Aperiodic tilings and entropy

Durand, Bruno; Gamard, Guilhem; Grandjean, Anaël

doi:10.1016/j.tcs.2016.12.013

Cited by 3 publications

(13 citation statements)

References 9 publications

(11 reference statements)

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“…i = Λ −j (i) . We've constructed t (2) and t (3) to be the values of t corresponding to where we "divide α by 2" or "divide α by 3" respectively.…”

Section: Andmentioning

confidence: 99%

“…Proposition 31 (Rhin [7]). For u 0 , u 1 , u 2 ∈ Z and H = max{|u 1 |, |u 2 Proof. Let x = log 2 log 6 − p q .…”

Section: Asymptotic Density Of Orbits Under Fmentioning

confidence: 99%

“…We have shown that z (3) exists and is unique. In an analogous way, construct z (2) ∈ lim ← − R/(2 n Z). Finally, since z…”

Section: Definition Letmentioning

confidence: 99%

“…In their proof, Eigen et al show the existence of Kari-Culik tilings arising from Sturmian sequences (defined in the next section). It was unknown if there were Kari-Culik configurations not arising from Sturmian sequences until Durand, Gamard, and Grandjean showed that the set of all Kari-Culik tilings has positive topological entropy [2]. Since the number of Sturmian sequences of length n is bounded by a polynomial p(n) [1], the fact that n log p(n)/n 2 → 0 shows that the subset of tilings arising from Sturmian sequences must have zero-entropy.…”

Section: Introductionmentioning

confidence: 99%

“…01, 11)| = |Φ −1 2 (01, 22)| = |Φ −1 2 (01, 21)| = 1 and |Φ −1 2 (01, 12)| = 2. Thus, Φ on a row of type 1 3 is only non-unique if α t = 1/2 which further implies α b = 3/2.Case r is of type 2.1: In this case, we know α t ∈[1,2] and α b ∈ [1/2, 1]. Transition graph for a type 2.1 row.…”

mentioning

confidence: 99%

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A minimal subsystem of the Kari–Culik tilings

Siefken¹

2016

Ergod. Th. Dynam. Sys.

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The Kari-Culik tilings are formed from a set of 13 Wang tiles that tile the plane only aperiodically. They are the smallest known set of Wang tiles to do so and are not as well understood as other examples of aperiodic Wang tiles. We show that the Z 2 action by translation on a certain subset of the Kari-Culik tilings, namely those whose rows can be interpreted as Sturmian sequences (rotation sequences), is minimal. We give a characterization of this space as a skew product as well as explicit bounds on the waiting time between occurrences of m × n configurations.

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“…i = Λ −j (i) . We've constructed t (2) and t (3) to be the values of t corresponding to where we "divide α by 2" or "divide α by 3" respectively.…”

Section: Andmentioning

confidence: 99%

“…Proposition 31 (Rhin [7]). For u 0 , u 1 , u 2 ∈ Z and H = max{|u 1 |, |u 2 Proof. Let x = log 2 log 6 − p q .…”

Section: Asymptotic Density Of Orbits Under Fmentioning

confidence: 99%

“…We have shown that z (3) exists and is unique. In an analogous way, construct z (2) ∈ lim ← − R/(2 n Z). Finally, since z…”

Section: Definition Letmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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A minimal subsystem of the Kari–Culik tilings

Siefken¹

2016

Ergod. Th. Dynam. Sys.

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A self-similar aperiodic set of 19 Wang tiles

Labbé

2018

Geom Dedicata

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We define a Wang tile set U of cardinality 19 and show that the set Ω U of all valid Wang tilings Z 2 → U is self-similar, aperiodic and is a minimal subshift of U Z 2 . Thus U is the second smallest self-similar aperiodic Wang tile set known after Ammann's set of 16 Wang tiles. The proof is based on the unique composition property. We prove the existence of an expansive, primitive and recognizable 2-dimensional morphism ω : Ω U → Ω U that is onto up to a shift. The proof of recognizability is done in two steps using at each step the same criteria (the existence of marker tiles) for proving the existence of a recognizable one-dimensional substitution that sends each tile either on a single tile or on a domino of two tiles.

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