  Let $F\in\mathrm{Diff}(\mathbb{C}^2,0)$ be a germ of a holomorphic
diffeomorphism and let $\Gamma$ be an invariant formal curve of $F$. Assume
that the restricted diffeomorphism $F|_{\Gamma}$ is either hyperbolic
attracting or rationally neutral non-periodic (these are the conditions that
the diffeomorphism $F|_{\Gamma}$ should satisfy, if $\Gamma$ 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 $\Gamma$. Our
results generalize to the case where $\Gamma$ is a formal periodic curve of
$F$.
