# Spherical normal forms for germs of parabolic line biholomorphisms

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.
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https://hal-cnrs.archives-ouvertes.fr/hal-02947287
Contributor : Loïc Jean Dit Teyssier Connect in order to contact the contributor
Submitted on : Saturday, September 26, 2020 - 10:31:26 AM
Last modification on : Tuesday, September 29, 2020 - 3:31:58 AM
Long-term archiving on: : Friday, December 4, 2020 - 5:55:01 PM

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realization_parabolic.pdf
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### Identifiers

• HAL Id : hal-02947287, version 1
• ARXIV : 2009.13127

### Citation

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

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