99 research outputs found
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
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 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 it has a prefix
which cannot be decomposed to a concatenation of at most palindromes.Comment: Submitted to European Journal of Combinatoric
On Theta-palindromic Richness
In this paper we study generalization of the reversal mapping realized by an
arbitrary involutory antimorphism . It generalizes the notion of a
palindrome into a -palindrome -- a word invariant under . For
languages closed under we give the relation between
-palindromic complexity and factor complexity. We generalize the notion
of richness to -richness and we prove analogous characterizations of
words that are -rich, especially in the case of set of factors
invariant under . A criterion for -richness of
-episturmian words is given together with other examples of
-rich words.Comment: 14 page
On the least number of palindromes contained in an infinite word
We investigate the least number of palindromic factors in an infinite word.
We first consider general alphabets, and give answers to this problem for
periodic and non-periodic words, closed or not under reversal of factors. We
then investigate the same problem when the alphabet has size two.Comment: Accepted for publication in Theoretical Computer Scienc
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
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