Realization of aperiodic subshifts and uniform densities in groups
Realization of aperiodic subshifts and uniform densities in groups
复制标题
实现组内非周期性子平移和均匀密度
DOI：
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发表时间：
2015
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通讯作者：
St'ephan Thomass'e
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作者：
Nathalie Aubrun;S. Barbieri;St'ephan Thomass'e
Gao， Jackson和Seward的一个定理，最初被Glasner和Uspenskij推测为假，断言每个可数群都有$2$-染色。这个结果的一个直接结果是，每个可数群在字母表$\{0,1\}$上都有一个强非周期子位移。在本文中，我们首先利用Lovasz局部引理给出了上述定理的一个新的简单证明，然后证明了对于任意有限生成群$G$有效闭强非周期子移的存在性。我们还研究了将Sturmian序列的一个性质推广到有限生成群的子移位的构造问题。更准确地说，在字母表$\{0,1\}$上的子移位具有均匀的密度$\alpha \in[0,1]$，如果对于每一个构形，在任何递增的球序列中$1$的密度收敛于$\alpha$。我们给出了一个更一般的结果，即这些子移在亚指数增长群中总是存在的。
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 Lovasz 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 $\alpha \in [0,1]$ if for every configuration the density of $1$'s in any increasing sequence of balls converges to $\alpha$. We show a slightly more general result which implies that these subshifts always exist in the case of groups of subexponential growth.