Avoshifts

TL;DR

This paper introduces avoshifts on polycyclic groups, showing they are a recursively enumerable class of subshifts of finite type with effective computability and decidability properties, extending known results to non-algebraic cases.

Contribution

It extends the theory of avoshifts to polycyclic groups, demonstrating their recursive enumerability and effective computability of subdynamics and factors, with decidability results.

Findings

Avoshifts form a recursively enumerable class of subshifts of finite type.

Effective computation of lower-dimensional subdynamics and factors is possible.

Decidability of inclusion and equality for avoshifts is established.

Abstract

An avoshift is a subshift where for each set $C$ from a suitable family of subsets of the shift group, the set of all possible valid extensions of a globally valid pattern on $C$ to the identity element is determined by a bounded subpattern. This property is shared (for various families of sets $C$) by for example cellwise quasigroup shifts, TEP subshifts, and subshifts of finite type with a safe symbol. In this paper we concentrate on avoshifts on polycyclic groups, when the sets $C$ are what we call ``inductive intervals''. We show that then avoshifts are a recursively enumerable subset of subshifts of finite type. Furthermore, we can effectively compute lower-dimensional projective subdynamics and certain factors (avofactors), and we can decide equality and inclusion for subshifts in this class. These results were previously known for group shifts, but our class also covers many non-algebraic examples as well as many SFTs without dense periodic points. The theory also yields new proofs of decidability of inclusion for SFTs on free groups, and SFTness of subshifts with the topological strong spatial mixing property.