Following Favre, we define a holomorphic germ $f:({ℂ}^d,0)\to ({ℂ}^d,0)$ to be rigid if the union of the critical set of all iterates has simple normal crossing singularities. We give a partial classification of contracting rigid germs in arbitrary dimensions up to holomorphic conjugacy. Interestingly enough, we find new resonance phenomena involving the differential of $f$ and its linear action on the fundamental group of the complement of the critical set.

Recent years have witnessed an increasing interest in coordinated control of distributed dynamic systems. In order to steer a distributed dynamic system to a desired state, it often becomes necessary to have a prior control over the graph which represents the coupling among interacting agents. In this paper, a simple but compelling model of distributed dynamical systems operating over a dynamic graph is considered. The structure of the graph is assumed to be relied on the underling system's states....

$M_1$ è un particolare operatore di minimizzazione per forme di Dirichlet definite su un sottoinsieme finito di un frattale $K$ che è, in un certo senso, una sorta di frontiera di $K$. Viene talvolta chiamato mappa di rinormalizzazione ed è stato usato per definire su $K$ un analogo del funzionale $u\mapsto \int {\left|\text{grad}u\right|}^2$ e un moto Browniano. In questo lavoro si provano alcuni risultati sull'unicità dell'autoforma (rispetto a $M_1$ ), e sulla convergenza dell'iterata di $M_1$ rinormalizzata. Questi risultati sono collegati con l'unicità...

We give a thorough study of Cui's control of distortion technique in the analysis of convergence of simple pinching deformations, and extend his result from geometrically finite rational maps to some subset of geometrically infinite maps. We then combine this with mating techniques for pairs of polynomials to establish existence and continuity results for matings of polynomials with parabolic points. Consequently, if two hyperbolic quadratic polynomials tend to their respective root polynomials...

Given a finitely generated subgroup G of the group of affine transformations acting on the complex line C, we are interested in the quotient Fix( G)/G. The purpose of this note is to establish when this quotient is finite and in this case its cardinality. We give an application to the qualitative study of polynomial planar vector fields at a neighborhood of a nilpotent singular point.

On montre l’existence d’applications rationnelles $f:{\mathbb{P}}^k\to {\mathbb{P}}^k$ telles que$f$ est algébriquement stable  : pour tout $n\ge 0$, $deg{f}^{\circ n}={(degf)}^n$,il existe un unique courant positif fermé $T$ de bidegré $(1,1)$ vérifiant $f^{*}T=d·T$ et ${\int }_{{\mathbb{P}}^k}T\wedge {\omega }^{k-1}=1$ où $\omega$ est la forme de Fubini-Study sur ${\mathbb{P}}^k$ et$T$ est pluripolaire  : il existe un ensemble pluripolaire $X\subset {\mathbb{P}}^k$ tel que ${\int }_{X}T\wedge {\omega }^{k-1}=1$

We extend the work of Bielefeld, Fisher and Hubbard on critical portraits to arbitrary postcritically finite polynomials. This gives the classification of such polynomials as dynamical systems in terms of their external ray behavior.