Generic smooth maps not robustly Turing-universal

Determine whether a C-infinity generic diffeomorphism f: M -> M on a compact computable manifold M can be extended to a robustly Turing-universal computational dynamical system, specifically by proving or refuting that no such generic f admits such an extension.

Background

The paper develops a framework for computational dynamical systems (CDSs) and proves strong non-universality results for broad classes such as Axiom A systems and measure-preserving dynamics. Building on these, the authors propose a conjecture about the typical (generic) behavior of smooth maps, suggesting that generic diffeomorphisms cannot robustly simulate universal Turing machines within their CDS framework.

They note they can prove this under stronger decoder assumptions and constant slowdown (Theorem 4.6), but the fully generic statement remains open and may require deep inputs from differentiable dynamics, potentially related to Palis’ conjectures.