Slopes of multidimensional subshifts

Emmanuel Jeandel (1) , Etienne Moutot (2, 3) , Pascal Vanier (4)

1 MOCQUA - Designing the Future of Computational Models
2 MC2 - Modèles de calcul, Complexité, Combinatoire
3 University of Turku
4 LACL - Laboratoire d'Algorithmique Complexité et Logique

In this paper we study the directions of periodicity of multidimen-sional 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 Σ 0 1 sets of non-commensurable Z 2 vectors are exactly the sets of slopes of 2D SFTs and that Σ 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.

Mots clés: slopes; periodicity; computability; SFTs

Domaines: Mathématique discrète [cs.DM]

Emmanuel Jeandel, Etienne Moutot, Pascal Vanier. Slopes of multidimensional subshifts. Theory of Computing Systems, 2020, 64 (1), pp.35-61. ⟨10.1007/s00224-019-09931-1⟩. ⟨hal-02158012⟩