Lie complexity of words

Abstract

Given a finite alphabet Σ and a right-infinite word w over Σ, we define the Lie complexity function Lw:N→N, whose value at n is the number of conjugacy classes (under cyclic shift) of length-n factors x of w with the property that every element of the conjugacy class appears in w.We show that the Lie complexity function is uniformly bounded for words with linear factor complexity. As a result, we show that words of linear factor complexity have at most finitely many primitive factors y with the property that yn 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 also k-automatic.