On stability and hyperbolicity for polynomial automorphisms of C²

2014/09/15 by Berger, Pierre, Dujardin, Romain
#Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.1409.4449

Abstract

Let (f_λ)_{λ∈ Λ} be a holomorphic family of polynomial automorphisms of ℂ². 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^*.

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