Quasiperiodic Sturmian words and morphisms

Episturmian words: a survey

“…A shorter proof of this fact was provided in [81] and this result has also been proven true for episturmian words in [64].…”

“…The notion was later extended to infinite words by Marcus [85] who opened some questions, particularly concerning quasiperiodicity of Sturmian words. After a brief answer to some of these questions in [79], the Sturmian case was fully studied by Levé and Richomme [81] who proved that a Sturmian word is non-quasiperiodic if and only if it is an infinite Lyndon word. The study of quasiperiodicity in Sturmian words was very recently extended to episturmian words by Glen, Levé, and Richomme [58,64,80], who have completely described all of the quasiperiods of an episturmian word, yielding a characterization of quasiperiodic episturmian words in terms of their directive words.…”

Quasiperiodic and Lyndon episturmian words

“…This idea allows an episturmian word to be defined uniquely by its so-called normalized directive word, defined by some factor avoidance. It can be seen as a generalization of a previous result by Berthé, Holton, and Zamboni [7], which was used in [29] to show that the directive word of a non-quasiperiodic Sturmian word can take only two possible (similar) forms. For non-binary episturmian words, even those defined on a ternary alphabet, this simplicity does not hold since a combinatorial explosion of the number of cases occurs.…”

“…Morphic decompositions of Sturmian words have been used in [29] to characterize quasiperiodic Sturmian words.…”

“…In [33], Marcus extended this notion to infinite words and opened some questions, particularly concerning quasiperiodicity of Sturmian words. After a brief answer to some of these questions in [27], the Sturmian case was fully studied in [29] where it was proved 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.…”