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--{É}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--{É}calle-Voronin modulus of the parabolic germ at \(\infty\) is the inverse \(\psi^{\circ-1}\) of that at 0.