Upper-bound for the Number of Robust Parabolic Curves for a Class of Maps Tangent to Identity

The Leau-Fatou flower theorem [8] completely describes the dynamic behavior of 1−dimensional maps tangent to the identity. In dimension two Hakim [12] and Abate [1] proved that if f is a holomorphic map tangent to the identity in C and ν(f) is the degree of the first non vanishing jet of f − Id then there exist ν(f)− 1 robust parabolic curves (RP curves for short), namely attractive petals at the origin which survive under by blow-up (see [3] and Section 3) . The set of the exponential of holomorphic vector fields (of order greater than or equal to two), Φ≥2(C , 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 ∈ Φ≥2(C , 0).