Mould calculus: from primary to secondary mould symmetries

Abstract

This article deals with a part of mould calculus, a powerful combinatorial environment developed by J. Ecalle in the 80s. Its main goal is to give a complete introduction to the secondary mould symmetries, as well as to develop the path from the primary symmetries to the secondary symmetries.

We first present in §2 all the classical results on moulds and prove them completely in §3. Then, we introduce the secondary symmetries in §4 and extend to them the Hopf algebraic interpretation of mould calculus. Using this, we finally give in §5 the path to go from primary to secondary symmetries.