Topological invariants for words of linear factor complexity

Abstract

Given a finite alphabet Σ and a right-infinite word w over the alphabet Σ, we construct a topological space 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 Rec(w) can be endowed with a natural topology and we show that if w is word of linear factor complexity then Rec(w) is a finite topological space. In addition, we note that there are examples which show that if f:N→N is a function that tends to infinity as n→∞ then there is a word whose factor complexity function is O(nf(n)) such that Rec(w) is an infinite set. Finally, we pose a realization problem: which finite topological spaces can arise as Rec(w) for a word of linear factor complexity?