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

Sturmian numeration systems and decompositions to palindromes

We extend the classical Ostrowski numeration systems, closely related to Sturmian words, by allowing a wider range of coefficients, so that possible representations of a number $n$ better reflect the structure of the associated Sturmian word. In particular, this extended numeration system helps to catch occurrences of palindromes in a characteristic Sturmian word and thus to prove for Sturmian words the following conjecture stated in 2013 by Puzynina, Zamboni and the author: If a word is not periodic, then for every $Q>0$ it has a prefix which cannot be decomposed to a concatenation of at most $Q$ palindromes.Comment: Submitted to European Journal of Combinatoric

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

Factors of characteristic words: Location and decompositions

AbstractLet α be an irrational number with 0<α<1, and let fα be the characteristic word of α. The location of a factor w of fα is defined to be the set of all positions in fα at which w occurs. In this paper, an explicit formula of the location of each factor w of fα is obtained. The decompositions REDα(w) and SODα(w) of fα associated with w are established. The former one involves all return words of w while the latter one involves all occurrences of w and the corresponding separate factors and overlap factors. These results generalize the work by the present authors (2005), Z.-X. Wen and Z.-Y. Wen (1994), G. Melançon (1999), W.-T. Cao and Z.-Y. Wen (2003), I.M. Araújo and V. Bruyère (2005), and A. Glen (2006)