Let \(F\) be a germ of biholomorphism of \(\mathbb{C}^2\) and tangent to the identity with a fixed point at the origin, such that \(F\neq\mathrm{id}\), and let \[ F(z) = z + \sum_{j\ge 2} F_j(z), \] be the expansion of \(F\) as sum of homogeneous polynomials, where \(\deg F_j = j\) (or \(F_j\equiv 0\)). The minimal \(k+1\) such that \(F_{k+1}(z)\ne 0\) is called the order of \(F\) at the origin and it is a holomorphic invariant.
The local dynamics of such germs was studied by several authors (see the references of the paper under review for instance) and one of the most important notions in such a study is that of characteristic direction, as it was introduced in [M. Hakim, Duke Math. J. 92, No. 2, 403–428 (1998; Zbl 0952.32012)]. A direction \([v]\in\mathbb{P}^1(\mathbb{C})\) is characteristic if there exists \(\lambda\in \mathbb{C}\) so that \(F_{k+1}(v)= \lambda v\); if \(\lambda\ne 0\) the direction \([v]\) is called non-degenerate, otherwise it is called degenerate. Moreover, it is possible to associate to any non-degenerate characteristic direction an invariant, called director, and M. Hakim showed in [Duke Math. J. 92, No. 2, 403–428 (1998; Zbl 0952.32012)] that if the real part of the director is strictly positive then there exists an attracting domain for the considered map tangent to the characteristic direction \([v]\).
It is natural to ask whether every characteristic direction \([v]\) of \(F\) has some stable dynamics associated to it and in this interesting paper the authors provide a complete positive answer to this question. The main result of the paper is the following:
Theorem 1. Let \(F\in\mathrm{Diff}(\mathbb{C}^2, 0)\) be diffeomorphism of order \(k+1\) a tangent to the identity, and let \([v]\) be a characteristic direction of \(F\). Then at least one of the following possibilities holds:

1.

There exists an analytic curve pointwise fixed by \(F\) and tangent to \([v]\).

2.

There exist at least \(k\) invariant sets \(\Omega_1, \dots, \Omega_k\), where each \(\Omega_i\) is either a parabolic curve tangent to \([v]\) or a parabolic domain along \([v]\) and such that all the orbits in \(\Omega_1\cup\dots\cup \Omega_k\) are mutually asymptotic. Moreover, at least one of the invariant sets \(\Omega_j\) is a parabolic curve.

3.

There exist at least \(k\) parabolic domains \(\Omega_1, \dots, \Omega_k\) along \([v]\), where each \(\Omega_i\) is foliated by parabolic curves and such that all the orbits in \(\Omega_1\cup\dots\cup \Omega_k\) are mutually asymptotic.

In particular, if \(F\) has an isolated fixed point then for any characteristic direction \([v]\) there is a parabolic curve tangent to \([v]\).