Introducing privileged words: Privileged complexity of Sturmian words

Introducing privileged words: Privileged complexity of Sturmian words

In this paper we study the class of so-called privileged wordswhich have been previously considered only a little. We develop the basic properties of privileged words, which turn out to share similar properties with palindromes. Privileged words are studied in relation to previously studied classes of words, rich words, Sturmian words and episturmian words. A new characterization of Sturmian wo...

Remarks on Privileged Words

We discuss the notion of privileged word, recently introduced by Peltomäki. A word w is privileged if it is of length ≤ 1, or has a privileged border that occurs exactly twice in w. We prove the following results: (1) if wk is privileged for some k ≥ 1, then wj is privileged for all j ≥ 0; (2) the language of privileged words is neither regular nor context-free; (3) there is a linear-time algor...

Sturmian Words and Constant Additive Complexity

Resolving a question of Banero, we show that for every integer K > 1, there exists a word with additive complexity identically K. This result is perhaps surprising in light of the rather strong restriction on the existence of words with constant abelian complexity, given in the work of Currie and Rampersad. To prove our result we generalize the notion of a sturmian word. We also pose some quest...

Return words in Sturmian and episturmian words

Considering each occurrence of a word w in a recurrent infinite word, we define the set of return words of w to be the set of all distinct words beginning with an occurrence of w and ending exactly just before the next occurrence of w in the infinite word. We give a simpler proof of the recent result (of the second author) that an infinite word is Sturmian if and only if each of its factors has...

Sturmian Words: Dynamical Systems and Derivated Words