A characterization of fine words over a finite alphabet

Abstract

To any infinite word w over a finite alphabet A we can associate two infinite words min(w) and max(w) such that any prefix of min(w) (resp. max(w)) is the lexicographically smallest (resp. greatest) amongst the factors of w of the same length. We say that an infinite word w over A is "fine" if there exists an infinite word u such that, for any lexicographic order, min(w) = au where a = min(A). In this paper, we characterize fine words; specifically, we prove that an infinite word w is fine if and only if w 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.Comment: 16 pages; presented at the conference on "Combinatorics, Automata and Number Theory", Liege, Belgium, May 8-19, 2006 (to appear in a special issue of Theoretical Computer Science

Morse and Hedlund’s Skew Sturmian Words Revisited

“…This suggests that there is also an extension of Theorem 3.2 in the same spirit which indeed is true as shown in [5] (see also [6]). …”

Extremal properties of (epi)Sturmian sequences and distribution modulo $1$

“…One is purely combinatorial and looks at generalizations of Sturmian sequences; in particular episturmian sequences, which share many properties with Sturmian sequences and have similar extremal properties. In this direction, characterizations of finite and infinite (epi)Sturmian sequences via lexicographic orderings have recently been studied (see [37,38,40,48,51,72,73,74]). The other type of generalization is number-theoretic and looks at distribution modulo 1 from a combinatorial point of view.…”

Classification of eventually periodic subshifts

“…Sturmian sequences, especially those which are not periodic nor eventually periodic, have attracted the attention of many mathematicians [2,4,5,6,10,12]. However, skew Sturmian sequences are important to study since they appear naturally in the generalization of properties of Sturmian sequences [1,3,7,16]. By showing in this paper how to compute the size of an anomaly word of a skew Sturmian sequence, we are able to apply our results to obtain a characterization for conjugacy of skew Sturmian subshifts.…”