Following results of McMullen concerning rational maps, we show that the limit set of matings between a certain class of representations of C₂ ∗ C₃ and quadratic polynomials carries δ-conformal measures, and that if the correspondence is geometrically finite then the real number δ is equal to the Hausdorff dimension of the limit set. Moreover, when f is the limit of a pinching deformation ${f_t}_{0\le t<1}$ we give sufficient conditions for the dynamical convergence of $f_t$.

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...

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.

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.

Soit µ la mesure d’équilibre d’un endomorphisme de $P^k\left(C\right)$. Nous montrons ici qu’elle est son unique mesure d’entropie maximale. Nous construisons directement µ comme distribution asymptotique des préimages

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 $\ell \&gt;1$ 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.

For n ≥ 2, the family of rational maps $F_{\lambda }\left(z\right)=zⁿ+\lambda /zⁿ$ contains a countably infinite set of parameter values for which all critical orbits eventually land after some number κ of iterations on the point at infinity. The Julia sets of such maps are Sierpiński curves if κ ≥ 3. We show that two such maps are topologically conjugate on their Julia sets if and only if they are Möbius or anti-Möbius conjugate, and we give a precise count of the number of topological conjugacy classes as a function of n and κ.

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...