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.

翻译：以限定的字母 $\ sigma$ 和一纯正的字母 $\ bf w$ 超过$\ sigma$, 我们定义了“ 谎言的复杂性” 函数 $L ⁇ bf w w} :\ mathbb{ N ⁇ to\ mathbb{N} $, 其值为 $ 美元是 长- 美元因子的复数类数( 周期性转变中) $x美元\ bf w$, 其属性为 等同等级的每个元素均以 $\ bf w $ 。 我们显示, 谎言的复杂性函数是用线性因子复杂性的单词一致捆绑的, 结果我们显示, 线性因子复杂的单词在最有限的几个原始因子 $y $y, 其属性是每美元的一个因子。 我们然后查看自动序列, 并显示 美元自动序列的精度功能是 $- 美元- 自动 。