Slopes of Multidimensional Subshifts

Abstract

In this paper we study the directions of periodicity of multidimensional subshifts of finite type (SFTs) and of multidimensional effectively closed and sofic subshifts. A configuration of a subshift has a slope of periodicity if it is periodic in exactly one direction, the slope representing that direction. In this paper, we prove that ${{\Sigma }^{0}_{1}}$ sets of non-commensurable $\mathbb {Z}^{2}$ vectors are exactly the sets of slopes of 2D SFTs and that ${{\Sigma }^{0}_{2}}$ sets of non-commensurable vectors are exactly the sets of slopes of 3D SFTs, and exactly the sets of slopes of 2D and 3D sofic and effectively closed subshifts.