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arXiv · 2608.01517

Computability of Julia sets for complex H\'enon maps: The role of attracting and neutral cycles

Suzanne Boyd·Christian Wolf
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Abstract

We study the computability of Julia sets for polynomial diffeomorphisms of $\mathbb{C}^2$ with dynamical degree $d>1$, whose prototypical examples are complex H\'enon maps. In previous work, we established computability under the assumption of hyperbolicity (Axiom A). Here, we extend this result to maps whose Fatou components are attracting basins, allowing for the possibility of no attracting basins or infinitely many basins. This yields computability of the Julia set for several classes of non-hyperbolic maps, including certain substantially dissipative maps in the Lyubich-Peters class, and certain quasi-hyperbolic maps. Our proof is based on an algorithm that separates the dynamics into escaping, attracting, and saddle regimes. A key ingredient is the use of stable manifolds of saddle periodic points to approximate the forward Julia set via backward iteration. We first prove the result for generalized H\'enon mappings and then extend it to arbitrary polynomial diffeomorphisms by expressing them as finite compositions of generalized H\'enon maps. Finally, we present examples of non-computability in the presence of neutral dynamics, including H\'enon maps with computable coefficients exhibiting semi-Siegel behavior. These examples show that the computability/non-computability dichotomy associated with neutral dynamics in one-dimensional complex dynamics in part persists in higher dimensions.

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Suzanne Boyd, Christian Wolf. 2026-08-02. Computability of Julia sets for complex H\'enon maps: The role of attracting and neutral cycles. https://arxiv.org/abs/2608.01517

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