On the resurgent approach to Ecalle-Voronin's invariants

Abstract

Given a holomorphic germ at the origin of C with a simple parabolic fixed point, the Fatou coordinates have a common asymptotic expansion whose formal Borel transform is resurgent. We show how to use Ecalle's alien operators to study the singularities in the Borel plane and relate them to the horn maps, providing each of Ecalle-Voronin's invariants in the form of a convergent numerical series. The proofs are original and self-contained, with ordinary Borel summability as the only prerequisite.