Realization of aperiodic subshifts and uniform densities in groups

Nathalie Aubrun (1) , Sebastián Barbieri (1) , Stéphan Thomassé (1, 2)

1 MC2 - Modèles de calcul, Complexité, Combinatoire
2 IUF - Institut universitaire de France

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.

Mots clés: Stur-mian sequences; Amenable groups; Symbolic dynamics; Countable groups; Aperiodic subshift

Domaines: Systèmes dynamiques [math.DS]; Théorie des groupes [math.GR]

Nathalie Aubrun, Sebastián Barbieri, Stéphan Thomassé. Realization of aperiodic subshifts and uniform densities in groups. Groups, Geometry, and Dynamics, 2019, 13 (1), pp.107-129. ⟨hal-01989763⟩