Initiation to mould calculus through the example of saddle-node singularities

TL;DR

This paper introduces Ecalle's mould calculus through the example of saddle-node singularities, providing explicit formulas for normalising series in the context of singular vector fields.

Contribution

It presents an accessible introduction to mould calculus and demonstrates its application to saddle-node singularities, highlighting the resurgent nature of the formal normalisation.

Findings

Explicit formulas for normalising series of saddle-node singularities

Demonstration of the resurgent property in the normalisation process

Application of mould calculus to singular vector fields

Abstract

This article proposes an initiation to \'Ecalle'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. This is illustrated on the case of saddle-node singularities, generated by two-dimensional vector fields which are formally conjugate to Euler's vector field $x^2\frac{\pa}{\pa x}+(x+y)\frac{\pa}{\pa y}$, and for which the formal normalisation proves to be resurgent in $1/x$.