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

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 $\mathit{\text{well-behaved}}\mathit{\text{averages}}\mathit{\text{theory}}$ 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.