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Given a germ of biholomorphism $F\in\mathrm{Diff}(\mathbb{C}^n,0)$ with aformal invariant curve $\Gamma$ such that the multiplier of the restrictedformal diffeomorphism $F|_\Gamma$ is a root of unity or satisfies$|(F|_\Gamma)'(0)|<1$, we prove that either $\Gamma$ is contained in the set ofperiodic points of $F$ or there exists a finite family of stable manifolds of$F$ where all the orbits are asymptotic to $\Gamma$ and whose union eventuallycontains every orbit asymptotic to $\Gamma$. This result generalizes to thecase where $\Gamma$ is a formal periodic curve.

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