Parametrization by Horizontal Constraints in the Study of Algorithmic Properties of $\mathbb{Z}^2$-Subshift of Finite Type

TL;DR

This paper investigates how fixing horizontal constraints affects the decidability of the Domino Problem and the entropy spectrum in $\

Contribution

It introduces a parametrized approach to horizontal constraints in $\

Findings

Undecidability of the Domino Problem depends on horizontal constraints.

All right-recursively enumerable numbers can be realized as entropy under certain conditions.

Different approaches to local rules influence the properties of the subshifts.

Abstract

The non-emptiness, called the Domino Problem, and the characterization of the possible entropies of $\mathbb{Z}^2$-subshifts of finite type are standard problems of symbolic dynamics. In this article we study these questions with horizontal constraints fixed beforehand as a parameter. We determine for which horizontal constraints the Domino Problem is undecidable and when all right-recursively enumerable numbers can be obtained as entropy, with two approaches: either the additional local rules added to the horizontal constraints can be of any shape, or they can only be vertical rules.