A notion of effectiveness for subshifts on finitely generated groups

Aubrun, Nathalie; Barbieri, Sebastián; Sablik, Mathieu

doi:10.1016/j.tcs.2016.11.033

Cited by 18 publications

(20 citation statements)

References 16 publications

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“…Let T be a Turing machine which accepts every inconsistent pattern coding and a maximal set of consistent pattern codings which generates F such that X = X F . The existence of such a machine in the case of a recursively presented group is given in [1].…”

Section: A Graph-oriented Construction and Some Computational Propertiesmentioning

confidence: 99%

Realization of aperiodic subshifts and uniform densities in groups

2018

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A theorem of Gao, Jackson and Seward, originally conjectured to be false by Glasner and Uspenskij, asserts that every countable group admits a 2-coloring. A direct consequence of this result is that every countable group has a strongly aperiodic subshift on the alphabet {0, 1}. In this article, we use Lovász local lemma to first give a new simple proof of said theorem, and second to prove the existence of a G-effectively closed strongly aperiodic subshift for any finitely generated group G. We also study the problem of constructing subshifts which generalize a property of Sturmian sequences to finitely generated groups. More precisely, a subshift over the alphabet {0, 1} has uniform density α ∈ [0, 1] if for every configuration the density of 1's in any increasing sequence of balls converges to α. We show a slightly more general result which implies that these subshifts always exist in the case of groups of subexponential growth.

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Section: A Graph-oriented Construction and Some Computational Propertiesmentioning

confidence: 99%

Realization of aperiodic subshifts and uniform densities in groups

2018

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“…A subshift is effectively closed if there exists a recursively enumerable coding of a set of forbidden patterns F such that X = X F . More details can be found in [2] or in Section 4.…”

Section: Preliminariesmentioning

confidence: 99%

“…Before presenting the simulation theorem for expansive actions, we must define with more detail effectively closed subshifts in groups. A longer survey of these concepts can be found in [2]. Given a group G generated by S and a finite alphabet A a pattern coding c is a finite set of tuples c = (w i , a i ) i∈I where w i ∈ S * and a i ∈ A.…”

Section: The Simulation Theorem For Expansive Effective Flowsmentioning

confidence: 99%

A generalization of the simulation theorem for semidirect products

Barbieri¹,

Sablik²

2018

Ergod. Th. Dynam. Sys.

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We generalize a result of Hochman in two simultaneous directions: Instead of realizing an effectively closed Z d action as a factor of a subaction of a Z d+2 -SFT we realize an action of a finitely generated group analogously in any semidirect product of the group with Z 2 . Let H be a finitely generated group and G = Z 2 ⋊ H a semidirect product. We show that for any effectively closed H-dynamical system (Y, f ) where Y is a Cantor set, there exists a G-subshift of finite type (X, σ) such that the H-subaction of (X, σ) is an extension of (Y, f ). In the case where f is an expansive action of a recursively presented group H, a subshift conjugated to (Y, f ) can be obtained as the H-projective subdynamics of a G-sofic subshift. As a corollary, we obtain that G admits a non-empty strongly aperiodic subshift of finite type whenever the word problem of H is decidable.

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“…In this section, we define some basic notions of of symbolic dynamics and computability. Some references on symbolic dynamics on general groups are [3,2], and a standard reference on Z is [8].…”

Section: Subshiftsmentioning

confidence: 99%

“…Some of the important classes studied are the SFTs (subshifts defined by a finite set of forbidden patterns), sofic shifts (the factors of SFTs) and the effective, or Π 0 1 subshifts (defined by a recursively enumerable set of forbidden patterns). SFTs and sofic shifts are natural objects to study on all groups, and a robust notion of effectiveness of subshifts on arbitrary groups is given in [2] (see also Section 5).…”

Section: Introductionmentioning

confidence: 99%