Strongly aperiodic subshifts on surface groups

Cohen, David B.; Goodman-Strauss, Chaim

doi:10.4171/ggd/421

Cited by 16 publications

(21 citation statements)

References 26 publications

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“…Compare this with the other results for several one-ended groups which admit a subshift of finite type on which the group acts freely (see e.g. the classical result for Z 2 in [4] and many other more recent developments in [20], [32], [19]).…”

Section: Introductionsupporting

confidence: 61%

Garden of Eden and weakly periodic points for certain expansive actions of groups

Doucha¹

2021

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We present several applications of the weak specification property and certain topological Markov properties, recently introduced by S. Barbieri, F. García-Ramos and H. Li, and implied by the pseudoorbit tracing property, for general expansive group actions on compact spaces.First we show that any expansive action of a countable amenable group on a compact metrizable space satisfying the weak specification and strong topological Markov properties satisfies the Moore property, i.e. every surjective automorphism of such dynamical system is preinjective. This together with an earlier result of H. Li (where the strong topological Markov property is not needed) of the Myhill property, which we also re-prove here, establishes the Garden of Eden theorem for all expansive actions of countable amenable groups on compact metrizable spaces satisfying the weak specification and strong topological Markov properties. We hint how to easily generalize this result even for uncountable amenable groups and general compact, not necessarily metrizable, spaces.Second, we generalize the recent result of D. B. Cohen that any subshift of finite type of a finitely generated group having at least two ends has weakly periodic points. We show that every expansive action of such a group having a certain Markov topological property, again implied by the pseudo-orbit tracing property, has a weakly periodic point. If it has additionally the weak specification property, the set of such points is dense.

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Section: Introductionsupporting

confidence: 61%

Garden of Eden and weakly periodic points for certain expansive actions of groups

Doucha¹

2021

Preprint

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“…Our weakly but not strongly aperiodic SFT will work by encoding specific substitutions into BS (1, n). Indeed, the Cayley graph of BS(1, n) is very similar to orbit graphs of constant-size substitutions (see for example [2,9] for a definition of orbit graphs and another example of a Cayley graph similar to an orbit graph). In this section, we start by creating artificially a set of substitutions that are easy to encode in BS(1, n) (Section 3.1), and show how to do it (Section 3.2).…”

Section: A Weakly But Not Strongly Aperiodic Sft On Bs(1 N)mentioning

confidence: 99%

Weakly and Strongly Aperiodic Subshifts of Finite Type on Baumslag-Solitar Groups

Esnay¹,

Moutot²

2020

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We study the periodicity of subshifts of finite type (SFT) on Baumslag-Solitar groups. We show that for residually finite Baumslag-Solitar groups there exist both strongly and weakly-but-not-strongly aperiodic SFTs. In particular, this shows that unlike Z 2 , but as Z 3 , the notions of strong and weak periodicity are different for residually finite BS groups. More precisely, we prove that the weakly aperiodic SFT on BS(m, n) presented by Aubrun and Kari [3] is in fact strongly aperiodic on BS(1, n). In addition, we exhibit an SFT which is weakly but not strongly aperiodic on BS (1, n). Finally, we show that there exists a strongly aperiodic SFT on BS(n, n).The use of tilings as a computational model was initiated by Wang in the 60s [20] as a tool to study specific classes of logical formulas. His construction consisted in square tiles with colors on each side, that can be placed next to each other if the colors match. Then Wang studied the tilings of the discrete infinite plane (Z 2 ) with these tiles. A similar model exists to tile the infinite line (Z) with two-sided dominoes. Wang quickly realized that a key property was the periodicity of these tilings. It was already known that if a set of dominoes tiled Z, it was always possible to do it in a periodic way, and Wang suspected that it was the same for Wang tiles on Z 2 . However a few years later, one of his students, Berger, proved otherwise by providing a set of Wang tiles tiling the plane but only in aperiodic ways [4]. Alternative aperiodic sets of Wang tiles have been provided by many since then [13,14,18].The study of tilings with Wang tiles was extended to tilings using other symbols with adjacency rules, and became a part of symbolic dynamics, a more general way to encode a smooth dynamical system into symbolic states and trajectories. This discretization approach, see [10] for a comprehensive historiography, is itself a subpart of the field of discrete dynamical systems. In this broader context, it is interesting to study the set of all possible tilings for a given set of symbols and adjacency rules, called a Subshift of Finite Type. It can be equivalently described by the set of Wang tiles producing it, or by a finite alphabet associated with a finite set of forbidden patterns. Z-subshifts (tilings of the integer line) have been studied extensively, and many of their properties are known (see for example [15]). Z d −subshifts

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“…Inspired by [7], we associate a tiling of R 2 with the orbit of an infinite word w ∈ A Z under the action of a substitution, in which every tile codes a production rule of the substitution.…”

Section: Substitutions Orbits and Tilingsmentioning

confidence: 99%

“…This class of structures includes the hyperbolic plane model of [13], which can be though of as an orbit graph of the one-letter substitution 0 → 00. Using an idea involving superposition of orbit graphs of substitutions, presented in [7] and whose idea they attribute to Lorenzo Sadun, we show that the domino problem is undecidable on all orbit graphs of non-deterministic substitutions that satisfy a technical property (Section 4).…”

Section: Introductionmentioning

confidence: 99%

The domino problem is undecidable on surface groups

Aubrun,

Barbieri,

Moutot

2018

Preprint

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Strongly aperiodic subshifts on surface groups

Cited by 16 publications

References 26 publications

Garden of Eden and weakly periodic points for certain expansive actions of groups

Garden of Eden and weakly periodic points for certain expansive actions of groups

Weakly and Strongly Aperiodic Subshifts of Finite Type on Baumslag-Solitar Groups

The domino problem is undecidable on surface groups

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