Computational Dynamical Systems

The paper proves that Axiom A diffeomorphisms cannot be extended to robustly Turing‑universal CDSs (Theorem 1.6/4.7), via a spectral‑decomposition/stable‑manifold argument that constructs too many invariant regions to fit finitely many basic sets . The candidate solution reaches the same conclusion but its proof has critical gaps: it (i) implicitly treats the variable‑lag map x ↦ f^{τ(x)} as if standard shadowing/closing lemmas for a fixed diffeomorphism applied; (ii) omits the paper’s requirement that τ be constant on connected components of decoding cells, needed to synchronize sampling times ; (iii) assumes a near‑return needed to close a periodic pseudo‑orbit without justification; and (iv) conflates periodicity of a constructed periodic orbit with periodicity of the original simulated run. These issues undermine the contradiction step, so the model’s proof is not sound.