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

TL;DR

This paper provides a Hopf-algebraic framework for Ecalle's mould-comould formalism, introducing new algebraic structures and factorizations that enhance understanding of arborification and coarborification processes in dynamical systems.

Contribution

It offers a complete Hopf-algebraic description of Ecalle's formalism, including a new Hopf algebra for normalization of local dynamical systems.

Findings

Characterization of homogeneous coarborification via duality

Factorization of characters using shuffle and quasishuffle Hopf algebras

Introduction of a new Hopf algebra for dynamical systems normalization

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.