Spherical normal forms for germs of parabolic line biholomorphisms

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.