Episturmian words: a survey

Abstract

In this paper, we survey the rich theory of infinite episturmian words which generalize to any finite alphabet, in a rather resembling way, the well-known family of Sturmian words on two letters. After recalling definitions and basic properties, we consider episturmian morphisms that allow for a deeper study of these words. Some properties of factors are described, including factor complexity, palindromes, fractional powers, frequencies, and return words. We also consider lexicographical properties of episturmian words, as well as their connection to the balance property, and related notions such as finite episturmian words, Arnoux-Rauzy sequences, and "episkew words" that generalize the skew words of Morse and Hedlund.

Directive words of episturmian words: equivalences and normalization

“…was tackled by Justin and Pirillo in [15] for bi-infinite episturmian words, that is, episturmian words with letters indexed by Z (and not by N as considered until now). Let us recall relations between right-infinite episturmian words and bi-infinite episturmian words (see [15, p. 332] and [9] for more details).…”

“…spinned infinite word∆) satisfying the conditions of the above theorem, we say that ∆ (resp.∆) is a (spinned) directive word for t or that t is directed by ∆ (resp.∆). Notice that this directive word is exactly the one that arises from the equivalent definition of epistandard words that uses palindromic closure [5,9,13] and, in the binary case, it is related to the continued fraction of the slope of the straight line represented by a standard word (see [20]). It follows immediately from Theorem 2.1 that, with the notation of case ii), each t (n) is an episturmian word directed by x n+1xn+2 · · · .…”

Lamination languages

“…Note also that there exist affine complexity languages which are not lamination languages. In particular, Arnoux-Rauzy words [4] (or strict episturmian words [20]) are minimal words with affine complexity whose associated shift invariant languages are generally not lamination languages (see Corollary 4.2.4).…”

ON THE WORDS BY $k$ TO $k$ INSERTION OF A LETTER IN STURMIAN WORDS

“…During the last thirty years, Sturmian words were the subject of intensive study. These investigations led to the discovery of numerous characterizations and various properties [3,6,7,11,14,15] on these words. During the last two decades, palindromes were used extensively in the literature of infinite words combinatorial study (see citealbaca, droupi, kab).…”