Non-density of stability for holomorphic mappings on $\protect \mathbb{P}^k$

Romain Dujardin

doi:10.5802/jep.57

Non-density of stability for holomorphic mappings on

ℙ^{k}

[Non-densité de la stabilité pour les applications holomorphes sur

ℙ^{k}

]

Romain Dujardin ¹

¹ Laboratoire de probabilités et modèles aléatoires, UMR 7599, Université Pierre et Marie Curie 4 place Jussieu, 75005 Paris, France

Journal de l’École polytechnique — Mathématiques, Tome 4 (2017), pp. 813-843.

Résumé
Abstract

Un théorème célèbre dû à Mañé-Sad-Sullivan et Lyubich affirme que les paramètres $J$ -stables forment un ouvert dense de toute famille holomorphe de systèmes dynamiques rationnels en dimension $1$ . Dans cet article nous montrons que ce résultat ne subsiste pas en dimension supérieure. Plus précisément nous construisons des ouverts contenus dans le lieu de bifurcation des applications holomorphes de degré $d$ de $ℙ^{k} (ℂ)$ pour tout $d \geq 2$ et $k \geq 2$ .

A well-known theorem due to Mañé-Sad-Sullivan and Lyubich asserts that $J$ -stable maps are dense in any holomorphic family of rational maps in dimension $1$ . In this paper we show that the corresponding result fails in higher dimension. More precisely, we construct open subsets in the bifurcation locus in the space of holomorphic mappings of degree $d$ of $ℙ^{k} (ℂ)$ for every $d \geq 2$ and $k \geq 2$ .

Reçu le : 2016-10-23
Accepté le : 2017-07-26
Publié le : 2017-08-16

MR

DOI : 10.5802/jep.57

Classification : 37F45, 37F10, 37F15
Keywords: Holomorphic dynamics in higher dimension, J-stability, bifurcations, blenders
Mot clés : Dynamique holomorphe en dimension supérieure, J-stabilité, bifurcations, mélangeurs

Affiliations des auteurs :

Romain Dujardin ¹

¹ Laboratoire de probabilités et modèles aléatoires, UMR 7599, Université Pierre et Marie Curie 4 place Jussieu, 75005 Paris, France

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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Romain Dujardin. Non-density of stability for holomorphic mappings on $\protect \mathbb{P}^k$. Journal de l’École polytechnique — Mathématiques, Tome 4 (2017), pp. 813-843. doi : 10.5802/jep.57. https://jep.centre-mersenne.org/articles/10.5802/jep.57/

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