The dynamics of holomorphic maps near curves of fixed points

Filippo Bracci

Displaying similar documents to “The dynamics of holomorphic maps near curves of fixed points”

Local dynamics of holomorphic diffeomorphisms

Filippo Bracci (2004)

Bollettino dell'Unione Matematica Italiana

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This is a survey about local holomorphic dynamics, from Poincaré's times to nowadays. Some new ideas on how to relate discrete dynamics to continuous dynamics are also introduced. It is the text of the talk given by the author at the XVII UMI Congress at Milano.

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. ...

Holomorphic foliations by curves on $ℙ^{3}$ with non-isolated singularities

Gilcione Nonato Costa (2006)

Annales de la faculté des sciences de Toulouse Mathématiques

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Let $ℱ$ be a holomorphic foliation by curves on $ℙ^{3}$ . We treat the case where the set $Sing (ℱ)$ consists of disjoint regular curves and some isolated points outside of them. In this situation, using Baum-Bott’s formula and Porteuos’theorem, we determine the number of isolated singularities, counted with multiplicities, in terms of the degree of $ℱ$ , the multiplicity of $ℱ$ along the curves and the degree and genus of the curves.

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...

The Łojasiewicz numbers and plane curve singularities

Evelia García Barroso, Tadeusz Krasiński, Arkadiusz Płoski (2005)

Annales Polonici Mathematici

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For every holomorphic function in two complex variables with an isolated critical point at the origin we consider the Łojasiewicz exponent ₀(f) defined to be the smallest θ > 0 such that ${| g r a d f (z) | \geq c | z |}^{θ}$ near 0 ∈ ℂ² for some c > 0. We investigate the set of all numbers ₀(f) where f runs over all holomorphic functions with an isolated critical point at 0 ∈ ℂ².

On vanishing inflection points of plane curves

Mauricio Garay (2002)

Annales de l’institut Fourier

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We study the local behaviour of inflection points of families of plane curves in the projective plane. We develop normal forms and versal deformation concepts for holomorphic function germs $f : (ℂ^{2}, 0) ⟶ (ℂ, 0)$ which take into account the inflection points of the fibres of $f$ . We give a classification of such function- germs which is a projective analog of Arnold’s A,D,E classification. We compute the versal deformation with respect to inflections of Morse function-germs.

Variations of complex structures on an open Riemann surface

M. S. Narasimhan (1961)

Annales de l'institut Fourier

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Soit $U_{1}$ un ouvert dans $C^{m}$ . Soit $π_{1} : S \to U_{1}$ une famille holomorphe de structures complexes sur une surface de Riemann non-compacte $M$ , avec $S_{t_{0}} = π_{1}^{- 1} (t_{0}) = M$ . ( $S = S (M \times U_{1})$ est une structure complexe sur le produit différentiable $M \times U_{1}$ ). Soit $M_{1}$ un domaine relativement compact dans $M$ . On démontre : pour tout voisinage de Stein $U$ de $t_{0}$ , assez petit, la famille $π_{1} : S (M_{1} \times U) \to U$ est isomorphe à la famille $π : Ω \to π (Ω)$ , où $Ω$ est un de la variété produit $M \times C^{m}$ , $π$ étant la projection $M \times C^{m} \to C^{m}$ . On donne aussi un résultat analogue pour le cas des variations différentiables. ...

Displaying similar documents to “The dynamics of holomorphic maps near curves of fixed points”

Local dynamics of holomorphic diffeomorphisms

Some remarks on indices of holomorphic vector fields.

Holomorphic foliations by curves on ℙ 3 with non-isolated singularities

On deformations of holomorphic foliations

The Łojasiewicz numbers and plane curve singularities

On vanishing inflection points of plane curves

Variations of complex structures on an open Riemann surface

Holomorphic foliations by curves on $ℙ^{3}$ with non-isolated singularities