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
C
2
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. The set of the exponential of holomorphic vector fields (of order greater than or equal to two),
Φ
≥2
(
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∈
Φ
≥2
(
C
2
,0).