Colloquium: Prof. Christian Wolf

Computability Properties of Hyperbolic H\’enon Maps

In this talk we address the question if the Julia set of a complex H\’enon map can be computed rigorously by an algorithm. More broadly, we study computability properties of polynomial diffeomorphisms of $\mathbb{C}^2$. We show that, whenever such a map is hyperbolic, its Julia set is computable to arbitrary prescribed accuracy. The approach uses finite box models of the chain recurrent set together with the density of saddle periodic points in the Julia set. We also describe an algorithm that semi-decides hyperbolicity by constructing rigorously verified stable and unstable cone fields. As a consequence, the hyperbolicity locus of generalized H\’enon families is lower semi-computable. We will explain the main ideas behind these algorithms and discuss further consequences, including the semi-decidability of disconnected Julia sets and extensions to higher-dimensional polynomial automorphisms. This material discussed in this talk is joint work with Suzanne Boyd, see https://arxiv.org/abs/2605.26306.