Slopes of 3-Dimensional Subshifts of Finite Type

Moutot, Etienne; Vanier, Pascal

doi:10.1007/978-3-319-90530-3_22

Cited by 2 publications

(5 citation statements)

References 16 publications

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“…The set of slopes of periodicity of a subshift is a conjugacy invariant. A consequence of Corollary 8 is that the sets of slopes of periodicity of Z 3 -SFTs is a Σ 0 2 -computable set, and together with [16] this implies the following caracterization:…”

Section: The Full Caracterization Of Slopes Of Z 3 -Sftsmentioning

confidence: 89%

“…Proof. We know from [16] that one can realize any such Σ 0 2 set S as a set of slopes of a Z 3 -subshift. Let us now show the remaining direction.…”

Section: The Full Caracterization Of Slopes Of Z 3 -Sftsmentioning

confidence: 99%

“…Sets of periods have also been studied and characterized through computability and complexity theory [9]. [16] recently proved that any Σ 0 2 set of (Q ∪ {∞})…”

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confidence: 99%

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Aperiodic points in $\mathbb Z^2$-subshifts

Grandjean¹,

Menibus²,

Vanier³

2018

Preprint

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We consider the structure of aperiodic points in Z 2 -subshifts, and in particular the positions at which they fail to be periodic. We prove that if a Z 2 -subshift contains points whose smallest period is arbitrarily large, then it contains an aperiodic point. This lets us characterise the computational difficulty of deciding if an Z 2 -subshift of finite type contains an aperiodic point. Another consequence is that Z 2 -subshifts with no aperiodic point have a very strong dynamical structure and are almost topologically conjugate to some Z-subshift. Finally, we use this result to characterize sets of possible slopes of periodicity for Z 3 -subshifts of finite type. ACM Subject ClassificationTheory of computation → Models of computation Keywords and phrases Subshifts of finite type, Wang tiles, periodicity, aperiodicity, computability, tilings Digital Object Identifier 10.4230/LIPIcs.ICALP.2018.496 Related Version hal-01722008v1Acknowledgements The authors wish to thank anonymous reviewers for many helpful remarks and improvements.A subshift on Z d is a set of colorings of Z d by a finite set of colors avoiding some family of forbidden patterns. When this family is finite, the subshift is called a subshift of finite type (SFT). In dimension 2, SFTs are equivalent to sets of tilings by Wang tiles: Wang tiles are unit squares with colored borders that cannot be rotated and may be placed next to each other only if the borders match.Wang tiles were introduced by Wang in order to study the decidability of some fragments of logic [18,19]. He thus introduced the Domino Problem: given a set of Wang tiles, do they tile the plane? (in other words, is the corresponding subshift nonempty?) Wang first conjectured that whenever a tileset tiles the plane, it can do so in a periodic manner, which would have implied the decidability of the Domino Problem.

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Section: The Full Caracterization Of Slopes Of Z 3 -Sftsmentioning

confidence: 89%

“…Proof. We know from [16] that one can realize any such Σ 0 2 set S as a set of slopes of a Z 3 -subshift. Let us now show the remaining direction.…”

Section: The Full Caracterization Of Slopes Of Z 3 -Sftsmentioning

confidence: 99%

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Aperiodic points in $\mathbb Z^2$-subshifts

Grandjean¹,

Menibus²,

Vanier³

2018

Preprint

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“…This article covers the results announced in [15] and [21] together with a treatment of the effective and sofic case. It also includes the full characterization allowed by [8].…”

Section: Theoremmentioning

confidence: 95%

“…Careful reader will notice that the definition of slope changed since [15] and [21]. The reason being that the former definition did not extend nicely to dimensions greater than two.…”

Section: Definition 1 (Periodicitymentioning

confidence: 99%

Slopes of Multidimensional Subshifts

Jeandel¹,

Moutot

Vanier³

2019

Theory Comput Syst

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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 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. Keywords subshifts • SFTs • computability • slopes • periodicity This work was supported by grants TARMAC ANR 12 BS02 007 01 and CoCoGro ANR 16 CE40 0005.

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Slopes of 3-Dimensional Subshifts of Finite Type

Cited by 2 publications

References 16 publications

Aperiodic points in $\mathbb Z^2$-subshifts

Aperiodic points in $\mathbb Z^2$-subshifts

Slopes of Multidimensional Subshifts

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