57 research outputs found
Quasiperiodic Sturmian words and morphisms
We 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
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
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
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
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
Quasicrystals, model sets, and automatic sequences
We survey mathematical properties of quasicrystals, first from the point of
view of harmonic analysis, then from the point of view of morphic and automatic
sequences.
Nous proposons un tour d'horizon de propri\'et\'es math\'ematiques des
quasicristaux, d'abord du point de vue de l'analyse harmonique, ensuite du
point de vue des suites morphiques et automatiques
Ten Conferences WORDS: Open Problems and Conjectures
In connection to the development of the field of Combinatorics on Words, we
present a list of open problems and conjectures that were stated during the ten
last meetings WORDS. We wish to continually update the present document by
adding informations concerning advances in problems solving
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