Lie complexity of words

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.