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

Powers in a class of A-strict standard episturmian words

This paper concerns a specific class of strict standard episturmian words whose directive words resemble those of characteristic Sturmian words. In particular, we explicitly determine all integer powers occurring in such infinite words, extending recent results of Damanik and Lenz (2003), who studied powers in Sturmian words. The key tools in our analysis are canonical decompositions and a generalization of singular words, which were originally defined for the ubiquitous Fibonacci word. Our main results are demonstrated via some examples, including the $k$-bonacci word: a generalization of the Fibonacci word to a $k$-letter alphabet ($k\geq2$).Comment: 26 pages; extended version of a paper presented at the 5th International Conference on Words, Montreal, Canada, September 13-17, 200

Conjugates of characteristic Sturmian words generated by morphisms

This article is concerned with characteristic Sturmian words of slope α and 1-α (denoted by cα and c1-α resp.), where α∈(0,1) is an irrational number such that α=[0;1+d1,d2,..., dn] with dn≥d1≥1. It is known that both cα and c1-α are fixed points of non-trivial (standard) morphisms σ and σ̂, respectively, if and only if α has a continued fraction expansion as above. Accordingly, such words cα and c1-α are generated by the respective morphisms σ and σ̂. For the particular case when α=[0;2,r̄] (r≥1), we give a decomposition of each conjugate of cα (and hence c1-α) into generalized adjoining singular words, by considering conjugates of powers of the standard morphism σ by which it is generated. This extends a recent result of Levé and Séébold on conjugates of the infinite Fibonacci word

Conjugates of characteristic Sturmian words generated by morphisms

This article is concerned with characteristic Sturmian words of slope α and 1 − α (denoted by cα and c1−α respectively), where α ∈ (0, 1) is an irrational number such that α = [0; 1 + d1, d2,..., dn] with dn ≥ d1 ≥ 1. It is known that both cα and c1−α are fixed points of non-trivial (standard) morphisms σ and ˆσ, respectively, if and only if α has a continued fraction expansion as above. Accordingly, such words cα and c1−α are generated by the respective morphisms σ and ˆσ. For the particular case when α = [0; 2, r] (r ≥ 1), we give a decomposition of each conjugate of cα (and hence c1−α) into generalized adjoining singular words, by considering conjugates of powers of the standard morphism σ by which it is generated. This extends a recent result of Levé and Séébold on conjugates of the infinite Fibonacci word