce a new Hopf algebra that systematically underlies the calculations for the normalization of local dynamical systems.

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• Explicit linearization of multi-dimensional germs and vector fields through Ecalle's tree expansions
• Construction of pre-and post-Lie algebras for stochastic PDEs

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