Topological invariants for words of linear factor complexity

TL;DR

This paper introduces a topological space associated with infinite words and shows that for words with linear factor complexity, this space is finite, exploring its properties and posing related open problems.

Contribution

It constructs the space ${ m Rec}(w)$ for infinite words and proves its finiteness for words with linear factor complexity, linking combinatorics on words with topology.

Findings

${ m Rec}(w)$ is finite for words with linear factor complexity.

Existence of words with factor complexity ${ m O}(nf(n))$ where ${ m Rec}(w)$ is infinite.

Open problem: which finite topological spaces can be realized as ${ m Rec}(w)$?

Abstract

Given a finite alphabet $\Sigma$ and a right-infinite word $w$ over the alphabet $\Sigma$, we construct a topological space ${\rm Rec}(w)$ consisting of all right-infinite recurrent words whose factors are all factors of $w$, where we work up to an equivalence in which two words are equivalent if they have the exact same set of factors (finite contiguous subwords). We show that ${\rm Rec}(w)$ can be endowed with a natural topology and we show that if $w$ is word of linear factor complexity then ${\rm Rec}(w)$ is a finite topological space. In addition, we note that there are examples which show that if $f:\mathbb{N}\to \mathbb{N}$ is a function that tends to infinity as $n\to \infty$ then there is a word whose factor complexity function is ${\rm O}(nf(n))$ such that ${\rm Rec}(w)$ is an infinite set. Finally, we pose a realization problem: which finite topological spaces can arise as ${\rm Rec}(w)$ for a word of linear factor complexity?