Universality in symbolic dynamics constrained by Medvedev degrees

Abstract: We define a weak notion of universality in symbolic dynamics and, by generalizing a proof of Mike Hochman, we prove that this yields necessary conditions on the forbidden patterns defining a universal subshift: These forbidden patterns are necessarily in a Medvedev degree greater or equal than the degree of the set of subshifts for which it is universal. We also show that this necessary condition is optimal by giving constructions of universal subshifts in the same Medvedev degree as the subshifts they simulate and prove that this universality can be achieved by the sofic projective subdynamics of such a subshift as soon as the necessary conditions are verified. This could be summarized as: There are obstructions for the existence of universal subshifts due to the theory of computability and they are the only ones.