arXiv subjects

Amy Glen

Publications and source records attributed to Amy Glen.

23 records · Page 2Linked to original sources

arXiv2007

Characterizations of finite and infinite episturmian words via lexicographic orderings

In this paper, we characterize by lexicographic order all finite Sturmian and episturmian words, i.e., all (finite) factors of such infinite words. Consequently, we obtain a characterization of infinite episturmian words in a "wide sense" (episturmian and episkew infinite words). That is, we characterize the set of all infinite words whose factors are (finite) episturmian. Similarly, we characterize by lexicographic order all balanced infinite words over a 2-letter alphabet; in other words, all Sturmian and skew infinite words, the factors of which are (finite) Sturmian.

math.CO↗

arXiv2007

Occurrences of palindromes in characteristic Sturmian words

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

math.CO↗

arXiv2007

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 ($k\geq2$).

math.CO↗

arXiv2007

A characterization of fine words over a finite alphabet

To any infinite word w over a finite alphabet A we can associate two infinite words min(w) and max(w) such that any prefix of min(w) (resp. max(w)) is the lexicographically smallest (resp. greatest) amongst the factors of w of the same length. We say that an infinite word w over A is "fine" if there exists an infinite word u such that, for any lexicographic order, min(w) = au where a = min(A). In this paper, we characterize fine words; specifically, we prove that an infinite word w is fine if and only if w is either a "strict episturmian word" or a strict "skew episturmian word''. This characterization generalizes a recent result of G. Pirillo, who proved that a fine word over a 2-letter alphabet is either an (aperiodic) Sturmian word, or an ultimately periodic (but not periodic) infinite word, all of whose factors are (finite) Sturmian.

math.CO↗

arXiv2007

A note on the Markoff condition and central words

We define Markoff words as certain factors appearing in bi-infinite words satisfying the Markoff condition. We prove that these words coincide with central words, yielding a new characterization of Christoffel words.

math.CO↗