## A characterization of fine words over a finite alphabet.(English)Zbl 1133.68065

Summary: To any infinite word $$\mathbf t$$ over a finite alphabet $$\mathcal A$$ we can associate two infinite words $$\min (\mathbf t)$$ and $$\max (\mathbf t)$$ such that any prefix of $$\min (\mathbf t)$$ (resp. $$\max (\mathbf t)$$) is the lexicographically smallest (resp. greatest) amongst the factors of $$\mathbf t$$ of the same length. We say that an infinite word $$\mathbf t$$ over $$\mathcal A$$ is fine if there exists an infinite word $$\mathbf s$$ such that, for any lexicographic order, $$\min (\mathbf t) = a \mathbf s$$ where $$a = \min (\mathcal A)$$. In this paper, we characterize fine words; specifically, we prove that an infinite word $$\mathbf t$$ is fine if and only if $$\mathbf t$$ is either a strict episturmian word or a strict “skew episturmian word”. This characterization generalizes a recent result of G. Pirillo, who proved that a fine word over a 2-letter alphabet is either an (aperiodic) Sturmian word, or an ultimately periodic (but not periodic) infinite word, all of whose factors are (finite) Sturmian.

### MSC:

 68R15 Combinatorics on words 68Q45 Formal languages and automata
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### References:

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