Characterizations of finite and infinite episturmian words via lexicographic orderings

Abstract

In this paper, we characterize by lexicographic order all finite Sturmian and episturmian words, i.e., all (finite) factors of such infinite words. Consequently, we obtain a characterization of infinite episturmian words in a wide sense (episturmian and episkew infinite words). That is, we characterize the set of all infinite words whose factors are (finite) episturmian. Similarly, we characterize by lexicographic order all balanced infinite words over a 2-letter alphabet; in other words, all Sturmian and skew infinite words, the factors of which are (finite) Sturmian.

Episturmian words: a survey

“…Such words were enumerated by Mignosi and Zamboni [88], who described a multi-dimensional generalization of the Euler phi-function that counts the number of finite Arnoux-Rauzy words of each length. Finite episturmian words have also been characterized with respect to lexicographic orderings in [62] (see Theorem 7.5 later).…”

“…When |A| = 2, Definition 2.1 gives the (aperiodic) Sturmian words, as well as the periodic balanced infinite words (also known as the periodic Sturmian words). See for instance [62] or Section 7.1.…”

“…In this section, we recall some basic definitions and properties relating to episturmian words which are needed throughout the paper. For the most part, we follow the notation and terminology of [43,73,75,62].…”

Directive words of episturmian words: equivalences and normalization

“…In this section, we recall some basic definitions and properties relating to episturmian words which are needed later in the paper. For the most part, we follow the notation and terminology of [5,10,13,15].…”

“…More precisely, the family of episturmian words is composed of the Arnoux-Rauzy sequences, images of the Arnoux-Rauzy sequences by episturmian morphisms, and certain periodic infinite words. In the binary case, Arnoux-Rauzy sequences are exactly the Sturmian words whereas episturmian words include all recurrent balanced words, that is, periodic balanced words and Sturmian words (see [10,22,29] for recent results relating episturmian words to the balanced property). See also [9] for a recent survey on episturmian theory.…”