Quasiperiodic and Lyndon episturmian words

TL;DR

This paper extends the characterization of quasiperiodic Sturmian words to episturmian words, describing their quasiperiods and Lyndon properties using directive words and morphisms, revealing a broader class of non-quasiperiodic episturmian words.

Contribution

It provides a complete description of quasiperiods in episturmian words and characterizes all episturmian Lyndon words, generalizing previous Sturmian results.

Findings

Characterization of quasiperiods of episturmian words

Identification of all Lyndon episturmian words

Broader class of non-quasiperiodic episturmian words

Abstract

Recently the second two authors characterized quasiperiodic Sturmian words, proving that a Sturmian word is non-quasiperiodic if and only if it is an infinite Lyndon word. Here we extend this study to episturmian words (a natural generalization of Sturmian words) by describing all the quasiperiods of an episturmian word, which yields a characterization of quasiperiodic episturmian words in terms of their "directive words". Even further, we establish a complete characterization of all episturmian words that are Lyndon words. Our main results show that, unlike the Sturmian case, there is a much wider class of episturmian words that are non-quasiperiodic, besides those that are infinite Lyndon words. Our key tools are morphisms and directive words, in particular "normalized" directive words, which we introduced in an earlier paper. Also of importance is the use of "return words" to characterize quasiperiodic episturmian words, since such a method could be useful in other contexts.