Characterization Of Sturmian Words Research Articles

On abelian closures of infinite non-binary words

Two finite words u and v are called abelian equivalent if each letter occurs equally many times in both u and v. The abelian closure A(x) of an infinite word x is the set of infinite words y such that, for each factor u of y, there exists a factor v of x which is abelian equivalent to u. The notion of an abelian closure gives a characterization of Sturmian words: among uniformly recurrent binary words, periodic and aperiodic Sturmian words are exactly those words for which A(x) equals the shift orbit closure Ω(x). In this paper, we investigate how this property extends to non-binary words. We consider the abelian closures of most natural generalizations of Sturmian words to non-binary alphabets, such as balanced words and minimal complexity words. We characterize the abelian closures of words in these families and show that in both families, there exist both words which satisfy the property A(x)=Ω(x) and which do not. We observe that for Arnoux-Rauzy words, we always have a strict inclusion Ω(x)⊂A(x). We also consider abelian closures of general subshifts and make some initial observations of their abelian closures and pose some related open questions.

On Extended Boundary Sequences of Morphic and Sturmian Words

Generalizing the notion of the boundary sequence introduced by Chen and Wen, the $n$th term of the $\ell$-boundary sequence of an infinite word is the finite set of pairs $(u,v)$ of prefixes and suffixes of length $\ell$ appearing in factors $uyv$ of length $n+\ell$ ($n\ge \ell\ge 1$). Otherwise stated, for increasing values of $n$, one looks for all pairs of factors of length $\ell$ separated by $n-\ell$ symbols.
 For the large class of addable abstract numeration systems $S$, we show that if an infinite word is $S$-automatic, then the same holds for its $\ell$-boundary sequence. In particular, they are both morphic (or generated by an HD0L system). To precise the limits of this result, we discuss examples of non-addable numeration systems and $S$-automatic words for which the boundary sequence is nevertheless $S$-automatic and conversely, $S$-automatic words with a boundary sequence that is not $S$-automatic. In the second part of the paper, we study the $\ell$-boundary sequence of a Sturmian word. We show that it is obtained through a sliding block code from the characteristic Sturmian word of the same slope.
 We also show that it is the image under a morphism of some other characteristic Sturmian word.

Combinatorial structure of Sturmian words and continued fraction expansion of Sturmian numbers

Let θ=[0;a 1 ,a 2 ,⋯] be the continued fraction expansion of an irrational real number θ∈(0,1). It is well-known that the characteristic Sturmian word of slope θ is the limit of a sequence of finite words (M k ) k≥0 , with M k of length q k (the denominator of the k-th convergent to θ) being a suitable concatenation of a k copies of M k-1 and one copy of M k-2 . Our first result extends this to any Sturmian word s. Let b≥2 be an integer. Our second result gives the continued fraction expansion of any real number ξ whose b-ary expansion is a Sturmian word s over the alphabet {0,b-1}. This extends a classical result of Böhmer who considered only the case where s is characteristic. As a consequence, we obtain a formula for the irrationality exponent of ξ in terms of the slope and the intercept of s.

Abelian closures of infinite binary words

Two finite words u and v are called Abelian equivalent if each letter occurs equally many times in both u and v. The abelian closure A(x) of (the shift orbit closure of) an infinite word x is the set of infinite words y such that, for each factor u of y, there exists a factor v of x which is abelian equivalent to u. The notion of an abelian closure gives a characterization of Sturmian words: among binary uniformly recurrent words, Sturmian words are exactly those words for which A(x) equals the shift orbit closure Ω(x). In this paper we show that, contrary to larger alphabets, the abelian closure of a uniformly recurrent aperiodic binary word which is not Sturmian contains infinitely many minimal subshifts.

An Inequality for the Number of Periods in a Word

We prove an inequality for the number of periods in a word [Formula: see text] in terms of the length of [Formula: see text] and its initial critical exponent. Next, we characterize all periods of the length-[Formula: see text] prefix of a characteristic Sturmian word in terms of the lazy Ostrowski representation of [Formula: see text], and use this result to show that our inequality is tight for infinitely many words [Formula: see text]. We propose two related measures of periodicity for infinite words. Finally, we also consider special cases where [Formula: see text] is overlap-free or squarefree.

Sturmian numeration systems and decompositions to palindromes

We extend classical Ostrowski numeration systems, closely related to Sturmian words, by allowing a wider range of coefficients, so that possible representations of a number n better reflect the structure of the associated characteristic Sturmian word. In particular, this extended numeration system helps to catch occurrences of palindromes in a characteristic Sturmian word and thus to prove for Sturmian words the following conjecture stated in 2013 by Puzynina, Zamboni and the author: If a word is not periodic, then for every Q>0 it has a prefix which cannot be decomposed to a concatenation of at most Q palindromes.

Sturmian images of non Sturmian words and standard morphisms

We prove that if a Sturmian word is the image by a morphism of a word which is a fixed point of another morphism, then this latter word is mostly a Sturmian word, and the involved morphisms are Sturmian. This gives a characterization of Sturmian words that are generated by HD0L systems.We also characterize the non Sturmian words which can be sent on Sturmian words by morphism, and the involved morphisms. We prove that the same Sturmian images can be obtained by using the standard morphisms of which the above morphisms are the conjugates, and we show how to obtain these Sturmian morphisms from their standard representatives.

Cyclic complexity of words

We introduce and study a complexity function on words cx(n), called 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 cx(n)=1 for some n≥1 implies x is periodic, it is natural to consider the quantity liminfn→∞cx(n). We show that if x is a Sturmian word, then liminfn→∞cx(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, liminfn→∞ct(n)=+∞.

On Christoffel and standard words and their derivatives

We introduce and study natural derivatives for Christoffel and standard words, as well as for characteristic Sturmian words. These derivatives, which are defined as inverse images under suitable morphisms, preserve the aforementioned classes of words. In the case of Christoffel words, the morphisms involved map a to ak+1b (resp., abk) and b to akb (resp., abk+1) for a suitable k>0. As long as derivatives are not just a single letter, higher-order derivatives are naturally obtained. We define the depth of a Christoffel or of a standard word as the smallest order for which the derivative is a single letter. We give several combinatorial and arithmetic descriptions of the depth, and (tight) lower and upper bounds for it.

Sturmian words and the Stern sequence

Central, standard, and Christoffel words are three strongly interrelated classes of binary finite words which represent a finite counterpart to characteristic Sturmian words. A natural arithmetization of the theory is obtained by representing central and Christoffel words by irreducible fractions labeling, respectively, two binary trees, the Raney (or Calkin–Wilf) tree and the Stern–Brocot tree. The sequence of denominators of the fractions in the Raney tree is the famous Stern diatomic numerical sequence. An interpretation of the terms s(n) of Stern's sequence as lengths of Christoffel words when n is odd, and as minimal periods of central words when n is even, allows one to interpret several results on Christoffel and central words in terms of Stern's sequence and, conversely, to obtain new insight into the combinatorics of Christoffel and central words. One of our main results is a non-commutative version of the “alternating bit sets theorem” by Calkin and Wilf. We also study the length distribution of Christoffel words corresponding to nodes of equal height in the tree, obtaining some interesting bounds and inequalities.

SOME EXTREMAL PROPERTIES OF THE FIBONACCI WORD

We prove that the Fibonacci word f satisfies among all characteristic Sturmian words, three interesting extremal properties. The first concerns the length and the second the minimal period of its palindromic prefixes. Each of these two properties characterizes f up to a renaming of its letters. A third property concerns the number of occurrences of the letter b in its palindromic prefixes. It characterizes uniquely f among all characteristic Sturmian words having the prefix abaa.

Introducing privileged words: Privileged complexity of Sturmian words

In this paper we introduce a new class of so-called privileged words which have been previously considered only a little. We develop the basic properties of privileged words, which turn out to share similar properties with palindromes. Privileged words are studied in relation to previously studied classes of words, rich words, Sturmian words and episturmian words. A new characterization of Sturmian words is given in terms of privileged complexity. The privileged complexity of the Thue–Morse word is also briefly studied.

Abelian returns in Sturmian words

Return words constitute a powerful tool for studying symbolic dynamical systems. They may be regarded as a discrete analogue of the first return map in dynamical systems. In this paper we investigate two abelian variants of the notion of return word, each of them gives rise to a new characterization of Sturmian words. We prove that a recurrent infinite word is Sturmian if and only if each of its factors has two or three abelian (or semi-abelian) returns. We study the structure of abelian returns in Sturmian words and give a characterization of those factors having exactly two abelian returns. Finally we discuss connections between abelian returns and periodicity in words.

Characteristic Sturmian words are extremal for the Critical Factorization Theorem

We prove that characteristic Sturmian words are extremal for the Critical Factorization Theorem (CFT) in the following sense. If px(n) denotes the local period of an infinite word x at point n, we prove that x is a characteristic Sturmian word if and only if px(n) is smaller than or equal to n+1 for all n≥1 and it is equal to n+1 for infinitely many integers n.This result is extremal with respect to the CFT since a consequence of the CFT is that, for any infinite recurrent word x, either the function px is bounded, and in such a case x is periodic, or px(n)≥n+1 for infinitely many integers n.As a byproduct of the techniques used in the paper we extend a result of Harju and Nowotka (2002) in [18] stating that any finite Fibonacci word fn,n≥5, has only one critical point. Indeed we determine the exact number of critical points in any finite standard Sturmian word.

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.

On z-factorization and c-factorization of standard episturmian words

Ziv–Lempel and Crochemore factorization are two kinds of factorizations of words related to text processing. In this paper, we find these factorizations for standard epiesturmian words. Thus the previously known c-factorization of characteristic Sturmian words is provided as a special case. Moreover, the two factorizations are compared.

Standard factors of Sturmian words

Among the various ways to construct a characteristic Sturmian word, one of the most used consists in defining an infinite sequence of prefixes that are standard. Nevertheless in any characteristic word c, some standard words occur that are not prefixes of c. We characterize all standard words occurring in any characteristic word (and so in any Sturmian word) using firstly morphisms, then standard prefixes and finally palindromes.

Powers in a class of [formula omitted]-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 [D. Damanik, D. Lenz, Powers in Sturmian sequences, European J. Combin. 24 (2003) 377–390, doi:10.1016/S0195-6698(03)00026-X], 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 k - bonacci word, a generalization of the Fibonacci word to a k -letter alphabet ( k ≥ 2 ) .

A little more about morphic Sturmian words

Among Sturmian words, some of them are morphic, i.e. fixed point of a non-identical morphism on words. Berstel and Seebold (1993) have shown that if a characteristic Sturmian word is morphic, then it can be extended by the left with one or two letters in such a way that it remains morphic and Sturmian. Yasutomi (1997) has proved that these were the sole possible additions and that, if we cut the first letters of such a word, it didn't remain morphic. In this paper, we give an elementary and combinatorial proof of this result.

Palindromic prefixes and episturmian words

Let w be an infinite word on an alphabet A . We denote by ( n i ) i ⩾ 1 the increasing sequence (assumed to be infinite) of all lengths of palindromic prefixes of w. In this text, we give an explicit construction of all words w such that n i + 1 ⩽ 2 n i + 1 for all i, and study these words. Special examples include characteristic Sturmian words, and more generally standard episturmian words. As an application, we study the values taken by the quantity lim sup n i + 1 / n i , and prove that it is minimal (among all nonperiodic words) for the Fibonacci word.