On stability and hyperbolicity for polynomial automorphisms of C^2

TL;DR

This paper investigates the stability of polynomial automorphisms of complex two-dimensional space, showing that weak stability ensures the holomorphic movement of regular points and preserves hyperbolicity on the Julia set.

Contribution

It introduces a notion of regular points inspired by Pesin theory and proves their holomorphic movement in weakly stable families, linking weak stability to probabilistic structural stability.

Findings

Regular points move holomorphically in weakly stable families

Weak stability preserves uniform hyperbolicity on the Julia set

Weak stability implies probabilistic structural stability

Abstract

Let $(f_\lambda)_{\lambda\in \Lambda}$ be a holomorphic family of polynomial automorphisms of $\mathbb{C}^2$. 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 $J^*$.