On the zeros and critical points of a rational map.
Ensembles de Julia de mesure positive
La mesure d’équilibre d’un endomorphisme de
Soit un endomorphisme holomorphe de . Je présenterai une construction géométrique, due à Briend et Duval, d’une mesure de probabilité ayant les propriétés suivantes : reflète la distribution des préimages des points en dehors d’un ensemble exceptionnel algébrique, les points périodiques répulsifs de s’équidistribuent par rapport à et est l’unique mesure d’entropie maximale de .
Virtually repelling fixed point.
In this article, we study the notion oí virtually repelling fixed point. We first give a definition and an interpretation of it. We then prove that most proper holomorphic mappings f: U -> V with U contained in V have at least one virtually repelling fixed point.
Courants dynamiques pluripolaires
On montre l’existence d’applications rationnelles telles que
Farey curves.
Perturbations of flexible Lattès maps
We prove that any Lattès map can be approximated by strictly postcritically finite rational maps which are not Lattès maps.
A new proof of a conjecture of Yoccoz
We give a new proof of the following conjecture of Yoccoz: where , is its Siegel disk if is linearizable (or otherwise), is the conformal radius of the Siegel disk of (or if there is none) and is Yoccoz’s Brjuno function. In a former article we obtained a first proof based on the control of parabolic explosion. Here, we present a more elementary proof based on Yoccoz’s initial methods. We then extend this result to some new families...
A parabolic Pommerenke-Levin-Yoccoz inequality
In a recent preprint [B], Bergweiler relates the number of critical points contained in the immediate basin of a multiple fixed point β of a rational map f: ℙ¹ → ℙ¹, the number N of attracting petals and the residue ι(f,β) of the 1-form dz/(z-f(z)) at β. In this article, we present a different approach to the same problem, which we were developing independently at the same time. We apply our method to answer a question raised by Bergweiler. In particular, we prove that when there are only...
Twisted matings and equipotential gluings
One crucial tool for studying postcritically finite rational maps is Thurston’s topological characterization of rational maps. This theorem is proved by iterating a holomorphic endomorphism on a certain Teichmüller space. The graph of this endomorphism covers a correspondence on the level of moduli space. In favorable cases, this correspondence is the graph of a map, which can be used to study matings. We illustrate this by way of example: we study the mating of the basilica with itself.
Questions about Polynomial Matings
We survey known results about polynomial mating, and pose some open problems.
Page 1