  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 $J^*$.
