Spherical normal forms for germs of parabolic line biholomorphisms

TL;DR

This paper constructs a unique parabolic map realizing a given Birkhoff–Écalle–Voronin modulus for germs of parabolic biholomorphisms, using Gevrey formal vector fields and analyzing their analytic continuation.

Contribution

It introduces a preferred family of parabolic maps with a specified modulus, proving their uniqueness and detailed analytic properties, including multivaluedness and branch points.

Findings

Constructed a parabolic map realizing a given modulus.

Proved the map's uniqueness within a certain functional class.

Analyzed the map's multivalued analytic continuation and branch points.

Abstract

We address the inverse problem for holomorphic germs of a tangent-to-identity mapping of the complex line near a fixed point. We provide a preferred (family of) parabolic map $\Delta$ realizing a given Birkhoff--{\'E}calle-Voronin modulus $\psi$ and prove its uniqueness in the functional class we introduce. The germ is the time-1 map of a Gevrey formal vector field admitting meromorphic sums on a pair of infinite sectors covering the Riemann sphere. For that reason, the analytic continuation of $\Delta$ is a multivalued map admitting finitely many branch points with finite monodromy. In particular $\Delta$ is holomorphic and injective on an open slit sphere containing 0 (the initial fixed point) and $\infty$, where sits the companion parabolic point under the involution $\frac{-1}{\id}$. It turns out that the Birkhoff--{\'E}calle-Voronin modulus of the parabolic germ at $\infty$ is the inverse $\psi^{\circ-1}$ of that at 0.