The dynamics of holomorphic maps near curves of fixed points

Filippo Bracci

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2003)

  • Volume: 2, Issue: 3, page 493-520
  • ISSN: 0391-173X

Abstract

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Let M be a two-dimensional complex manifold and f : M → M a holomorphic map. Let S ⊂ M be a curve made of fixed points of f , i.e.  Fix ( f ) = 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 ∈ 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 .

How to cite

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Bracci, Filippo. "The dynamics of holomorphic maps near curves of fixed points." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 2.3 (2003): 493-520. <http://eudml.org/doc/84510>.

@article{Bracci2003,
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. $\{\rm \{Fix\}\} (f)=S$. 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 $S\cdot S&lt;0$ 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 $S$.},
author = {Bracci, Filippo},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
language = {eng},
number = {3},
pages = {493-520},
publisher = {Scuola normale superiore},
title = {The dynamics of holomorphic maps near curves of fixed points},
url = {http://eudml.org/doc/84510},
volume = {2},
year = {2003},
}

TY - JOUR
AU - Bracci, Filippo
TI - The dynamics of holomorphic maps near curves of fixed points
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 2003
PB - Scuola normale superiore
VL - 2
IS - 3
SP - 493
EP - 520
AB - 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. ${\rm {Fix}} (f)=S$. 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 $S\cdot S&lt;0$ 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 $S$.
LA - eng
UR - http://eudml.org/doc/84510
ER -

References

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