The dynamics near quasi-parabolic fixed points of holomorphic diffeomorphisms in $C^2$

Let $F$ be a germ of holomorphic diffeomorphism of $\C^2$ fixing $O$ and such that $dF_O$ has eigenvalues $1$ and $e^{i\theta}$ with $|e^{i\theta}|= 1$ and $e^{i\theta}\neq 1$. Introducing suitable normal forms for $F$ we define an invariant, $\nu(F)\geq 2$, and a generic condition, that of being {\sl dynamically separating}. In the case $F$ is dynamically separating, we prove that there exist $\nu(F)-1$ parabolic curves for $F$ at $O$ tangent to the eigenspace of $1$.