Quasiperiodic Sturmian words and morphisms

European Journal of Combinatorics, 2008

There is a natural involution on Christoffel words, originally studied by the second author in [dL2]. We show that it has several equivalent definitions: one of them uses the slope of the word, and changes the numerator and the denominator respectively in their inverses modulo the length; another one uses the cyclic graph allowing the construction of the word, by interpreting it in two ways (one as a permutation and its ascents and descents, coded by the two letters of the word, the other in the setting of the Fine and Wilf periodicity theorem); a third one uses central words and generation through iterated palindromic closure, by reversing the directive word. We show further that this involution extends to Sturmian morphisms, in the sense that it preserves conjugacy classes of these morphisms, which are in bijection with Christoffel words. The involution on morphisms is the restriction of some conjugation of the automorphisms of the free group. Finally, we show that, through the geometrical interpretation of substitutions of Arnoux and Ito, our involution is the same thing as duality of endomorphisms (modulo some conjugation).

In this paper we define Sturmian words with balanced construction. We formulate a fixed-point theorem for Sturmian words and analyze the set of all fixed points. The inspiration for this work came from the Kolakoski word and the general idea of self-reading sequences by Pȃun and Salomaa. The basis for this article is the author's earlier research on the influence of the continued fraction elements in the expansion of a ∈ ]0, 1[\Q on the construction of runs for the upper mechanical word with slope a and intercept 0.

Discrete Mathematics, 2000

We express any general characteristic sturmian word as a unique infinite non-increasing product of Lyndon words. Using this identity, we give a new ω-division for characteristic sturmian words. We also give a short proof of a result by Berstel and de Luca (Sturmian words, Lyndon words and trees, Theoret. Comput. Sci., 178 (1997) 171–203.); more precisely, we show that the set of factors of sturmian words that qualify as Lyndon words is the set of primitive Christoffel words.

Information Processing Letters, 2006

Counting the number of distinct factors in the words of a language gives a measure of complexity for that language similar to the factor-complexity of infinite words. Similarly as for infinite words, we prove that this complexity functions f (n) is either bounded or f (n) ≥ n+1. We call languages with bounded complexity periodic and languages with complexity f (n) = n + 1 Sturmian. We describe the structure of periodic languages and we characterize the Sturmian languages as the sets of factors of (one-or two-way) infinite Sturmian words.

Acta Informatica, 2000

Let w be a finite word and n the least non-negative integer such that w has no right special factor of length n and its right factor of length n is unrepeated. We prove that if all the factors of another word v up to the length n + 1 are also factors of w, then v itself is a factor of w. A similar result for ultimately periodic infinite words is established. As a consequence, some 'uniqueness conditions' for ultimately periodic words are obtained as well as an upper bound for the rational exponents of the factors of uniformly recurrent non-periodic infinite words. A general formula is derived for the 'critical exponent' of a power-free Sturmian word. In particular, we effectively compute the 'critical exponent' of any Sturmian sequence whose slope has a periodic development in a continued fraction.

Lecture Notes in Computer Science, 2013

A word is closed if it contains a proper factor that occurs both as a prefix and as a suffix but does not have internal occurrences, otherwise it is open. We deal with the sequence of open and closed prefixes of Sturmian words and prove that this sequence characterizes every finite or infinite Sturmian word up to isomorphisms of the alphabet. We then characterize the combinatorial structure of the sequence of open and closed prefixes of standard Sturmian words. We prove that every standard Sturmian word, after swapping its first letter, can be written as an infinite product of squares of reversed standard words.

Discrete Applied Mathematics, 2002

An inÿnite word x on the {0; 1} alphabet is balanced if, given two factors of x; w and w , having the same length, the di erence between the number of 0's in w (denoted by |w|0) and the number of 0's in w is at most 1, i.e. ||w|0 − |w |0| 6 1. It is well known that an aperiodic word is Sturmian if and only if it is balanced.

Theoretical Computer Science, 1997

We prove some new results concerning the structure, the combinatorics and the arithmetics of the set PER of all the words w having two periods p and q, p <q, which are coprimes and such that ]w] = pfq-2.

International Journal of Foundations of Computer Science, 2008

In this paper, we study some periodicity concepts on words. First, we extend the notion of full tilings which was recently introduced by Karhumäki, Lifshits, and Rytter to partial tilings. Second, we investigate the notion of quasiperiods and show in particular that the set of quasiperiodic words is a context-sensitive language that is not context-free, answering a conjecture by Dömösi, Horváth and Ito.

Theoretical Computer Science, 1994

de Luca, A. and F. Mignosi, Some combinatorial properties of Sturmian words, Theoretical Computer Science 136 (1994) 361-385.