38 research outputs found
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
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
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
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
Clustering and Arnoux-Rauzy words
We characterize the clustering of a word under the Burrows-Wheeler transform
in terms of the resolution of a bounded number of bispecial factors belonging
to the language generated by all its powers. We use this criterion to compute,
in every given Arnoux-Rauzy language on three letters, an explicit bound
such that each word of length at least is not clustering; this bound is
sharp for a set of Arnoux-Rauzy languages including the Tribonacci one. In the
other direction, we characterize all standard Arnoux-Rauzy clustering words,
and all perfectly clustering Arnoux-Rauzy words. We extend some results to
episturmian languages, characterizing those which produce infinitely many
clustering words, and to larger alphabets
Characterization of infinite LSP words and endomorphisms preserving the LSP property
Answering a question of G. Fici, we give an -adic characterization of
thefamily of infinite LSP words, that is, the family of infinite words having
all their left special factors as prefixes.More precisely we provide a finite
set of morphisms and an automaton such that an infinite word is
LSP if and only if it is -adic and one of its directive words is
recognizable by .Then we characterize the endomorphisms that preserve
the property of being LSP for infinite words.This allows us to prove that there
exists no set of endomorphisms for which the set of infinite LSP words
corresponds to the set of -adic words. This implies that an automaton is
required no matter which set of morphisms is used.Comment: arXiv admin note: text overlap with arXiv:1705.0578
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