The supports of higher bifurcation currents

Romain Dujardin

doi:10.5802/afst.1378

Romain Dujardin ¹

¹ CMLS, École Polytechnique, 91128 Palaiseau, France. Nouvelle adresse : LAMA, Université Paris Est Marne-la-Vallée, Cité Descartes 77454 Marne-la-Vallée cedex France.

Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 22 (2013) no. 3, pp. 445-464.

Abstract
Abstract

Let ${(f_{λ})}_{λ \in Λ}$ be a holomorphic family of rational mappings of degree $d$ on $ℙ^{1} (ℂ)$ , with $k$ marked critical points $c_{1}, ..., c_{k}$ . To this data is associated a closed positive current $T_{1} \land \dots \land T_{k}$ of bidegree $(k, k)$ on $Λ$ , aiming to describe the simultaneous bifurcations of the marked critical points. In this note we show that the support of this current is accumulated by parameters at which $c_{1}, ..., c_{k}$ eventually fall on repelling cycles. Together with results of Buff, Epstein and Gauthier, this leads to a complete characterization of $Supp (T_{1} \land \dots \land T_{k})$ .

Soit ${(f_{λ})}_{λ \in Λ}$ une famille holomorphe d’applications rationnelles de degré $d$ de $ℙ^{1} (ℂ)$ , avec $k$ points critiques marqués $c_{1}, ..., c_{k}$ . À cette donnée est associée un courant $T_{1} \land \dots \land T_{k}$ de bidegré $(k, k)$ sur l’espace des paramètres $Λ$ , visant à décrire les bifurcations simultanées des points critiques marqués. Dans cette note nous montrons que le support de ce courant est accumulé par des paramètres en lesquels $c_{1}, ..., c_{k}$ tombent sur des cycles répulsifs. En combinant ceci avec des résultats de Buff, Epstein et Gauthier, on obtient ainsi une caractérisation complète du support de $T_{1} \land \dots \land T_{k}$ .

EuDML MR Zbl

DOI: 10.5802/afst.1378

Author's affiliations:

Romain Dujardin ¹

¹ CMLS, École Polytechnique, 91128 Palaiseau, France. Nouvelle adresse : LAMA, Université Paris Est Marne-la-Vallée, Cité Descartes 77454 Marne-la-Vallée cedex France.

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     title = {The supports of higher bifurcation currents},
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     pages = {445--464},
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Romain Dujardin. The supports of higher bifurcation currents. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 22 (2013) no. 3, pp. 445-464. doi : 10.5802/afst.1378. https://afst.centre-mersenne.org/articles/10.5802/afst.1378/

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