Jarkko Peltomäki

Introducing privileged words: Privileged complexity of Sturmian words

In this paper we introduce a new class of so-called privileged words which 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 words is given in terms of privileged complexity. The privileged complexity of the Thue-Morse word is also briefly studied.

Characterization of repetitions in Sturmian words: A new proof

Abstract We present a new, dynamical way to study powers (that is, repetitions) in Sturmian words based on results from Diophantine approximation theory. As a result, we provide an alternative and shorter proof of a result by Damanik and Lenz characterizing powers in Sturmian words [6] . Further, as a consequence, we obtain a previously known formula for the fractional index of a Sturmian word based on the continued fraction expansion of its slope.

Remarks on Privileged Words

We discuss the notion of privileged word, recently introduced by Peltomaki. A word w is privileged if it is of length =1, then w^j is privileged for all j >= 0; (2) the language of privileged words is neither regular nor context-free; (3) there is a linear-time algorithm to check if a given word is privileged; and (4) there are at least 2^{n-5}/n^2 privileged binary words of length n.

Privileged factors in the Thue–Morse word—A comparison of privileged words and palindromes

In this paper we continue the studies of [J. Peltomäki, Introducing Privileged Words: Privileged Complexity of Sturmian Words, Theor. Comp. Sci. (2013)] on so-called privileged words. In this earlier work the basic properties of privileged words were proved along with a result that relates privileged words with so-called rich words (see [A. Glen et al, Palindromic Richness, Eur. Jour. of Comb. 30 (2009) 510–531]) and a result characterizing Sturmian words using privileged words. The main result of this paper is a derivation of a recursive formula for the privileged complexity function of the Thue-Morse word. Using the formula it’s proven that this function is unbounded, but its values still have arbitrary large gaps of zeros. All in all the behavior of this complexity function differs radically from the Sturmian case investigated in the earlier work. In addition we further study the relation between rich words and privileged words and compare the behavior of palindromes and privileged words in infinite words.Abstract In this paper we study the privileged complexity function of the Thue–Morse word. We prove a recursive formula describing this function, and using the formula we show that the function is unbounded and that the values of the function have arbitrarily large gaps of zeros. This demonstrates that the privileged complexity function of an infinite word can drastically differ from its palindromic complexity function, even though there are relations between these functions. Further we study the behavior of palindromes and privileged words in infinite words and the relation between rich words and privileged words.

A Square Root Map on Sturmian Words

We introduce a square root map on Sturmian words and study its properties. Given a Sturmian word of slope \(\alpha \), there exists exactly six minimal squares in its language. A minimal square does not have a square as a proper prefix. A Sturmian word s of slope \(\alpha \) can be written as a product of these six minimal squares: \(s = X_1^2 X_2^2 X_3^2 \cdots \). The square root of s is defined to be the word \(\sqrt{s} = X_1 X_2 X_3 \cdots \). We prove that \(\sqrt{s}\) is also a Sturmian word of slope \(\alpha \). Moreover, we describe how to find the intercept of \(\sqrt{s}\) and an occurrence of any prefix of \(\sqrt{s}\) in s. Related to the square root map, we characterize the solutions of the word equation \(X_1^2 X_2^2 \cdots X_m^2 = (X_1 X_2 \cdots X_m)^2\) in the language of Sturmian words of slope \(\alpha \) where the words \(X_i^2\) are minimal squares of slope \(\alpha \).