Upper-bound for the number of robust parabolic curves for a class of maps tangent to identity

Abstract

The Leau-Fatou flower theorem completely describes the dynamic behavior of $1-$dimensional maps tangent to the identity. In dimension two Hakim and Abate proved that if $f$ is a holomorphic map tangent to the identity in ${\mathbb{C}}^2$ and $\nu (f)$ is the degree of the first non vanishing jet of $f-Id$ then there exist $\nu (f)-1$ robust parabolic curves (RP curves for short), namely attractive petals at the origin which survive under by blow-up. The set of the exponential of holomorphic vector fields (of order greater than or equal to two), ${\Phi }_{\ge 2}({\mathbb{C}}^2,0)$, is dense in the space of germs of maps tangent to the identity. In this paper we give an upper-bound for the number of robust parabolic curves of $f\in {\Phi }_{\ge 2}({\mathbb{C}}^2,0).$