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La mesure d’équilibre d’un endomorphisme de k ( )

Xavier Buff

Séminaire Bourbaki

Soit f un endomorphisme holomorphe de k ( ) . Je présenterai une construction géométrique, due à Briend et Duval, d’une mesure de probabilité μ ayant les propriétés suivantes : μ reflète la distribution des préimages des points en dehors d’un ensemble exceptionnel algébrique, les points périodiques répulsifs de f s’équidistribuent par rapport à μ et μ est l’unique mesure d’entropie maximale de f .

Virtually repelling fixed point.

Xavier Buff — 2003

Publicacions Matemàtiques

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

Xavier Buff — 2011

Annales de la faculté des sciences de Toulouse Mathématiques

On montre l’existence d’applications rationnelles f : k k telles que f est algébriquement stable  : pour tout n 0 , deg f n = ( deg f ) n , il existe un unique courant positif fermé T de bidegré ( 1 , 1 ) vérifiant f * T = d · T et k T ω k - 1 = 1 ω est la forme de Fubini-Study sur k et ...

A new proof of a conjecture of Yoccoz

Xavier BuffArnaud Chéritat — 2011

Annales de l’institut Fourier

We give a new proof of the following conjecture of Yoccoz: ( C ) ( θ ) log rad Δ ( Q θ ) - Y ( θ ) + C , where Q θ ( z ) = e 2 π i θ z + z 2 , Δ ( Q θ ) is its Siegel disk if Q θ is linearizable (or otherwise), rad Δ ( Q θ ) is the conformal radius of the Siegel disk of Q θ (or 0 if there is none) and Y ( θ ) 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

Xavier BuffAdam L. Epstein — 2002

Fundamenta Mathematicae

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

Xavier BuffAdam L. EpsteinSarah Koch — 2012

Annales de la faculté des sciences de Toulouse Mathématiques

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.

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