Ecalle's averages, Rota-Baxter algebras and the construction of moulds

TL;DR

This paper explores how Rota-Baxter algebras and Atkinson's recursion can be used to understand Ecalle's well-behaved averages theory through character factorization in Hopf algebras.

Contribution

It demonstrates how Atkinson recursion offers an alternative approach to Ecalle's character constructions within the framework of Rota-Baxter algebras.

Findings

Atkinson recursion retrieves Ecalle's characters

Rota-Baxter algebras connect to Ecalle's averages

Character factorization is central to the theory

Abstract

Rota-Baxter algebras and Atkinson's method are powerful tools for the factorization of characters on Hopf algebras. The theory of real resummation discovered by J. Ecalle and known as \textit{well-behaved averages theory} can be reformulated in terms of character factorization. The aim of this article is to explain how Atkinson recursion provides an alternative way to retrieve characters already discovered by Ecalle.