Abstract

Let $M$ be a two-dimensional complex manifold and $f:M \rightarrow M$ a holomorphic map. Let $S \subset M$ be a curve made of fixed points of $f$, i.e. ${\rm {Fix}} =S$. We study the dynamics near $S$ in case $f$ acts as the identity on the normal bundle of the regular part of $S$. Besides results of local nature, we prove that if $S$ is a globally and locally irreducible compact curve such that $S\cdot S<0$ then there exists a point $p \in S$ and a holomorphic $f$-invariant curve with $p$ on the boundary which is attracted by $p$ under the action of $f$. These results are achieved introducing and studying a family of local holomorphic foliations related to $f$ near $S$