Occurrences of palindromes in characteristic Sturmian words

This paper is concerned with palindromes occurring in characteristic Sturmian words $c_\alpha$ of slope $\alpha$, where $\alpha \in (0,1)$ is an irrational. As $c_\alpha$ is a uniformly recurrent infinite word, any (palindromic) factor of $c_\alpha$ occurs infinitely many times in $c_\alpha$ with bounded gaps. Our aim is to completely describe where palindromes occur in $c_\alpha$. In particular, given any palindromic factor $u$ of $c_\alpha$, we shall establish a decomposition of $c_\alpha$ with respect to the occurrences of $u$. Such a decomposition shows precisely where $u$ occurs in $c_\alpha$, and this is directly related to the continued fraction expansion of $\alpha$.Comment: 17 page

On the Number of Balanced Words of Given Length and Height over a Two-Letter Alphabet

We exhibit a recurrence on the number of discrete line segments joining two integer points in the plane using an encoding of such segments as balanced words of given length and height over the two-letter alphabet $\{0,1\}$. We give generating functions and study the asymptotic behaviour. As a particular case, we focus on the symmetrical discrete segments which are encoded by balanced palindromes.Comment: 24 page

Palindromic complexity of infinite words associated with simple Parry numbers

A simple Parry number is a real number \beta>1 such that the R\'enyi expansion of 1 is finite, of the form d_\beta(1)=t_1...t_m. We study the palindromic structure of infinite aperiodic words u_\beta that are the fixed point of a substitution associated with a simple Parry number \beta. It is shown that the word u_\beta contains infinitely many palindromes if and only if t_1=t_2= ... =t_{m-1} \geq t_m. Numbers \beta satisfying this condition are the so-called confluent Pisot numbers. If t_m=1 then u_\beta is an Arnoux-Rauzy word. We show that if \beta is a confluent Pisot number then P(n+1)+ P(n) = C(n+1) - C(n)+ 2, where P(n) is the number of palindromes and C(n) is the number of factors of length n in u_\beta. We then give a complete description of the set of palindromes, its structure and properties.Comment: 28 pages, to appear in Annales de l'Institut Fourie

On Theta-palindromic Richness

In this paper we study generalization of the reversal mapping realized by an arbitrary involutory antimorphism $\Theta$. It generalizes the notion of a palindrome into a $\Theta$-palindrome -- a word invariant under $\Theta$. For languages closed under $\Theta$ we give the relation between $\Theta$-palindromic complexity and factor complexity. We generalize the notion of richness to $\Theta$-richness and we prove analogous characterizations of words that are $\Theta$-rich, especially in the case of set of factors invariant under $\Theta$. A criterion for $\Theta$-richness of $\Theta$-episturmian words is given together with other examples of $\Theta$-rich words.Comment: 14 page