Computability properties of hyperbolic complex H\'{e}non maps

In this article, we provide the first theoretical framework guaranteeing that computers can, in principle, be used to analyze the parameter space of complex H\'{e}maps. More precisely, we obtain computability results for hyperbolic polynomial diffeomorphisms of $\mathbb{C}^2$, for which H\'{e}non maps are prototypical examples. Specifically, we establish computability of the Julia set for hyperbolic maps, semi-decidability of hyperbolicity, and lower computability of the hyperbolicity locus in the parameter space of generalized H\'{e}non mappings of fixed degree at least two. Our approach builds upon techniques developed in our's recent previous works on polynomial maps of $\mathbb{C}$ and polynomial skew products of $\mathbb{C}^2$. In the setting of polynomial diffeomorphisms of $\mathbb{C}^2$, however, establishing hyperbolicity for the Julia set is considerably more difficult, as it requires identifying unstable (and stable) cone fields that are preserved and expanded by $Df$ (respectively $Df^{-1}$), and also due to the lack of algorithmically detectable quantitative shadowing.