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Singularités des flots holomorphes. II

Étienne Ghys, Julio C. Rebelo (1997)

Annales de l'institut Fourier

Dans un article précédent [Singularité des flots holomorphes, Ann. Inst. Fourier, Grenoble, 46-2 (1996), 411-428], le deuxième auteur démontrait, en particulier, qu’un champ de vecteurs holomorphe complet sur une surface complexe ne peut posséder une singularité isolée dont le deuxième jet est nul. Nous nous proposons ici de donner une description précise des champs de vecteurs holomorphes complets sur les surfaces complexes qui possèdent une singularité isolée dont le premier jet est nul. Dans...

Singularités nilpotentes et intégrales premières.

Rafik Meziani, Paulo Sad (2007)

Publicacions Matemàtiques

This paper presents a classification of plane dicritical nilpotent singularities, i.e. singularities which have nilpotent linear part and infinitely many separatrices. In particular the existence of meromorphic first integrals is discussed. The same ideas are applied to other kind of dicritical singularities.

Small divisors and large multipliers

Boele Braaksma, Laurent Stolovitch (2007)

Annales de l’institut Fourier

We study germs of singular holomorphic vector fields at the origin of n of which the linear part is 1 -resonant and which have a polynomial normal form. The formal normalizing diffeomorphism is usually divergent at the origin but there exists holomorphic diffeomorphisms in some “sectorial domains” which transform these vector fields into their normal form. In this article, we study the interplay between the small divisors phenomenon and the Gevrey character of the sectorial normalizing diffeomorphisms....

Smooth Gevrey normal forms of vector fields near a fixed point

Laurent Stolovitch (2013)

Annales de l’institut Fourier

We study germs of smooth vector fields in a neighborhood of a fixed point having an hyperbolic linear part at this point. It is well known that the “small divisors” are invisible either for the smooth linearization or normal form problem. We prove that this is completely different in the smooth Gevrey category. We prove that a germ of smooth α -Gevrey vector field with an hyperbolic linear part admits a smooth β -Gevrey transformation to a smooth β -Gevrey normal form. The Gevrey order β depends on...

Some novel ways of generating Cantor and Julia type sets

Marta Kosek (2012)

Annales Polonici Mathematici

It is a survey article showing how an enhanced version of the Banach contraction principle can lead to generalizations of attractors of iterated function systems and to Julia type sets.

Some pinching deformations of the Fatou function

Patricia Domínguez, Guillermo Sienra (2015)

Fundamenta Mathematicae

We are interested in deformations of Baker domains by a pinching process in curves. In this paper we deform the Fatou function F ( z ) = z + 1 + e - z , depending on the curves selected, to any map of the form F p / q ( z ) = z + e - z + 2 π i p / q , p/q a rational number. This process deforms a function with a doubly parabolic Baker domain into a function with an infinite number of doubly parabolic periodic Baker domains if p = 0, otherwise to a function with wandering domains. Finally, we show that certain attracting domains can be deformed by a pinching...

Stability of foliations induced by rational maps

F. Cukierman, J. V. Pereira, I. Vainsencher (2009)

Annales de la faculté des sciences de Toulouse Mathématiques

We show that the singular holomorphic foliations induced by dominant quasi-homogeneous rational maps fill out irreducible components of the space q ( r , d ) of singular foliations of codimension q and degree d on the complex projective space r , when 1 q r - 2 . We study the geometry of these irreducible components. In particular we prove that they are all rational varieties and we compute their projective degrees in several cases.

Stabilization of monomial maps in higher codimension

Jan-Li Lin, Elizabeth Wulcan (2014)

Annales de l’institut Fourier

A monomial self-map f on a complex toric variety is said to be k -stable if the action induced on the 2 k -cohomology is compatible with iteration. We show that under suitable conditions on the eigenvalues of the matrix of exponents of f , we can find a toric model with at worst quotient singularities where f is k -stable. If f is replaced by an iterate one can find a k -stable model as soon as the dynamical degrees λ k of f satisfy λ k 2 > λ k - 1 λ k + 1 . On the other hand, we give examples of monomial maps f , where this condition...

Strong almost reducibility for analytic and Gevrey quasi-periodic cocycles

Claire Chavaudret (2013)

Bulletin de la Société Mathématique de France

This article is about almost reducibility of quasi-periodic cocycles with a diophantine frequency which are sufficiently close to a constant. Generalizing previous works by L.H. Eliasson, we show a strong version of almost reducibility for analytic and Gevrey cocycles, that is to say, almost reducibility where the change of variables is in an analytic or Gevrey class which is independent of how close to a constant the initial cocycle is conjugated. This implies a result of density, or quasi-density,...

Strong bifurcation loci of full Hausdorff dimension

Thomas Gauthier (2012)

Annales scientifiques de l'École Normale Supérieure

In the moduli space d of degree  d rational maps, the bifurcation locus is the support of a closed ( 1 , 1 ) positive current T bif which is called the bifurcation current. This current gives rise to a measure μ bif : = ( T bif ) 2 d - 2 whose support is the seat of strong bifurcations. Our main result says that supp ( μ bif ) has maximal Hausdorff dimension 2 ( 2 d - 2 ) . As a consequence, the set of degree  d rational maps having ( 2 d - 2 ) distinct neutral cycles is dense in a set of full Hausdorff dimension.

Structure of leaves and the complex Kupka-Smale property

Tanya Firsova (2013)

Annales de l’institut Fourier

We study topology of leaves of 1 -dimensional singular holomorphic foliations of Stein manifolds. We prove that for a generic foliation all leaves, except for at most countably many, are contractible, the rest are topological cylinders. We show that a generic foliation is complex Kupka-Smale.

Structure of the McMullen domain in the parameter planes for rational maps

Robert L. Devaney (2005)

Fundamenta Mathematicae

We show that, for the family of functions F λ ( z ) = z + λ / z where n ≥ 3 and λ ∈ ℂ, there is a unique McMullen domain in parameter space. A McMullen domain is a region where the Julia set of F λ is homeomorphic to a Cantor set of circles. We also prove that this McMullen domain is a simply connected region in the plane that is bounded by a simple closed curve.

Sulla stabilità di un punto fisso per funzioni di n variabili complesse. Problema del Centro di Schröder-Siegel

Timoteo Carletti (2005)

Bollettino dell'Unione Matematica Italiana

Viene considerato il problema della stabilità di un punto fisso per un germe di diffeomorfismo di più variabili complesse cercando un coniugio con la sua parte lineare: Problema del centro di Schröder-Siegel. Dopo aver formulato il problema e ricordato i principali risultati nel caso di diffeomorfismi olomorfi, mostriamo come estendere il problema ad alcune situazioni non olomorfe, in particolare ci interesseremo al caso di germi Gevrey. Concluderemo con un'applicazione rivolta a mostrare la stabilità...

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