A characterization of fine words over a finite alphabet

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