On the resurgent approach to Ecalle-Voronin's invariants

TL;DR

This paper explores the resurgent analysis of Ecalle-Voronin invariants for holomorphic germs with parabolic fixed points, using alien calculus to relate Borel plane singularities to horn maps, resulting in convergent series representations.

Contribution

It introduces a novel application of Ecalle's alien operators to explicitly compute Ecalle-Voronin invariants via convergent series, with self-contained proofs.

Findings

Relates Borel plane singularities to horn maps.

Provides convergent series for invariants.

Uses alien calculus for explicit computations.

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.