The Fatou coordinate for parabolic Dulac germs

TL;DR

This paper establishes the existence of a unique Fatou coordinate for parabolic Dulac germs of hyperbolic polycycles, providing a constructive proof and describing its asymptotic expansion in a specialized scale.

Contribution

It offers the first constructive proof of the Fatou coordinate's existence for this class of germs, with detailed asymptotic analysis.

Findings

Existence of a unique Fatou coordinate for parabolic Dulac germs.

Asymptotic expansion of the Fatou coordinate in power-iterated log scale.

Constructive method for deriving the Fatou coordinate.

Abstract

We study the class of parabolic Dulac germs of hyperbolic polycycles. For such germs we give a constructive proof of the existence of a unique Fatou coordinate, admitting an asymptotic expansion in the power-iterated log scale.