In this survey we give geometric interpretations of some standard results on boundary behaviour of holomorphic self-maps in the unit disc of ℂ and generalize them to holomorphic self-maps of some particular domains of ℂⁿ.

We first introduce the class of quasi-algebraically stable meromorphic maps of Pk. This class is strictly larger than that of algebraically stable meromorphic self-maps of Pk. Then we prove that all maps in the new class enjoy a recurrent property. In particular, the algebraic degrees for iterates of these maps can be computed and their first dynamical degrees are always algebraic integers.

In this paper we study the semigroups $\mathrm{\Phi }:{ℝ}^{+}\to Hol(D,D)$ of holomorphic maps of a strictly convex domain $D\subset {𝐂}^n$ into itself. In particular, we characterize the semigroups converging, uniformly on compact subsets, to a holomorphic map $h:D\to {𝐂}^n$.

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 ${\int }_{X}T\wedge {\omega }^{k-1}=1$

Let $f$ be a meromorphic self-mapping of a compact Kähler manifold. We study the rate of decreasing of volumes under the iteration of $f$. We use these volume estimates to construct the Green current of $f$ in a quite general setting.

A small perturbation of a rational function causes only a small perturbation of its periodic orbits. We show that the situation is different for transcendental maps. Namely, orbits may escape to infinity under small perturbations of parameters. We show examples where this "diffusion to infinity" occurs and prove certain conditions under which it does not.

Nous construisons pour toute correspondance polynomiale $F$ d’exposant de Lojasiewicz $\ell \>1$ une mesure d’équilibre $\mu$. Nous montrons que $\mu$ est approximable par les préimages d’un point générique et que les points périodiques répulsifs sont équidistribués sur le support de $\mu$. En utilisant ces résultats, nous donnons une caractérisation des ensembles d’unicité pour les polynômes.

We continue our study of the dynamics of mappings with small topological degree on projective complex surfaces. Previously, under mild hypotheses, we have constructed an ergodic “equilibrium” measure for each such mapping. Here we study the dynamical properties of this measure in detail: we give optimal bounds for its Lyapunov exponents, prove that it has maximal entropy, and show that it has product structure in the natural extension. Under a natural further assumption, we show that saddle points...

Our first main result is a construction of a simple formal normal form for holomorphic diffeomorphisms in $C^n$ whose differentials have one-dimensional family of resonances in the first $m$ eigenvalues, $m\le n$ (but more resonances are allowed for other eigenvalues). Next, we provide invariants and give conditions for the existence of basins of attraction. Finally, we give applications and examples demonstrating the sharpness of our conditions.

We consider complex dynamics of a critically finite holomorphic map from Pk to Pk, which has symmetries associated with the symmetric group Sk+2 acting on Pk, for each k ≥1. The Fatou set of each map of this family consists of attractive basins of superattracting points. Each map of this family satisfies Axiom A.

Let $f:{\mathbb{P}}^k\to {\mathbb{P}}^k$ be a dominating rational mapping of first algebraic degree $\lambda \ge 2$. If $S$ is a positive closed current of bidegree $(1,1)$ on ${\mathbb{P}}^k$ with zero Lelong numbers, we show – under a natural dynamical assumption – that the pullbacks ${\lambda }^{-n}{\left(f^n\right)}^{*}S$ converge to the Green current $T_f$. For some families of mappings, we get finer convergence results which allow us to characterize all $f^{*}$-invariant currents.

Let $f$ be a non-invertible holomorphic endomorphism of a projective space and $f^n$ its iterate of order $n$. We prove that the pull-back by $f^n$ of a generic (in the Zariski sense) hypersurface, properly normalized, converges to the Green current associated to $f$ when $n$ tends to infinity. We also give an analogous result for the pull-back of positive closed $(1,1)$-currents and a similar result for regular polynomial automorphisms of ${ℂ}^k$.

Let f: ℙ → ℙ be a holomorphic endomorphism of a complex projective space ${\mathbb{P}}^k$, k ≥ 1, and let J be the Julia set of f (the topological support of the unique maximal entropy measure). Then there exists a positive number ${\kappa }_f>0$ such that if ϕ: J → ℝ is a Hölder continuous function with $sup\left(\varphi \right)-inf\left(\varphi \right)<{\kappa }_f$, then ϕ admits a unique equilibrium state ${\mu }_{\varphi }$ on J. This equilibrium state is equivalent to a fixed point of the normalized dual Perron-Frobenius operator. In addition, the dynamical system $(f,{\mu }_{\varphi })$ is K-mixing, whence ergodic. Proving...