The dynamics of maps tangent to the identity and with nonvanishing index

Abstract:

Let $f$ be a germ of a holomorphic self-map of $\mathbb {C}^2$ at the origin $O$ tangent to the identity, and with $O$ as a nondicritical isolated fixed point. A parabolic curve for $f$ is a holomorphic $f$-invariant curve, with $O$ on the boundary, attracted by $O$ under the action of $f$. It has been shown by M. Abate (2001) that if the characteristic direction $[v]\in \mathbb {P}(T_O\mathbb {C}^2)$ has residual index not belonging to $\mathbb {Q}^+$, then there exist parabolic curves for $f$ tangent to $[v]$. In this paper we prove, using a different method, that the conclusion still holds just assuming that the residual index is not vanishing (at least when $f$ is regular along $[v]$).