Computability for Axiom A Polynomial Skew Products of C^2

The paper proves that for any Axiom A polynomial skew product f, J2 is computable and there is an algorithm that, for each hyperbolic component (X0, X1, Λ), either computes it or determines emptiness; it does so via a uniform box-chain/pseudo-orbit framework with derivative-based hyperbolicity detection and a halting proof exploiting shadowing and density of saddle periodic points (Theorem 1.1 and Section 3) . The candidate solution gives a correct but non-uniform construction: it assumes isolating neighborhoods and hyperbolicity constants are hard-wired per f and then uses certified box images, preimage propagation for the repeller J2, forward images for attractors X0, and two-sided graph pruning for saddle sets Λ and X1, achieving 2^{-n}-Hausdorff approximations and emptiness detection. It does not show how to derive the hyperbolicity data from coefficients or provide the shadowing-based halting argument, but for a fixed Axiom A map with computable coefficients, its algorithms are sound. Hence both are correct, with substantially different proof frameworks (uniform algorithmic detection and pseudo-orbits in the paper vs. fixed-constant interval/graph methods in the model).