More on Square-free Words Obtained from Prefixes by Permutations

Abstract

An infinite square-free word w over the alphabet Σ3 = {0, 1, 2} is said to have a k-stem σ if |σ| = k and w = σw1w2$\dots$ where for each i, there exists a permutation πi of Σ3 which extended to a morphism gives wi = πiσ. Harju proved that there exists an infinite k-stem word for k = 1, 2, 3, 9 and 13 ≤ k ≤ 19, but not for 4 ≤ k ≤ 8 and 10 ≤ k ≤ 12. He asked whether k-stem words exist for each k ≥ 20. We give a positive answer to this question. Currie has found another construction that answers Harju's question.