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## Spherical normal forms for germs of parabolic line biholomorphisms

Loïc Teyssier

#### 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–Écalle-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–Écalle-Voronin modulus of the parabolic germ at $\infty$ is the inverse $\psi^{\circ-1}$ of that at 0.

### Dates and versions

hal-02947287 , version 1 (26-09-2020)
hal-02947287 , version 2 (27-10-2022)

### Identifiers

• HAL Id : hal-02947287 , version 2
• ARXIV :

### Cite

Loïc Teyssier. Spherical normal forms for germs of parabolic line biholomorphisms. 2020. ⟨hal-02947287v2⟩

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