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Exponentially long time stability for non-linearizable analytic germs of ( ℂ n , 0 ) .

Timoteo Carletti (2004)

Annales de l’institut Fourier

We study the Siegel-Schröder center problem on the linearization of analytic germs of diffeomorphisms in several complex variables, in the Gevrey- s , s > 0 category. We introduce a new arithmetical condition of Bruno type on the linear part of the given germ, which ensures the existence of a Gevrey- s formal linearization. We use this fact to prove the effective stability, i.e. stability for finite but long time, of neighborhoods of the origin, for the analytic germ.

Farey curves.

Buff, Xavier, Henriksen, Christian, Hubbard, John H. (2001)

Experimental Mathematics

Feuilletages singuliers de codimension un, groupoïde de Galois et intégrales premières

Guy Casale (2006)

Annales de l’institut Fourier

Dans cet article, nous étudions le groupoïde de Galois d’un germe de feuilletage holomorphe de codimension un. Nous associons à ce 𝒟 -groupoïde de Lie un invariant biméromorphe  : le rang transverse. Nous étudions en détails les relations entre cet invariant, l’existence de suites de Godbillon-Vey particulières et l’existence d’une intégrale première dans une extension fortement normale du corps différentiel des germes de fonctions méromorphes. Nous obtenons ainsi une généralisation d’un théorème...

Fibrations of compact Kähler manifolds in terms of cohomological properties of their fundamental groups

Ngaiming Mok (2000)

Annales de l'institut Fourier

We prove fibration theorems on compact Kähler manifolds with conditions on first cohomology groups of fundamental groups with respect to unitary representations into Hilbert spaces. If the fundamental group T of compact Kähler manifold X violates Property (T) of Kazhdan’s, then H 1 ( G a m m a , Φ ) ≠ 0 for some unitary representation Φ . By our earlier work there exists a d -closed holomorphic 1-form with coefficients twisted by some unitary representation Φ ' , possibly non-isomorphic to Φ . Taking norms we obtains a positive...

Finite determinacy of dicritical singularities in ( ℂ 2 , 0 )

Gabriel Calsamiglia-Mendlewicz (2007)

Annales de l’institut Fourier

For germs of singularities of holomorphic foliations in ( ℂ 2 , 0 ) which are regular after one blowing-up we show that there exists a functional analytic invariant (the transverse structure to the exceptional divisor) and a finite number of numerical parameters that allow us to decide whether two such singularities are analytically equivalent. As a result we prove a formal-analytic rigidity theorem for this kind of singularities.

Flat 3-webs of degree one on the projective plane

A. Beltrán, M. Falla Luza, D. Marín (2014)

Annales de la faculté des sciences de Toulouse Mathématiques

The aim of this work is to study global 3 -webs with vanishing curvature. We wish to investigate degree 3 foliations for which their dual web is flat. The main ingredient is the Legendre transform, which is an avatar of classical projective duality in the realm of differential equations. We find a characterization of degree 3 foliations whose Legendre transform are webs with zero curvature.

Fonctions et feuilletages Levi-Flat. Étude locale

Dominique Cerveau, Paulo R. Sad (2004)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We define the notion of CR equivalence for Levi-flat foliations and compare in a local setting these foliations to their linear parts. We study also the situation where the foliation has a first integral ; a condition is given so that this integral is the real part of a holomorphic function.

From Newton’s method to exotic basins Part I: The parameter space

Krzysztof Barański (1998)

Fundamenta Mathematicae

This is the first part of the work studying the family 𝔉 of all rational maps of degree three with two superattracting fixed points. We determine the topological type of the moduli space of 𝔉 and give a detailed study of the subfamily ℱ 2 consisting of maps with a critical point which is periodic of period 2. In particular, we describe a parabolic bifurcation in ℱ 2 from Newton maps to maps with so-called exotic basins.

From Newton's method to exotic basins Part II: Bifurcation of the Mandelbrot-like sets

Krzysztof Barański (2001)

Fundamenta Mathematicae

This is a continuation of the work [Ba] dealing with the family of all cubic rational maps with two supersinks. We prove the existence of the following parabolic bifurcation of Mandelbrot-like sets in the parameter space of this family. Starting from a Mandelbrot-like set in cubic Newton maps and changing parameters in a continuous way, we construct a path of Mandelbrot-like sets ending in the family of parabolic maps with a fixed point of multiplier 1. Then it bifurcates into two paths of Mandelbrot-like...

Generalized iterated function systems, multifunctions and Cantor sets

Maciej Klimek, Marta Kosek (2009)

Annales Polonici Mathematici

Using a construction similar to an iterated function system, but with functions changing at each step of iteration, we provide a natural example of a continuous one-parameter family of holomorphic functions of infinitely many variables. This family is parametrized by the compact space of positive integer sequences of prescribed growth and hence it can also be viewed as a parametric description of a trivial analytic multifunction.

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