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

Abstract

The non-emptiness, called the Domino Problem, and the characterization of the possible entropies of 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.Note that a Rauzy graph can be made of one or several strongly connected components (SCC for short). We recall that it is constituted of one unique strongly connected components if and only if the SFT associated is transitive; that is, for any u, w ∈ L H there exists v ∈ L H so that uvw ∈ L H . If the Rauzy graph has several SCCs it can also contain transient vertices, that are vertices with no path from themselves to themselves. We refer to [LM95] for more details.