Matrix Characterization of Multidimensional Subshifts of Finite Type

TL;DR

This paper introduces a matrix-based characterization of multidimensional subshifts of finite type, providing new criteria for their non-emptiness and conditions for periodic points, extending to arbitrary dimensions.

Contribution

It develops a novel matrix representation for multidimensional subshifts of finite type and establishes criteria for non-emptiness and periodicity in higher dimensions.

Findings

Characterization of 2D subshifts using infinite matrices

Criteria for non-emptiness of multidimensional shift spaces

Conditions for the existence of periodic points

Abstract

Let $X\subset A^{Z^d}$ be a $2$-dimensional subshift of finite type. We prove that any $2$-dimensional multidimensional subshift of finite type can be characterized by a square matrix of infinite dimension. We extend our result to a general $d$-dimensional case. We prove that the multidimensional shift space is non-empty if and only if the matrix obtained is of positive dimension. In the process, we give an alternative view of the necessary and sufficient conditions obtained for the non-emptiness of the multidimensional shift space. We also give sufficient conditions for the shift space $X$ to exhibit periodic points.