754 research outputs found
Various Properties of Sturmian Words
This overview paper is devoted to Sturmian words. The first part summarizes different characterizations of Sturmian words. Besides the well known theorem of Hedlund and Morse it also includes recent results on the characterization of Sturmian words using return words or palindromes. The second part deals with substitution invariant Sturmian words, where we present our recent results. We generalize one-sided Sturmian words using the cut-and-project scheme and give a full characterization of substitution invariant Sturmian words.
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 -bonacci word: a generalization of the
Fibonacci word to a -letter alphabet ().Comment: 26 pages; extended version of a paper presented at the 5th
International Conference on Words, Montreal, Canada, September 13-17, 200
Extremal properties of (epi)Sturmian sequences and distribution modulo 1
Starting from a study of Y. Bugeaud and A. Dubickas (2005) on a question in
distribution of real numbers modulo 1 via combinatorics on words, we survey
some combinatorial properties of (epi)Sturmian sequences and distribution
modulo 1 in connection to their work. In particular we focus on extremal
properties of (epi)Sturmian sequences, some of which have been rediscovered
several times
The sequence of open and closed prefixes of a Sturmian word
A finite word is closed if it contains a factor that occurs both as a prefix
and as a suffix but does not have internal occurrences, otherwise it is open.
We are interested in the {\it oc-sequence} of a word, which is the binary
sequence whose -th element is if the prefix of length of the word is
open, or if it is closed. We exhibit results showing that this sequence is
deeply related to the combinatorial and periodic structure of a word. In the
case of Sturmian words, we show that these are uniquely determined (up to
renaming letters) by their oc-sequence. Moreover, we prove that the class of
finite Sturmian words is a maximal element with this property in the class of
binary factorial languages. We then discuss several aspects of Sturmian words
that can be expressed through this sequence. Finally, we provide a linear-time
algorithm that computes the oc-sequence of a finite word, and a linear-time
algorithm that reconstructs a finite Sturmian word from its oc-sequence.Comment: Published in Advances in Applied Mathematics. Journal version of
arXiv:1306.225
Directive words of episturmian words: equivalences and normalization
Episturmian morphisms constitute a powerful tool to study episturmian words.
Indeed, any episturmian word can be infinitely decomposed over the set of pure
episturmian morphisms. Thus, an episturmian word can be defined by one of its
morphic decompositions or, equivalently, by a certain directive word. Here we
characterize pairs of words directing a common episturmian word. We also
propose a way to uniquely define any episturmian word through a normalization
of its directive words. As a consequence of these results, we characterize
episturmian words having a unique directive word.Comment: 15 page
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.Comment: 18 pages; to appear in the European Journal of Combinatoric
Enumeration and Structure of Trapezoidal Words
Trapezoidal words are words having at most distinct factors of length
for every . They therefore encompass finite Sturmian words. We give
combinatorial characterizations of trapezoidal words and exhibit a formula for
their enumeration. We then separate trapezoidal words into two disjoint
classes: open and closed. A trapezoidal word is closed if it has a factor that
occurs only as a prefix and as a suffix; otherwise it is open. We investigate
open and closed trapezoidal words, in relation with their special factors. We
prove that Sturmian palindromes are closed trapezoidal words and that a closed
trapezoidal word is a Sturmian palindrome if and only if its longest repeated
prefix is a palindrome. We also define a new class of words, \emph{semicentral
words}, and show that they are characterized by the property that they can be
written as , for a central word and two different letters .
Finally, we investigate the prefixes of the Fibonacci word with respect to the
property of being open or closed trapezoidal words, and show that the sequence
of open and closed prefixes of the Fibonacci word follows the Fibonacci
sequence.Comment: Accepted for publication in Theoretical Computer Scienc
Open and Closed Prefixes of Sturmian Words
A word is closed if it contains a proper factor that occurs both as a prefix
and as a suffix but does not have internal occurrences, otherwise it is open.
We deal with the sequence of open and closed prefixes of Sturmian words and
prove that this sequence characterizes every finite or infinite Sturmian word
up to isomorphisms of the alphabet. We then characterize the combinatorial
structure of the sequence of open and closed prefixes of standard Sturmian
words. We prove that every standard Sturmian word, after swapping its first
letter, can be written as an infinite product of squares of reversed standard
words.Comment: To appear in WORDS 2013 proceeding
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