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arXiv · 1409.4449

On stability and hyperbolicity for polynomial automorphisms of C^2

Pierre Berger·Romain Dujardin
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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 $J^*$.

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Pierre Berger, Romain Dujardin. 2014-09-15. On stability and hyperbolicity for polynomial automorphisms of C^2. https://arxiv.org/abs/1409.4449

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