The dynamics of holomorphic maps near curves of fixed points

Bracci, Filippo

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 2 (2003) no. 3, pp. 493-520.

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 introducing and studying a family of local holomorphic foliations related to $f$ near $S$.

Classification : 32H50, 37F99, 32S45, 32S65

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