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

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 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

Rich Words and Balanced Words

This thesis is mostly focused on palindromes. Palindromes have been studied extensively, in recent years, in the field of combinatorics on words.Our main focus is on rich words, also known as full words. These are words which have maximum number of distinct palindromes as factors.We shed some more light on these words and investigate certain restricted problems. Finite rich words are known to be extendable to infinite rich words. We study more closely how many different ways, and in which situations, rich words can be extended so that they remain rich.The defect of a ord is defined to be the number of palindromes the word is lacking.We will generalize the definition of defect with respect to extending the word to be infinite.The number of rich words, on an alphabet of size $n$, is given an upper and a lower bound. Hof, Knill and Simon presented (Commun. Math. Phys. 174, 1995) a well-known question whether all palindromic subshifts which are enerated by primitive substitutions arise from substitutions which are in class P. Over the years, this question has transformed a bit and is nowadays called the class P conjecture. The main point of the conjecture is to attempt to explain how an infinite word can contain infinitely many palindromes.We will prove a partial result of the conjecture. Rich square-free words are known to be finite (Pelantov\'a and Sarosta, Discrete Math. 313, 2013). We will give another proof for that result. Since they are finite, there exists a longest such word on an $n$-ary alphabet.We give an upper and a lower bound for the length of that word. We study also balanced words. Oliver Jenkinson proved (Discrete Math., Alg. and Appl. 1(4), 2009) that if we take the partial sum of the lexicographically ordered orbit of a binary word, then the balanced word gives the least partial sum. The balanced word also gives the largest product. We will show that, at the other extreme, there are the words of the form $0^{q-p}1^p$ ($p$ and $q$ are integers with $1\leq p<q$), which we call the most unbalanced words. They give the greatest partial sum and the smallest product.Tässä väitöskirjassa käsitellään pääasiassa palindromeja. Palindromeja on tutkittu viime vuosina runsaasti sanojen kombinatoriikassa.Suurin kiinnostuksen kohde tässä tutkielmassa on rikkaissa sanoissa. Nämä ovat sanoja joissa on maksimaalinen määrä erilaisia palindromeja tekijöinä.Näitä sanoja tutkitaan monesta eri näkökulmasta. Äärellisiä rikkaita sanoja voidaan tunnetusti jatkaa äärettömiksi rikkaiksi sanoiksi.Työssä tutkitaan tarkemmin sitä, miten monella tavalla ja missä eri tilanteissa rikkaita sanoja voidaan jatkaa siten, että ne pysyvät rikkaina.Sanan vajauksella tarkoitetaan puuttuvien palindromien lukumäärää.Vajauksen käsite yleistetään tapaukseen, jossa sanaa on jatkettava äärettömäksi sanaksi.Rikkaiden sanojen lukumäärälle annetaan myös ylä- ja alaraja. Hof, Knill ja Simon esittivät kysymyksen (Commun. Math. Phys. 174, 1995), saadaanko kaikki äärettömät sanat joissa on ääretön määrä palindromeja tekijöinä ja jotka ovat primitiivisen morfismin generoimia, morfismeista jotka kuuluvat luokkaan P. Nykyään tätä ongelmaa kutsutaan luokan P konjektuuriksi ja sen tarkoitus on saada selitys sille,millä tavalla äärettömässä sanassa voi olla tekijöinä äärettömän monta palindromia. Osittainen tulos tästä konjektuurista todistetaan. Rikkaiden neliövapaiden sanojen tiedetään olevan äärellisiä (Pelantov\'a ja Starosta, Discrete Math. 313, 2013). Tälle tulokselle annetaan uudenlainen todistus.Koska kyseiset sanat ovat äärellisiä, voidaan selvittää mikä niistä on pisin.Ylä- ja alaraja annetaan tällaisen pisimmän sanan pituudelle. Työssä tutkitaan myös tasapainotettuja sanoja.Tasapainotetut sanat antavat pienimmän osittaissumman binäärisille sanoille (Jenkinson, Discrete Math., Alg. and Appl. 1(4), 2009).Lisäksi ne antavat suurimman tulon.Muotoa $0^{q-p}1^p$ ($p$ ja $q$ ovat kokonaislukuja joille $1\leq p<q$) olevien sanojen todistetaan vastaavasti antavan suurimman osittaissumman ja pienimmän tulon.Ne muodostavat täten toisen ääripään tasapainotetuille sanoille, ja asettavat kaikki muut sanat näiden väliin.Siirretty Doriast

Abelian returns in Sturmian words

In this paper we study an abelian version of the notion of return word. Our main result is a new characterization of Sturmian words via abelian returns. Namely, we prove that a word is Sturmian if and only if each of its factors has two or three abelian returns. In addition, we describe the structure of abelian returns in Sturmian words, and discuss connections between abelian returns and periodicity

Cyclic Complexity of Words

We introduce and study a complexity function on words $c_x(n),$ called \emph{cyclic complexity}, which counts the number of conjugacy classes of factors of length $n$ of an infinite word $x.$ We extend the well-known Morse-Hedlund theorem to the setting of cyclic complexity by showing that a word is ultimately periodic if and only if it has bounded cyclic complexity. Unlike most complexity functions, cyclic complexity distinguishes between Sturmian words of different slopes. We prove that if $x$ is a Sturmian word and $y$ is a word having the same cyclic complexity of $x,$ then up to renaming letters, $x$ and $y$ have the same set of factors. In particular, $y$ is also Sturmian of slope equal to that of $x.$ Since $c_x(n)=1$ for some $n\geq 1$ implies $x$ is periodic, it is natural to consider the quantity $\liminf_{n\rightarrow \infty} c_x(n).$ We show that if $x$ is a Sturmian word, then $\liminf_{n\rightarrow \infty} c_x(n)=2.$ We prove however that this is not a characterization of Sturmian words by exhibiting a restricted class of Toeplitz words, including the period-doubling word, which also verify this same condition on the limit infimum. In contrast we show that, for the Thue-Morse word $t$, $\liminf_{n\rightarrow \infty} c_t(n)=+\infty.$Comment: To appear in Journal of Combinatorial Theory, Series