Universality in symbolic dynamics constrained by Medvedev degrees

TL;DR

This paper explores the limits of universality in symbolic dynamics imposed by computability constraints, establishing necessary and sufficient conditions related to Medvedev degrees for the existence of universal subshifts.

Contribution

It introduces a weak notion of universality in symbolic dynamics and characterizes the Medvedev degree conditions necessary and sufficient for universal subshifts to exist.

Findings

Forbidden patterns must have Medvedev degree at least that of the subshifts they simulate.

Universal subshifts can be constructed within the same Medvedev degree as the subshifts they simulate.

Universality can be achieved via sofic projective subdynamics when conditions are met.

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.