Let $M$ be a two-dimensional complex manifold and $f:M\to M$ a holomorphic map. Let $S\subset M$ be a curve made of fixed points of $f$, i.e. $\mathrm{Fix}\left(f\right)=S$. 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 $S·S\<0$ 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...

Little is known about the global topology of the Fatou set U(f) for holomorphic endomorphisms $f:ℂ{\mathbb{P}}^k\to ℂ{\mathbb{P}}^k$, when k >1. Classical theory describes U(f) as the complement in $ℂ{\mathbb{P}}^k$ of the support of a dynamically defined closed positive (1,1) current. Given any closed positive (1,1) current S on $ℂ{\mathbb{P}}^k$, we give a definition of linking number between closed loops in $ℂ{\mathbb{P}}^k\setminus suppS$ and the current S. It has the property that if lk(γ,S) ≠ 0, then γ represents a non-trivial homology element in $H₁\left(ℂ{\mathbb{P}}^k\setminus suppS\right)$. As an application, we use these linking...