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Slopes of multidimensional subshifts. (English) Zbl 1440.37025
Summary: 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.
MSC:
 37B51 Multidimensional shifts of finite type 37B10 Symbolic dynamics
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References:
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