We prove that the one-parameter group of holomorphic automorphisms induced on a strictly geometrically bounded domain by a biholomorphism with a model domain is parabolic. This result is related to the Greene-Krantz conjecture and more generally to the classification of domains having a non compact automorphisms group. The proof relies on elementary estimates on the Kobayashi pseudo-metric.

Harvey and Lawson introduced the Kähler rank and computed it in connection to the cone of positive exact currents of bidimension $(1,1)$ for many classes of compact complex surfaces. In this paper we extend these computations to the only further known class of surfaces not considered by them, that of Kato surfaces. Our main tool is the reduction to the dynamics of associated holomorphic contractions $({ℂ}^2,0)\to ({ℂ}^2,0)$.

Nous donnons une condition suffisante pour l’existence de points périodiques pour une application birationnelle de $ℂ{\mathbb{P}}^2$. Sous cette hypothèse, une estimation précise du nombre de points périodiques de période fixée est obtenue. Nous donnons une application de ce résultat à l’étude dynamique de ces applications, en calculant explicitement l’auto-intersection de leur courant invariant naturellement associé. Nos résultats reposent essentiellement sur le théorème de Bézout donnant le cardinal de l’intersection...

Let $X$ and $Y$ be compact Kähler manifolds, and let $f:X\to Y$ be a dominant meromorphic map. Based upon a regularization theorem of Dinh and Sibony for DSH currents, we define a pullback operator $f^{♯}$ for currents of bidegrees $(p,p)$ of finite order on $Y$ (and thus foranycurrent, since $Y$ is compact). This operator has good properties as may be expected. Our definition and results are compatible to those of various previous works of Meo, Russakovskii and Shiffman, Alessandrini and Bassanelli, Dinh and Sibony, and can...

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.

A monomial self-map $f$ on a complex toric variety is said to be $k$-stable if the action induced on the $2k$-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 ${\lambda }_k$ of $f$ satisfy ${\lambda }_k^2\>{\lambda }_{k-1}{\lambda }_{k+1}$. On the other hand, we give examples of monomial maps $f$, where this condition...

Nous construisons des mesures selles (dans un sens faible) pour les endomorphismes holomorphes de ${\mathbb{P}}^2\left(ℂ\right)$.

We try to find a geometric interpretation of the wedge product of positive closed laminar currents in C2. We say such a wedge product is geometric if it is given by intersecting the disks filling up the currents. Uniformly laminar currents do always intersect geometrically in this sense. We also introduce a class of strongly approximable laminar currents, natural from the dynamical point of view, and prove that such currents intersect geometrically provided they have continuous potentials.

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. $\mathrm{Fix}\left(f\right)=S$. 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 $S·S\<0$ 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 are achieved...

Little is known about the global topology of the Fatou set U(f) for holomorphic endomorphisms $f:ℂ{\mathbb{P}}^k\to ℂ{\mathbb{P}}^k$, when k >1. Classical theory describes U(f) as the complement in $ℂ{\mathbb{P}}^k$ of the support of a dynamically defined closed positive (1,1) current. Given any closed positive (1,1) current S on $ℂ{\mathbb{P}}^k$, we give a definition of linking number between closed loops in $ℂ{\mathbb{P}}^k\setminus suppS$ and the current S. It has the property that if lk(γ,S) ≠ 0, then γ represents a non-trivial homology element in $H₁\left(ℂ{\mathbb{P}}^k\setminus suppS\right)$. As an application, we use these linking...