Dong Pyo Chi, DoYong Kwon
Consider the minimal $β$-shift containing the shift space generated by given Sturmian word. In this paper we characterize such $β$ and investigate its combinatorial, dynamical and topological properties and prove that such $β$ are transcendental numbers.
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$α$-numbers, diophantine exponent and factorisations of sturmian words
该研究引入Sturmian字的$α$-数和形式截距概念，通过Rauzy图构建形式体系，推导其一般分解公式，还计算了重复函数与丢番图指数，为Sturmian字分析提供新工具。
Generalised polynomials are maps constructed by applying the floor function, addition, and multiplication to polynomials. Despite superficial similarity, generalised polynomials exhibit many phenomena which are impossible for polynomials. In particular, there exist generalised polynomial sequences which take only finitely many values without being periodic; examples of such sequences include the Sturmian words, as well as more complicated sequences like $[ 2\{ πn^2 + \sqrt{2}n[\sqrt{3}n] \}]$. The purpose of this paper is to investigate letter-to-letter codings of finitely-valued generalised p…
In this paper we use the relation of the index of an infinite aperiodic word and its recurrence function to give another characterization of Sturmian words. As a byproduct, we give a new proof of theorem describing the index of a Sturmian word in terms of the continued fraction expansion of its slope. This theorem was independently proved by Carpi and de Luca, and Damanik and Lenz.
Let $θ= [0; a_1, a_2, \dots]$ be the continued fraction expansion of an irrational real number $θ\in (0, 1)$. It is well-known that the characteristic Sturmian word of slope $θ$ is the limit of a sequence of finite words $(M_k)_{k \ge 0}$, 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. Let $b \ge 2$ be an integer. Our second result gives the continued fraction expansion of any real number $ξ$ whose $b$-ary expansion is a Sturmi…
We give a combinatorial definition of the second parameter describing Sturmian words, that we call formal intercept, as an infinite Ostrowski expansion. We recall the well-known properties of Sturmian words, study factors graphs and repetition function, and end with the definition of formal intercepts.
We introduce a new geometric approach to Sturmian words by means of a mapping that associates certain lines in the n x n -grid and sets of finite Sturmian words of length n. Using this mapping, we give new proofs of the formulas enumerating the finite Sturmian words and the palindromic finite Sturmian words of a given length. We also give a new proof for the well-known result that a factor of a Sturmian word has precisely two return words.
We say $x \in \{0,1,2 \}^{\NN}$ is a word with Sturmian erasures if for any $a\in \{0,1,2 \}$ the word obtained erasing all $a$ in $x$ is a Sturmian word. A large family of such words is given coding trajectories of balls in the game of billiards in the cube. We prove that the monoid of morphisms mapping all words with Sturmian erasures to words with Sturmian erasures is not finitely generated.
Aperiodic tile sets from Sturmian lattices
研究基于二次斜率的斯图姆单词构造非周期平铺集，核心方法是给出显式算法，关键要素有斯图姆晶格和德洛内集的有界位移等价性，能生成基础缩放常数为实二次域单位的非周期平铺集。
The properties characterizing Sturmian words are considered for words on multiliteral alphabets. We summarize various generalizations of Sturmian words to multiliteral alphabets and enlarge the list of known relationships among these generalizations. We also collect many examples of infinite words to illustrate differences in the generalized definitions of Sturmian words.
Quasi-Sturmian colorings on regular trees
本文研究正则树上的拟Sturmian着色，将其分为有界与无界两类，提出类似Sturmian着色的归纳算法，并利用回归函数区分这两类着色。
Balanced rectangles over Sturmian words and minimal discrepancy intervals
研究Sturmian单词斜率为α的m×n矩形矩阵的平衡性质，通过Ostrowski表示描述其特性，扩展了Anselmo等人和Shallit的研究，采用nα mod 1的分布方法。
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. …
In this paper we study the 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.
For an infinite word $x$, Bugeaud and Kim introduced a quantity $\mathrm{rep}(x)$ called the exponent of repetition of $x$. We prove that $\mathrm{rep}(x) = \mathrm{rep}(y)$ holds for a Sturmian word $x$ and every suffix $y$ of $x$. Let $c$ be the characteristic Sturmian word of slope $θ$. When $θ$ is a quadratic irrational, a formula of $\mathrm{rep}(c)$ is given. This formula shows that $\mathrm{rep}(c) = \mathrm{rep}(c')$ if $c'$ is the characteristic Sturmian word whose slope is a quadratic irrational equivalent to $θ$.
In this paper we explore various interconnections between rich words, Sturmian words, and trapezoidal words. Rich words, first introduced in arXiv:0801.1656 by the second and third authors together with J. Justin and S. Widmer, constitute a new class of finite and infinite words characterized by having the maximal number of palindromic factors. Every finite Sturmian word is rich, but not conversely. Trapezoidal words were first introduced by the first author in studying the behavior of the subword complexity of finite Sturmian words. Unfortunately this property does not characterize finite Stu…
In this paper we present three new characterizations of Sturmian words based on the lexicographic ordering of their factors.
We follow the works of Puzynina and Zamboni, and Rigo et al. on abelian returns in Sturmian words. We determine the cardinality of the set $\mathcal{APR}_u$ of abelian returns of all prefixes of a Sturmian word $u$ in terms of the coefficients of the continued fraction of the slope, dependingly on the intercept. We provide a simple algorithm for finding the set $\mathcal{APR}_u$ and we determine it for the characteristic Sturmian words.
We take a global view at substitution invariant Sturmian sequences. We show that homogeneous substitution invariant Sturmian sequences $s_{α,α}$ can be indexed by two binary trees, associated directly to Johannes Kepler's tree of harmonic fractions from 1619. We obtain similar results for the inhomogeneous sequences $s_{α,1-α}$ and $s_{α,0}$.
By measuring second occurring times of factors of an infinite word $x$, Bugeaud and Kim introduced a new quantity ${\rm rep}(x)$ called the exponent of repetition of $x$. It was proved by Bugeaud and Kim that $1 \leq {\rm rep}(x) \leq r_{\max} = \sqrt{10} - 3/2$ if $x$ is a Sturmian words. In this paper, we determine the value $r_1$ such that there is no Sturmian word $x$ satisfying $r_1 < {\rm rep}(x) < r_{\max}$ and $r_1$ is an accumulation point of the set of ${\rm rep}(x)$ when $x$ runs over the 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.
A word is closed if it contains a proper factor that occurs both as a prefix and as a suffix but does not have internal occurrences, otherwise it is open. We deal with the sequence of open and closed prefixes of Sturmian words and prove that this sequence characterizes every finite or infinite Sturmian word up to isomorphisms of the alphabet. We then characterize the combinatorial structure of the sequence of open and closed prefixes of standard Sturmian words. We prove that every standard Sturmian word, after swapping its first letter, can be written as an infinite product of squares of rever…
We consider the following open question in the spirit of Ramsey theory: Given an aperiodic infinite word $w$, does there exist a finite coloring of its factors such that no factorization of $w$ is monochromatic? We show that such a coloring always exists whenever $w$ is a Sturmian word or a standard episturmian word.
Any infinite uniformly recurrent word ${\bf u}$ can be written as concatenation of a finite number of return words to a chosen prefix $w$ of ${\bf u}$. Ordering of the return words to $w$ in this concatenation is coded by derivated word $d_{\bf u}(w)$. In 1998, Durand proved that a fixed point ${\bf u}$ of a primitive morphism has only finitely many derivated words $d_{\bf u}(w)$ and each derivated word $d_{\bf u}(w)$ is fixed by a primitive morphism as well. In our article we focus on Sturmian words fixed by a primitive morphism. We provide an algorithm which to a given Sturmian morphism $ψ$ …
Consecutive Power Occurrences in Sturmian Words
本文证明了Sturmian字中立方体连续结束位置距离的最优界为10，扩展了Fibonacci字的界9，并推广证明了对特定范围内的e-幂次距离存在一致有界。
Alternating geometric progressions modulo one and Sturmian words
研究了模一的交替几何级数中，通过斯特姆词描述所有无理数ξ，使得序列(ξ(-b)^n)的像包含在长度为b^{-1}+b^{-2}-b^{-3}的区间内。
Sturmian lattices and Aperiodic tile sets
本文提出基于二次斜率斯特鲁米安词构造非周期性瓷砖集的算法，并利用斯特鲁米安晶格和Delone集的有界位移等价性进行分类和构造。
Decidability for Sturmian words
该研究证明Presburger算术上Sturmian字的一阶理论可判定，借助通用加法器论证其一致ω-自动性，并用Pecan算法实现复现经典定理、得到特征Sturmian字中反平方与反回文的新结果。
Permutation complexity of images of Sturmian words by marked morphisms
本文证明了 Sturmian 词经二元标记态射作用后的像，其置换复杂度在足够长时呈线性形式 n+k。
Frid, Puzynina and Zamboni (2013) defined the palindromic length of a finite word $w$ as the minimal number of palindromes whose concatenation is equal to $w$. For an infinite word $u$ we study $PL_{u}$, that is, the function that assigns to each positive integer $n$, the maximal palindromic length of factors of length $n$ in $u$. Recently, Frid (2018) proved that $\limsup_{n\to\infty} PL_{u}(n)=+\infty$ for any Sturmian word $u$. We show that there is a constant $K>0$ such that $PL_{u}(n)\leq K\ln n$ for every Sturmian word $u$, and that for each non-decreasing function $f$ with property $\li…
Let p be a maximal palindrome in a Sturmian word s=ul_1pl_2v so that p is a palindrome and l_1pl_2 is not for letters l_1 and l_2. Let α(p,p') be a morphism mapping letters a and b respectively to a^pb and a^p'b, |p-p'|=1. In this paper, we characterize the palindromes in a Sturmian word and show that the number of maximal palindromes in a Sturmian word X= α(p,p')(Y) for finite Y and thus X is 2|X|-2|Y|. We show that the set of maximal palindromes in a finite Sturmian word X has the cardinality Σ i=1..n max(pi,p'i) where X is characterized by subsequent mappings of i=1..n.
Infinite string bricks and Sturmian words over some gentle algebras
本文分类了双Kronecker温和代数上的无限串砖模，证明其与Sturmian词双射，并推广至更广的温和代数。
In this paper, we analyze the periodic factors of Sturmian words for the findings to lead to a linear-time algorithm for the computation of runs in this class of words which, to our best knowledge, is an open problem in literature.
Bell and Shallit recently introduced the Lie complexity of an infinite word $s$ as the function counting for each length the number of conjugacy classes of words whose elements are all factors of $s$. They proved, using algebraic techniques, that the Lie complexity is bounded above by the first difference of the factor complexity plus one; hence, it is uniformly bounded for words with linear factor complexity, and, in particular, it is at most 2 for Sturmian words, which are precisely the words with factor complexity $n+1$ for every $n$. In this note, we provide an elementary combinatorial pro…
Sturmian and infinitely desubstitutable words accepted by an ω-automaton
本文通过元ω-自动机与去替换方法，研究ω-自动机接受词在替换与同态序列下的结构，并判定纯替换词、不动点及Sturmian词等无限可去替换词的存在性。
A finite word is closed if it contains a factor that occurs both as a prefix and as a suffix but does not have internal occurrences, otherwise it is open. We are interested in the {\it oc-sequence} of a word, which is the binary sequence whose $n$-th element is $0$ if the prefix of length $n$ of the word is open, or $1$ if it is closed. We exhibit results showing that this sequence is deeply related to the combinatorial and periodic structure of a word. In the case of Sturmian words, we show that these are uniquely determined (up to renaming letters) by their oc-sequence. Moreover, we prove th…
A binary word is a map W : N --> {0,1}, and the set of factors of W with length n is F_n(W):={(W(i),W(i+1),...,W(i+n-1)) : i >= 0}. A word is Sturmian if |F_n(W)|=n+1 for every n>0. We show that the sum of the heights (also known as hamming weights) of the n+1 factors with length n of a binary Sturmian word has the same parity as n, independent of W.
We prove new results concerning the relation between bifix codes, episturmian words and subgroups offree groups. We study bifix codes in factorial sets of words. We generalize most properties of ordinary maximal bifix codes to bifix codes maximal in a recurrent set $F$ of words ($F$-maximal bifix codes). In the case of bifix codes contained in Sturmian sets of words, we obtain several new results. Let $F$ be a Sturmian set of words, defined as the set of factors of a strict episturmian word. Our results express the fact that an $F$-maximal bifix code of degree $d$ behaves just as the set of wo…
A Sturmian word is a map W from the natural numbers into {0,1} for which the set of {0,1}-vectors F_n(W):={(W(i),W(i+1),...,W(i+n-1))^T : i \ge 0} has cardinality exactly n+1 for each positive integer n. Our main result is that the volume of the simplex whose n+1 vertices are the n+1 points in F_n(W) does not depend on W. Our proof of this motivates studying algebraic properties of the permutation $π$ (depending on an irrational x and a positive integer n) that orders the fractional parts {1 x}, {2 x}, ..., {n x}, i.e., 0 < {π(1) x} < {π(2) x} < ... < {π(n) x} < 1. We give a formula for the si…
This paper is concerned with palindromes occurring in characteristic Sturmian words $c_α$ of slope $α$, where $α\in (0,1)$ is an irrational. As $c_α$ is a uniformly recurrent infinite word, any (palindromic) factor of $c_α$ occurs infinitely many times in $c_α$ with bounded gaps. Our aim is to completely describe where palindromes occur in $c_α$. In particular, given any palindromic factor $u$ of $c_α$, we shall establish a decomposition of $c_α$ with respect to the occurrences of $u$. Such a decomposition shows precisely where $u$ occurs in $c_α$, and this is directly related to the continued…
Trapezoidal words are finite words having at most n+1 distinct factors of length n, for every n>=0. They encompass finite Sturmian words. We distinguish trapezoidal words into two disjoint subsets: open and closed trapezoidal words. A trapezoidal word is closed if its longest repeated prefix has exactly two occurrences in the word, the second one being a suffix of the word. Otherwise it is open. We show that open trapezoidal words are all primitive and that closed trapezoidal words are all Sturmian. We then show that trapezoidal palindromes are closed (and therefore Sturmian). This allows us t…
Richomme, Saari and Zamboni (J. Lond. Math. Soc. 83: 79-95, 2011) proved that at every position of a Sturmian word starts an abelian power of exponent $k$ for every $k > 0$. We improve on this result by studying the maximum exponents of abelian powers and abelian repetitions (an abelian repetition is an analogue of a fractional power) in Sturmian words. We give a formula for computing the maximum exponent of an abelian power of abelian period $m$ starting at a given position in any Sturmian word of rotation angle $α$. vAs an analogue of the critical exponent, we introduce the abelian critical …
We investigate abelian repetitions in Sturmian words. We exploit a bijection between factors of Sturmian words and subintervals of the unitary segment that allows us to study the periods of abelian repetitions by using classical results of elementary Number Theory. We prove that in any Sturmian word the superior limit of the ratio between the maximal exponent of an abelian repetition of period $m$ and $m$ is a number $\geq\sqrt{5}$, and the equality holds for the Fibonacci infinite word. We further prove that the longest prefix of the Fibonacci infinite word that is an abelian repetition of pe…
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…
Christoffel Matrices and Sturmian Determinants
本研究探讨与Christoffel词相关的矩阵，证明其具备群结构，计算了矩阵行列式并建立其与数论Zolotareff符号的联系，丰富了相关领域的代数与数论关联研究。
Central, standard, and Christoffel words are three strongly interrelated classes of binary finite words which represent a finite counterpart of 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 Raney's tree is the famous Stern diatomic numerical sequence. An interpretation of the terms $s(n)$ of Stern's sequence as lengths of Christoffel words wh…
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 …
This article is concerned with characteristic Sturmian words of slope $α$ and $1-α$ (denoted by $c_α$ and $c_{1-α}$ respectively), where $α\in (0,1)$ is an irrational number such that $α= [0;1+d_1,\bar{d_2,...,d_n}]$ with $d_n \geq d_1 \geq 1$. It is known that both $c_α$ and $c_{1-α}$ are fixed points of non-trivial (standard) morphisms $σ$ and $\hatσ$, respectively, if and only if $α$ has a continued fraction expansion as above. Accordingly, such words $c_α$ and $c_{1-α}$ are generated by the respective morphisms $σ$ and $\hatσ$. For the particular case when $α= [0;2,\bar{r}]$ ($r\geq1$), we…
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.
Dorst-Smeulders Coding for Arbitrary Binary Words
本文提出一种线性时间算法，计算任意二进制词的最长 Sturmian 前缀的 Dorst-Smeulders 编码，并基于最小分解将其推广到任意二进制词，可用于压缩方案。
Following earlier work by Aldo de Luca and others, we study trapezoidal words and their prefixes, with respect to their characteristic parameters $K$ and $R$ (length of shortest unrepeated suffix, and shortest length without right special factors, respectively), as well as their symmetric versions $H$ and $L$. We consider the distinction between closed (i.e., periodic-like) and open prefixes, and between Sturmian and non-Sturmian ones. Our main results characterize right special and strictly bispecial trapezoidal words, as done by de Luca and Mignosi for Sturmian 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 …
We give a presentation by generators and relations of a certain monoid generating a subgroup of index two in the group Aut(F_2) of automorphisms of the rank two free group F_2 and show that it can be realized as a monoid in the group B_4 of braids on four strings. In the second part we use Christoffel words to construct an explicit basis of F_2 lifting any given basis of the free abelian group Z^2. We further give an algorithm allowing to decide whether two elements of F_2 form a basis or not. We also show that, under suitable conditions, a basis has a unique conjugate consisting of two palind…
A proof of Rauzy's conjecture on abelian complexity
本文证明了拉乌兹猜想，表明不存在具有有理无关字母频率和常数阿贝尔复杂度为3的无限三元词。
We consider rotations on the torus $\mathbb{T}^2$, and we classify them with respect to the complexity functions. In dimension one, a minimal rotation can be coded by a sturmian word. A sturmian word has complexity $n+1$ by the Morse-Hedlund theorem. Here we make a generalization in dimension two.
We show that the coding of rotation by $α$ on $m$ intervals with rationally independent lengths can be recoded over $m$ Sturmian words of angle $α.$ More precisely, for a given $m$ an universal automaton is constructed such that the edge indexed by the vector of values of the $i$th letter on each Sturmian word gives the value of the $i$th letter of the coding of rotation.
Sturmian sequences are well-known as the ones having minimal complexity over a 2-letter alphabet. They are also the balanced sequences over a 2-letter alphabet and the sequences describing discrete lines. They are famous and have been extensively studied since the 18th century. One of the {extensions} of these sequences over a $k$-letter alphabet, with $k\geq 3$, are the episturmian sequences, which generalizes a construction of Sturmian sequences using the palindromic closure operation. There exists a finite version of the Sturmian sequences called the Christoffel words. They are known since …
We define a family of natural decompositions of Sturmian words in Christoffel words, called *reversible Christoffel* (RC) factorizations. They arise from the observation that two Sturmian words with the same language have (almost always) arbitrarily long Abelian equivalent prefixes. Using the three gap theorem, we prove that in each RC factorization, only 2 or 3 distinct Christoffel words may occur. We begin the study of such factorizations, considered as infinite words over 2 or 3 letters, and show that in the general case they are either Sturmian words, or obtained by a three-interval exchan…
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 $\mathcal{A}(\mathbf{x})$ of (the shift orbit closure of) an infinite word $\mathbf{x}$ is the set of infinite words $\mathbf{y}$ such that, for each factor $u$ of $\mathbf{y}$, there exists a factor $v$ of $\mathbf{x}$ 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 $\mathcal{A}(\mathbf{x})$ equals the …
In this book chapter, written in French, we consider the classical family of Sturmian words, defined as the aperiodic infinite words containing only $n+1$ factors of a length $n$, which is the minimal possible value. We will discuss several techniques for Sturmian words which have not been described in monographs. In particular, we will consider the geometric dual method by Berstel and Pocchiola and the link between palindrome factors and Ostrowski numeration systems. We will also discuss a conjecture on palindromic length which was proved on Sturmian words by two completely different methods …
For an infinite word $\mathbf{x}$, Bugeaud and Kim introduced a new complexity function $\text{rep}(\mathbf{x})$ which is called the exponent of repetition of $\mathbf{x}$. They showed $1\le \text{rep}(\mathbf{x}) \le \sqrt{10}-\frac{3}{2}$ for any Sturmian word $\mathbf{x}$. Ohnaka and Watanabe found a gap in the set of the exponents of repetition of Sturmian words. For an irrational number $θ\in(0,1)$, let \[ \mathscr{L}(θ):=\{\text{rep}(\mathbf{x}):\textrm{$\mathbf{x}$ is an Sturmian word of slope $θ$}\}.\] In this article, we look into $\mathscr{L}(θ)$. The minimum of $\mathscr{L}(θ)$ is d…
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