35 research outputs found
On the number of return words in infinite words with complexity 2n+1
In this article, we count the number of return words in some infinite words
with complexity 2n+1. We also consider some infinite words given by codings of
rotation and interval exchange transformations on k intervals. We prove that
the number of return words over a given word w for these infinite words is
exactly k.Comment: see also http://liafa.jussieu.fr/~vuillon/articles.htm
A new geometric approach to Sturmian words
We introduce a new geometric approach to Sturmian words by means of a mapping
that associates certain lines in the n x n -grid and sets of finite Sturmian
words of length n. Using this mapping, we give new proofs of the formulas
enumerating the finite Sturmian words and the palindromic finite Sturmian words
of a given length. We also give a new proof for the well-known result that a
factor of a Sturmian word has precisely two return words.Comment: 12 pages, 7 figures. A preprint of a paper to appear in Theoretical
Computer Scienc
A local balance property of episturmian words
We prove that episturmian words and Arnoux-Rauzy sequences can be
characterized using a local balance property. We also give a new
characterization of epistandard words and show that the set of finite words
that are not factors of an episturmian word is not context-free
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.Comment: 16 pages; presented at the conference on "Combinatorics, Automata and
Number Theory", Liege, Belgium, May 8-19, 2006 (to appear in a special issue
of Theoretical Computer Science
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
Return Words and Recurrence Function of a Class of Infinite Words
Many combinatorial and arithmetical properties have been studied for infinite words ub associated with Ă-integers. Here, new results describing return words and recurrence function for a special case of ub will be presented. The methods used here can be applied to more general infinite words, but the description then becomes rather technical.
Occurrences of palindromes in characteristic Sturmian words
This paper is concerned with palindromes occurring in characteristic Sturmian
words of slope , where is an irrational.
As is a uniformly recurrent infinite word, any (palindromic) factor
of occurs infinitely many times in with bounded gaps. Our
aim is to completely describe where palindromes occur in . In
particular, given any palindromic factor of , we shall establish
a decomposition of with respect to the occurrences of . Such a
decomposition shows precisely where occurs in , and this is
directly related to the continued fraction expansion of .Comment: 17 page
Palindromic complexity of codings of rotations
International audienceWe study the palindromic complexity of infinite words obtained by coding rotations on partitions of the unit circle by inspecting the return words. The main result is that every coding of rotations on two intervals is full, that is, it realizes the maximal palindromic complexity. As a byproduct, a slight improvement about return words in codings of rotations is obtained: every factor of a coding of rotations on two intervals has at most 4 complete return words, where the bound is realized only for a finite number of factors. We also provide a combinatorial proof for the special case of complementary-symmetric Rote sequences by considering both palindromes and antipalindromes occurring in it