A characterization of fine words over a finite alphabet

To any infinite word over a finite alphabet A we can associate two infinite words and such that any prefix of (resp. ) is the lexicographically smallest (resp. greatest) amongst the factors of of the same length. We say that an infinite word over A is fine if there exists an infinite word such that, for any lexicographic order, where a=min(A). In this paper, we characterize fine words; specifically, we prove that an infinite word is fine if and only if 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.