17 research outputs found
Quasiperiodic and Lyndon episturmian words
Recently the second two authors characterized quasiperiodic Sturmian words,
proving that a Sturmian word is non-quasiperiodic if and only if it is an
infinite Lyndon word. Here we extend this study to episturmian words (a natural
generalization of Sturmian words) by describing all the quasiperiods of an
episturmian word, which yields a characterization of quasiperiodic episturmian
words in terms of their "directive words". Even further, we establish a
complete characterization of all episturmian words that are Lyndon words. Our
main results show that, unlike the Sturmian case, there is a much wider class
of episturmian words that are non-quasiperiodic, besides those that are
infinite Lyndon words. Our key tools are morphisms and directive words, in
particular "normalized" directive words, which we introduced in an earlier
paper. Also of importance is the use of "return words" to characterize
quasiperiodic episturmian words, since such a method could be useful in other
contexts.Comment: 33 pages; minor change
On Quasiperiodic Morphisms
Weakly and strongly quasiperiodic morphisms are tools introduced to study
quasiperiodic words. Formally they map respectively at least one or any
non-quasiperiodic word to a quasiperiodic word. Considering them both on finite
and infinite words, we get four families of morphisms between which we study
relations. We provide algorithms to decide whether a morphism is strongly
quasiperiodic on finite words or on infinite words.Comment: 12 page
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
Episturmian words: a survey
In this paper, we survey the rich theory of infinite episturmian words which
generalize to any finite alphabet, in a rather resembling way, the well-known
family of Sturmian words on two letters. After recalling definitions and basic
properties, we consider episturmian morphisms that allow for a deeper study of
these words. Some properties of factors are described, including factor
complexity, palindromes, fractional powers, frequencies, and return words. We
also consider lexicographical properties of episturmian words, as well as their
connection to the balance property, and related notions such as finite
episturmian words, Arnoux-Rauzy sequences, and "episkew words" that generalize
the skew words of Morse and Hedlund.Comment: 36 pages; major revision: improvements + new material + more
reference
Determining Sets of Quasiperiods of Infinite Words
A word is quasiperiodic if it can be obtained by concatenations and overlaps of a smaller word, called a quasiperiod. Based on links between quasiperiods, right special factors and square factors, we introduce a method to determine the set of quasiperiods of a given right infinite word. Then we study the structure of the sets of quasiperiods of right infinite words and, using our method, we provide examples of right infinite words with extremal sets of quasiperiods (no quasiperiod is quasiperiodic, all quasiperiods except one are quasiperiodic, ...). Our method is also used to provide a short proof of a recent characterization of quasiperiods of the Fibonacci word. Finally we extend this result to a new characterization of standard Sturmian words using a property of their sets of quasiperiods
On sets of indefinitely desubstitutable words
The stable set associated to a given set S of nonerasing endomorphisms or
substitutions is the set of all right infinite words that can be indefinitely
desubstituted over S. This notion generalizes the notion of sets of fixed
points of morphisms. It is linked to S-adicity and to property preserving
morphisms. Two main questions are considered. Which known sets of infinite
words are stable sets? Which ones are stable sets of a finite set of
substitutions? While bringing answers to the previous questions, some new
characterizations of several well-known sets of words such as the set of binary
balanced words or the set of episturmian words are presented. A
characterization of the set of nonerasing endomorphisms that preserve
episturmian words is also provided
Quasiperiodic Sturmian words and morphisms
AbstractWe characterize all quasiperiodic Sturmian words: A Sturmian word is not quasiperiodic if and only if it is a Lyndon word. Moreover, we study links between Sturmian morphisms and quasiperiodicity