Infinite ternary square-free words concatenated from permutations of a single word

We answer a question of Harju: An infinite square-free ternary word with an $n$-stem factorization exists for any $n\ge 13$. We show that there are uniform ternary morphisms of length $k$ for every $k\ge 23$. This resolves almost completely a problem of the author and Rampersad.

💡 Research Summary

The paper addresses a long‑standing question posed by Harju concerning the existence of infinite square‑free ternary words that admit an $n$‑stem factorisation, i.e., an infinite word that can be written as a concatenation of permutations of a single fixed block $s$ of length $n$. Harju asked for the smallest $n$ for which such a word exists. The authors prove that for every integer $n\ge13$ there indeed exists an infinite ternary word that is both square‑free and admits an $n$‑stem factorisation.

The central technical tool is the construction of uniform ternary morphisms (also called uniform morphisms) of arbitrary length $k\ge23$ that preserve square‑freeness. A uniform morphism $\varphi:{0,1,2}\to{0,1,2}^k$ maps each letter to a word of the same length $k$. The authors design $\varphi_k$ so that (i) each image contains no double letter (e.g., “00”), (ii) the three images are mutually related by a 3‑cycle permutation, and (iii) any possible occurrence of a square $xx$ in $\varphi_k(u)$ for a square‑free $u$ is ruled out by a careful analysis of boundary overlaps. For each $k\ge23$ they give an explicit description; for instance, when $k=23$ the morphism is

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