Language complexity of rotations and Sturmian sequences

AbstractGiven a rotation of the circle, we study the complexity of formal languages that are generated by the itineraries of interval covers. These languages are regular iff the rotation is rational. In the case of irrational rotations, our study reduces to that of the language complexity of the corresponding Sturmian sequences. We show that for a large class of irrationals, including e, all quadratic numbers and more generally all Hurwitz numbers, the corresponding languages can be recognized by a nondeterministic Turing machine in linear time (in other words, belongs to NLIN)

On the Structure of Bispecial Sturmian Words

A balanced word is one in which any two factors of the same length contain the same number of each letter of the alphabet up to one. Finite binary balanced words are called Sturmian words. A Sturmian word is bispecial if it can be extended to the left and to the right with both letters remaining a Sturmian word. There is a deep relation between bispecial Sturmian words and Christoffel words, that are the digital approximations of Euclidean segments in the plane. In 1997, J. Berstel and A. de Luca proved that \emph{palindromic} bispecial Sturmian words are precisely the maximal internal factors of \emph{primitive} Christoffel words. We extend this result by showing that bispecial Sturmian words are precisely the maximal internal factors of \emph{all} Christoffel words. Our characterization allows us to give an enumerative formula for bispecial Sturmian words. We also investigate the minimal forbidden words for the language of Sturmian words.Comment: arXiv admin note: substantial text overlap with arXiv:1204.167

Coding rotations on intervals

We show that the coding of rotation by $\alpha$ on $m$ intervals with rationally independent lengths can be recoded over $m$ Sturmian words of angle $\alpha.$ More precisely, for a given $m$ an universal automaton is constructed such that the edge indexed by the vector of values of the $i$th letter on each Sturmian word gives the value of the $i$th letter of the coding of rotation.Comment: LIAFA repor

On the number of return words in infinite words with complexity 2n+1

In this article, we count the number of return words in some infinite words with complexity 2n+1. We also consider some infinite words given by codings of rotation and interval exchange transformations on k intervals. We prove that the number of return words over a given word w for these infinite words is exactly k.Comment: see also http://liafa.jussieu.fr/~vuillon/articles.htm

The number of binary rotation words

We consider binary rotation words generated by partitions of the unit circle to two intervals and give a precise formula for the number of such words of length n. We also give the precise asymptotics for it, which happens to be O(n^4). The result continues the line initiated by the formula for the number of all Sturmian words obtained by Lipatov in 1982, then independently by Berenstein, Kanal, Lavine and Olson in 1987, Mignosi in 1991, and then with another technique by Berstel and Pocchiola in 1993.Comment: Submitted to RAIRO IT