Generic smooth maps are not robustly Turing‑universal

Prove that for any compact computable manifold M, a C^∞ generic diffeomorphism f: M → M cannot be extended to a robustly Turing‑universal computational dynamical system (CDS) with any choice of encoder, decoder, and slowdown function.

Background

The paper establishes strong obstructions to robust Turing universality for broad classes of smooth dynamical systems, including Axiom A systems and measure‑preserving (integrable) systems. Motivated by these results and expectations from differentiable dynamics, the authors propose a conjecture asserting a generic non‑universality statement for smooth diffeomorphisms.

They also note partial progress: under a strong condition on the decoder and a constant slowdown, they prove a non‑universality result (Theorem 4.3), and suggest that Palis’ conjectures might help resolve the general case. The conjecture seeks a definitive, unconditional generic non‑universality statement.