Local dynamics of holomorphic diffeomorphisms

Filippo Bracci

Displaying similar documents to “Local dynamics of holomorphic diffeomorphisms”

The dynamics of holomorphic maps near curves of fixed points

Filippo Bracci (2003)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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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. $Fix (f) = S$ . 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 $S \cdot S < 0$ 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...

Some remarks on indices of holomorphic vector fields.

Marco Brunella (1997)

Publicacions Matemàtiques

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One can associate several residue-type indices to a singular point of a two-dimensional holomorphic vector field. Some of these indices depend also on the choice of a separatrix at the singular point. We establish some relations between them, especially when the singular point is a generalized curve and the separatrix is the maximal one. These local results have global consequences, for example concerning the construction of logarithmic forms defining a given holomorphic foliation. ...

Deformations of transversely holomorphic flows on spheres and deformations of hopf manifolds

A. Haefliger (1985)

Compositio Mathematica

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Holomorphic foliations in certain holomorphically convex domains of $ℂ^{2}$

Marco Brunella, Paulo Sad (1995)

Bulletin de la Société Mathématique de France

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A topological characterization of holomorphic parabolic germs in the plane

Frédéric Le Roux (2008)

Fundamenta Mathematicae

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J.-M. Gambaudo and É. Pécou introduced the "linking property" in the study of the dynamics of germs of planar homeomorphisms in order to provide a new proof of Naishul's theorem. In this paper we prove that the negation of the Gambaudo-Pécou property characterizes the topological dynamics of holomorphic parabolic germs. As a consequence, a rotation set for germs of surface homeomorphisms around a fixed point can be defined, and it turns out to be non-trivial except for countably many...

On deformations of holomorphic foliations

Joan Girbau, Marcel Nicolau (1989)

Annales de l'institut Fourier

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Given a non-singular holomorphic foliation $ℱ$ on a compact manifold $M$ we analyze the relationship between the versal spaces $K$ and $K^{tr}$ of deformations of $ℱ$ as a holomorphic foliation and as a transversely holomorphic foliation respectively. With this purpose, we prove the existence of a versal unfolding of $ℱ$ parametrized by an analytic space $K^{f}$ isomorphic to $π^{- 1} (0) \times Σ$ where $Σ$ is smooth and $π$ : $K \to K^{tr}$ is the forgetful map. The map $π$ is shown to be an epimorphism in two situations: (i) if $H^{2} (M, Θ_{ℱ}^{f}) = 0$ , where $Θ_{ℱ}^{f}$ is...

Parabolic curves in $ℂ^{3}$ .

Abate, Marco, Tovena, Francesca (2003)

Abstract and Applied Analysis

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On fixed points of holomorphic type

Ewa Ligocka (2002)

Colloquium Mathematicae

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We study a linearization of a real-analytic plane map in the neighborhood of its fixed point of holomorphic type. We prove a generalization of the classical Koenig theorem. To do that, we use the well known results concerning the local dynamics of holomorphic mappings in ℂ².