Holomorphic dynamics near germs of singular curves

Abstract

Let M be a two dimensional complex manifold, p ∈ M and Open image in new window a germ of holomorphic foliation of M at p. LetOpen image in new window be a germ of an irreducible, possibly singular, curve at p in M which is a separatrix for Open image in new window . We prove that if the Camacho-Sad-Suwa index Ind Open image in new window then there exists another separatrix for Open image in new window at p. A similar result is proved for the existence of parabolic curves for germs of holomorphic diffeomorphisms near a curve of fixed points.