Characterizations of finite and infinite episturmian words via lexicographic orderings

TL;DR

This paper characterizes finite and infinite episturmian words using lexicographic orderings, providing a comprehensive framework for understanding their structure and relation to balanced words over two-letter alphabets.

Contribution

It introduces a lexicographic characterization of all finite and infinite episturmian words, extending to a wide class of balanced infinite words over two-letter alphabets.

Findings

Lexicographic orderings characterize all finite episturmian words.

Infinite episturmian words are characterized by their factors being finite episturmian.

Balanced infinite words over two-letter alphabets are characterized by their factors being Sturmian or skew.

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.