Polymorphisms and linearization of nonlinear polynomials.
Porosity of Collet–Eckmann Julia sets
We prove that the Julia set of a rational map of the Riemann sphere satisfying the Collet-Eckmann condition and having no parabolic periodic point is mean porous, if it is not the whole sphere. It follows that the Minkowski dimension of the Julia set is less than 2.
Pull-back of currents by meromorphic maps
Let and be compact Kähler manifolds, and let be a dominant meromorphic map. Based upon a regularization theorem of Dinh and Sibony for DSH currents, we define a pullback operator for currents of bidegrees of finite order on (and thus foranycurrent, since 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...
Quelques résultats sur la dimension de Hausdorff des ensembles de Julia des polynômes quadratiques
This paper deals with the Hausdorff dimension of the Julia set of quadratic polynomials. It is divided in two parts. The first aims to compute good numerical approximations of the dimension for hyperbolic points. For such points, Ruelle’s thermodynamical formalism applies, hence computing the dimension amounts to computing the zero point of a pressure function. It is this pressure function that we approximate by a Monte-Carlo process combined with a shift method that considerably decreases the computational...
Relating Hausdorff measures and harmonic measures on parabolic Jordan curves.
Representing real numbers in Möbius number systems
Rigidity and inflexibility in conformal dynamics.
Rigidity of harmonic measure
Let J be the Julia set of a conformal dynamics f. Provided that f is polynomial-like we prove that the harmonic measure on J is mutually absolutely continuous with the measure of maximal entropy if and only if f is conformally equivalent to a polynomial. This is no longer true for generalized polynomial-like maps. But for such dynamics the coincidence of classes of these two measures turns out to be equivalent to the existence of a conformal change of variable which reduces the dynamical system...
Snapback Repellers and Semiconjugacy for Iterated Entire Mappings: global aspects of an old Theorem of Poincaré.
Sur la dynamique des difféomorphismes birationnels des surfaces algébriques réelles : ensemble de Fatou et lieu réel
On s’intéresse aux difféomorphismes birationnels des surfaces algébriques réelles qui possèdent une dynamique réelle simple et une dynamique complexe riche. On donne un exemple d’une telle transformation sur , mais on montre qu’une telle situation est exceptionnelle et impose des conditions fortes à la fois sur la topologie du lieu réel et sur la dynamique réelle.
Sur les ensembles de Julia et Fatou des fonctions entières ultramétriques
Soit un nombre premier rationnel. Le sujet de l’article est l’étude de la dynamique des fonctions entières -adiques. On démontre des résultats analogues à ceux connus dans le domaine complexe, en particulier si deux fonctions entières -adiques qui ont un point répulsif commun commutent, alors leurs ensembles de Julia et de Fatou sont les mêmes.
The capacity of parabolic Julia sets.
The dynamics of holomorphic maps near curves of fixed points
Let be a two-dimensional complex manifold and a holomorphic map. Let be a curve made of fixed points of , i.e. . We study the dynamics near in case acts as the identity on the normal bundle of the regular part of . Besides results of local nature, we prove that if is a globally and locally irreducible compact curve such that then there exists a point and a holomorphic -invariant curve with on the boundary which is attracted by under the action of . These results are achieved...
The Equichordal Point Problem.
Topological invariance of the Collet–Eckmann property for S-unimodal maps
We prove that if f, g are smooth unimodal maps of the interval with negative Schwarzian derivative, conjugated by a homeomorphism of the interval, and f is Collet-Eckmann, then so is g.
Un théorème sur les fonctions itératives
Uniformization of the leaves of a rational vector field
We study the analytic structure of the leaves of a holomorphic foliation by curves on a compact complex manifold. We show that if every leaf is a hyperbolic surface then they can be simultaneously uniformized in a continuous manner. In case the manifold is complex projective space a sufficient condition is that there are no algebraic leaf.
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Wandering domains and invariant conformal structures for mappings of the 2-torus.