Stable manifolds of two-dimensional biholomorphisms asymptotic to formal curves

Let $F\in \mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}({\mathbb{C}}^2,0)$ be a germ of a holomorphic diffeomorphism and let $\mathrm{\Gamma }$ be an invariant formal curve of $F$. Assume that the restricted diffeomorphism $F{\mathrm{\mid }}_{\mathrm{\Gamma }}$ is either hyperbolic attracting or rationally neutral non-periodic (these are the conditions that the diffeomorphism $F{\mathrm{\mid }}_{\mathrm{\Gamma }}$ should satisfy, if $\mathrm{\Gamma }$ were convergent, in order to have orbits converging to the origin). Then we prove that $F$ has finitely many stable manifolds, either open domains or parabolic curves, consisting of and containing all converging orbits asymptotic to $\mathrm{\Gamma }$. Our results generalize to the case where $\mathrm{\Gamma }$ is a formal periodic curve of $F$.