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.