Stable manifolds of biholomorphisms in ${\mathbb{C}}^n$ asymptotic to formal curves

Given a germ of biholomorphism $F\in \mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}({\mathbb{C}}^n,0)$ with a formal invariant curve $\mathrm{\Gamma }$ such that the multiplier of the restricted formal diffeomorphism $F{\mathrm{\mid }}_{\mathrm{\Gamma }}$ is a root of unity or satisfies $\mathrm{\mid }(F{\mathrm{\mid }}_{\mathrm{\Gamma }}{)}^{\mathrm{\prime }}(0)\mathrm{\mid }<1$, we prove that either $\mathrm{\Gamma }$ is contained in the set of periodic points of $F$ or there exists a finite family of stable manifolds of $F$ where all the orbits are asymptotic to $\mathrm{\Gamma }$ and whose union eventually contains every orbit asymptotic to $\mathrm{\Gamma }$. This result generalizes to the case where $\mathrm{\Gamma }$ is a formal periodic curve.