Given a finite alphabet $\Sigma$ and a right-infinite word $\bf w$ over $\Sigma$, we define the Lie complexity function $L_{\bf w}:\mathbb{N}\to \mathbb{N}$, whose value at $n$ is the number of conjugacy classes (under cyclic shift) of length-$n$ factors $x$ of $\bf w$ with the property that every element of the conjugacy class appears in $\bf w$. We show that the Lie complexity function is uniformly bounded for words with linear factor complexity, and as a result we show that words of linear factor complexity have at most finitely many primitive factors $y$ with the property that $y^n$ is again a factor for every $n$. We then look at automatic sequences and show that the Lie complexity function of a $k$-automatic sequence is again $k$-automatic.

中文翻译：

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谎言的复杂性

给定一个有限字母$ \ Sigma $和一个右无穷单词$ \ bf w $超过$ \ Sigma $，我们定义了Lie复杂度函数$ L _ {\ bf w}：\ mathbb {N} \到\ mathbb {N } $，其值$ n $是长度的共轭类别（在循环移位下）-$ n $的系数$ x $的$ \ bf w $，其属性是，共轭类别的每个元素都出现在$ \ bf w $。我们证明了Lie复杂度函数对于具有线性因子复杂度的单词是一致有界的，结果表明，线性因子复杂度的单词最多具有有限的许多本原因子$ y $，而属性$ y ^ n $又是每$ n $一个因素。然后，我们查看自动序列，并显示$ k $-自动序列的Lie复杂度函数再次是$ k $ -automatic。