Mould expansions for the saddle-node and resurgence monomials

Abstract

This article is an introduction to some aspects of Écalle’s mould calculus, a powerful combinatorial tool which yields surprisingly explicit formulas for the normalising series attached to an analytic germ of singular vector field or of map. This is illustrated on the case of the saddle-node, a two-dimensional vector field which is formally conjugate to Euler’s vector field x2 ∂/∂x + (x + y) ∂/∂y, and for which the formal normalisation is shown to be resurgent in 1/x. Resurgence monomials adapted to alien calculus are also described as another application of mould calculus.