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

Dujardin, Romain

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

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 ${\mathbb{P}}^k\left(ℂ\right)$ pour tout $d\ge 2$ et $k\ge 2$.

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 ${\mathbb{P}}^k\left(ℂ\right)$ for every $d\ge 2$ and $k\ge 2$.

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

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