Given a germ of biholomorphism F ∈ Diff(C n , 0) with a formal invariant curve Γ such that the multiplier of the restricted formal diffeomorphism F | Γ is a root of unity or satisfies |(F | Γ ) ′ (0)| < 1, we prove that either Γ 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 Γ and whose union eventually contains every orbit asymptotic to Γ. This result generalizes to the case where Γ is a formal periodic curve.

Abstract. We give a simple proof of the existence of parabolic curves for tangent to the identity diffeomorphisms in (C 2 , 0) with isolated fixed point.

Abstract. We prove that if F is a tangent to the identity diffeomorphism at 0 ∈ C 2 and Γ is a formal invariant curve of F then there exists a parabolic curve (attracting or repelling) of F asymptotic to Γ. The result is a consequence of a more general one in arbitrary dimension, where we prove the existence of parabolic curves of a tangent to the identity diffeomorphism F at 0 ∈ C n asymptotic to a given formal invariant curve under some additional conditions, expressed in terms of a reduction of F to a special normal form by means of blow-ups and ramifications along the formal curve.

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

We prove that for each characteristic direction $[v]$ of a tangent to the identity diffeomorphism of order $k+1$ in $(\mathbb{C}^{2},0)$ there exist either an analytic curve of fixed points tangent to $[v]$ or $k$ parabolic manifolds where all the orbits are tangent to $[v]$, and that at least one of these parabolic manifolds is or contains a parabolic curve.

Let F be a tangent to the identity diffeomorphism in (ℂ2,0) and X its infinitesimal generator. We prove that Camacho and Sad’s formal invariant curves of X give summable formal power series, whose sums correspond to the parabolic curves found by Hakim for F and F−1.

No abstract

We prove a two-dimensional analog of Leau-Fatou flower theorem for non-degenerate reduced tangent to the identity biholomorphisms.