论文标题
关于无限非二元词的亚伯封闭
On Abelian Closures of Infinite Non-binary Words
论文作者
论文摘要
如果每个字母在$ u $和$ v $中同样多次出现，则两个有限的单词$ u $和$ v $称为Abelian同等数字。无限单词$ \ mathbf {x} $的Abelian封闭$ \ Mathcal {a}（\ Mathbf {x}）$是无限单词$ \ MATHBF {y} $的集合，因此，对于每个因素$ u $ u $ us of $ \ mathbf {y} $ a $相当于$ u $。亚洲封闭的概念给出了斯特里亚语单词的特征：在统一的二进制单词中，周期性和上的sturmian单词在$ \ mathcal {a}（\ mathbf {x}} $ quars y Mathbf {x}）$ quars of shift orbit oorbit oorbit cloture $ω（\ mathbf {x x}）$。此外，对于一个不是Sturmian的基质二进制词，其Abelian闭合包含无限的许多微型乘坐。在本文中，我们考虑了众所周知的非二进制单词家庭的封闭，例如平衡的单词和最小的复杂性单词。我们还考虑了亚伯封闭的一般次要换档，并对其亚伯封闭的最初观察到了一些相关的开放问题。
Two finite words $u$ and $v$ are called abelian equivalent if each letter occurs equally many times in both $u$ and $v$. The abelian closure $\mathcal{A}(\mathbf{x})$ of an infinite word $\mathbf{x}$ is the set of infinite words $\mathbf{y}$ such that, for each factor $u$ of $\mathbf{y}$, there exists a factor $v$ of $\mathbf{x}$ which is abelian equivalent to $u$. The notion of an abelian closure gives a characterization of Sturmian words: among uniformly recurrent binary words, periodic and aperiodic Sturmian words are exactly those words for which $\mathcal{A}(\mathbf{x})$ equals the shift orbit closure $Ω(\mathbf{x})$. Furthermore, for an aperiodic binary word that is not Sturmian, its abelian closure contains infinitely many minimial subshifts. In this paper we consider the abelian closures of well-known families of non-binary words, such as balanced words and minimal complexity words. We also consider abelian closures of general subshifts and make some initial observations of their abelian closures and pose some related open questions.
