The resurgent character of the Fatou coordinates of a simple parabolic germ

TL;DR

This paper revisits the construction of Fatou coordinates for simple parabolic germs, providing a new proof of their resurgent nature using Borel-Laplace summation, which is crucial for complex dynamics classification.

Contribution

It offers an original, self-contained proof of the resurgent character of Fatou coordinates, enhancing understanding of local dynamics near parabolic fixed points.

Findings

Fatou coordinates are shown to be resurgent functions.

The proof relies on Borel-Laplace summation techniques.

Results deepen the understanding of parabolic germ dynamics.

Abstract

Given a holomorphic germ at the origin of C with a simple parabolic fixed point, the local dynamics is classically described by means of pairs of attracting and repelling Fatou coordinates and the corresponding pairs of horn maps, of crucial importance for \'Ecalle-Voronin's classification result and the definition of the parabolic renormalization operator. We revisit \'Ecalle's approach to the construction of Fatou coordinates, which relies on Borel-Laplace summation, and give an original and self-contained proof of their resurgent character.