The expressiveness of quasiperiodic and minimal shifts of finite type

TL;DR

This paper investigates the complexity and structure of multidimensional minimal and quasiperiodic shifts of finite type, revealing new results about their computability, Turing degrees, and subdynamics representations.

Contribution

It extends known properties of shifts of finite type to the quasiperiodic and minimal cases, including computability and Turing degree characterizations.

Findings

Some quasiperiodic shifts of finite type admit only non-computable configurations.

Characterization of Turing degrees representable by quasiperiodic shifts of finite type.

Effective minimal and quasiperiodic shifts can be represented as projections of subdynamics of higher-dimensional shifts.

Abstract

We study multidimensional minimal and quasiperiodic shifts of finite type. We prove for these classes several results that were previously known for the shifts of finite type in general, without restriction. We show that some quasiperiodic shifts of finite type admit only non-computable configurations; we characterize the classes of Turing degrees that can be represented by quasiperiodic shifts of finite type. We also transpose to the classes of minimal/quasiperiodic shifts of finite type some results on subdynamics previously known for the effective shifts without restrictions: every effective minimal (quasiperiodic) shift of dimension $d$ can be represented as a projection of a subdynamics of a minimal (respectively, quasiperiodic) shift of finite type of dimension $d+1$.