14 research outputs found
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
Extremal properties of (epi)Sturmian sequences and distribution modulo 1
Starting from a study of Y. Bugeaud and A. Dubickas (2005) on a question in
distribution of real numbers modulo 1 via combinatorics on words, we survey
some combinatorial properties of (epi)Sturmian sequences and distribution
modulo 1 in connection to their work. In particular we focus on extremal
properties of (epi)Sturmian sequences, some of which have been rediscovered
several times
A Coloring Problem for Sturmian and Episturmian Words
We consider the following open question in the spirit of Ramsey theory: Given
an aperiodic infinite word , does there exist a finite coloring of its
factors such that no factorization of is monochromatic? We show that such a
coloring always exists whenever is a Sturmian word or a standard
episturmian word
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
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
Critical Exponents and Stabilizers of Infinite Words
This thesis concerns infinite words over finite alphabets. It contributes to two topics in this area: critical exponents and stabilizers.
Let w be a right-infinite word defined over a finite alphabet. The critical exponent of w is the supremum of the set of exponents r such that w contains an r-power as a subword. Most of the thesis (Chapters 3 through 7) is devoted to critical exponents.
Chapter 3 is a survey of previous research on critical exponents and repetitions in morphic words. In Chapter 4 we prove that every real number greater than 1 is the critical exponent of some right-infinite word over some finite alphabet. Our proof is constructive. In Chapter 5 we characterize critical exponents of pure morphic words generated by uniform binary morphisms. We also give an explicit formula to compute these critical exponents, based on a well-defined prefix of the infinite word. In Chapter 6 we generalize our results to pure morphic words generated by non-erasing morphisms over any finite alphabet. We prove that critical exponents of such words are algebraic, of a degree bounded by the alphabet size. Under certain conditions, our proof implies an algorithm for computing the critical exponent. We demonstrate our method by computing the critical exponent of some families of infinite words. In particular, in Chapter 7 we
compute the critical exponent of the Arshon word of order n for n ≥ 3.
The stabilizer of an infinite word w defined over a finite alphabet Σ is the set of morphisms f: Σ*→Σ* that fix w. In Chapter 8 we study various problems related to stabilizers and their generators. We show that over a binary alphabet, there exist stabilizers with at least n generators for all n. Over a ternary alphabet, the monoid of morphisms generating a given infinite word by iteration can be infinitely generated, even when the word is generated by iterating an invertible primitive morphism. Stabilizers of strict epistandard words are cyclic when non-trivial, while stabilizers of ultimately strict epistandard words are always non-trivial. For this latter family of words, we give a characterization of stabilizer elements.
We conclude with a list of open problems, including a new problem that has not been addressed yet: the D0L repetition threshold
On minimal critical exponent of balanced sequences
We study the threshold between avoidable and unavoidable repetitions in infinite balanced sequences over finite alphabets. The conjecture stated by Rampersad, Shallit and Vandomme says that the minimal critical exponent of balanced sequences over the alphabet of size d≥5 equals [Formula presented]. This conjecture is known to hold for d∈{5,6,7,8,9,10}. We refute this conjecture by showing that the picture is different for bigger alphabets. We prove that critical exponents of balanced sequences over an alphabet of size d≥11 are lower bounded by [Formula presented] and this bound is attained for all even numbers d≥12. According to this result, we conjecture that the least critical exponent of a balanced sequence over d letters is [Formula presented] for all d≥11. © 2022075-02-2021-1387, 075-02-2022-877; České Vysoké Učení Technické v Praze, ČVUT: SGS20/183/OHK4/3T/14; Ministerstvo Školství, Mládeže a Tělovýchovy, MŠMT: CZ.02.1.01/0.0/0.0/16_019/0000778; Ministry of Education and Science of the Russian Federation, MinobrnaukaThe second author was supported by Czech Technical University in Prague , through the project SGS20/183/OHK4/3T/14 . The first and the third authors were supported by The Ministry of Education, Youth and Sports of the Czech Republic through the project CZ.02.1.01/0.0/0.0/16_019/0000778 . The fourth author acknowledges the support by the Ministry of Science and Higher Education of the Russian Federation (Ural Mathematical Center project No. 075-02-2021-1387 ) and by Ural Mathematical Center , project No. 075-02-2022-877