Lie complexity of words

TL;DR

This paper introduces the Lie complexity function for infinite words, explores its boundedness in words with linear factor complexity, and demonstrates that for automatic sequences, the Lie complexity is also automatic.

Contribution

It defines the Lie complexity function for infinite words, establishes its boundedness for words with linear factor complexity, and proves that the Lie complexity of automatic sequences is itself automatic.

Findings

Lie complexity is bounded for words with linear factor complexity.

Words with linear factor complexity have finitely many primitive factors with all powers as factors.

Lie complexity of a $k$-automatic sequence is itself $k$-automatic.

Abstract

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.