On stability and hyperbolicity for polynomial automorphisms of ${ℂ}^2$

Berger, Pierre; Dujardin, Romain

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 50 (2017) no. 2, pp. 449-477.

Soit ${\left(f_{\lambda }\right)}_{\lambda \in \Lambda }$ une famille holomorphe d'automorphismes polynomiaux de ${ℂ}^2$. En accord avec un travail précédent de Dujardin et Lyubich, nous disons qu'une telle famille est faiblement stable si ses points périodiques ne bifurquent pas. La question est ouverte de savoir si cette notion équivaut à celle de stabilité structurelle sur l'ensemble de Julia $J^{*}$ (qui est par définition l'adhérence de l'ensemble des points périodiques selles).

Dans cet article nous introduisons une notion de point régulier pour un tel automorphisme, inspirée par la théorie de Pesin, et montrons que dans une famille faiblement stable, l'ensemble des points réguliers se déplace selon un mouvement holomorphe. Nous en déduisons qu'une famille faiblement stable est structurellement stable en un sens probabiliste. Une autre conséquence de cette étude est que la stabilité faible préserve l'hyperbolicité uniforme sur $J^{*}$.

Let ${\left(f_{\lambda }\right)}_{\lambda \in \Lambda }$ be a holomorphic family of polynomial automorphisms of ${ℂ}^2$. Following previous work of Dujardin and Lyubich, we say that such a family is weakly stable if saddle periodic orbits do not bifurcate. It is an open question whether this property is equivalent to structural stability on the Julia set $J^{*}$ (that is, the closure of the set of saddle periodic points).

In this paper we introduce a notion of regular point for a polynomial automorphism, inspired by Pesin theory, and prove that in a weakly stable family, the set of regular points moves holomorphically. It follows that a weakly stable family is probabilistically structurally stable, in a very strong sense. Another consequence of these techniques is that weak stability preserves uniform hyperbolicity on $J^{*}$.

Classification : 37F15, 37F45, 37F10
Keywords: Automorphismes polynomiaux de ${\mathbb{C}}^2$, mouvements holomorphes, stabilité structurelle, stabilité faible, hyperbolicité uniforme et non uniforme.
Mot clés : Polynomial automorphisms of ${\mathbb{C}}^2$, holomorphic motions, structural stability, weak stability, uniform and non-uniform hyperbolicity.

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