J. Bernat, Z. Masáková, E. Pelantová
In this article, we consider the factor complexity of a fixed point of a primitive substitution canonically defined by a beta-numeration system. We provide a necessary and sufficient condition on the Renyi expansion of 1 for having an affine factor complexity map C(n), that is, such that C(n)=an+b for any integer n.
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Commutation classes of reduced words and higher Bruhat orders for affine permutations
本文研究了仿射排列中交换类的简化解和更高布里哈特序的结构，提出了一种推广弱序区间[ id, w ]的通用方法，揭示了交换类与布里哈特序之间的关系。
Affine standard Lyndon words
该研究推广了A型相关结果，建立了所有类型仿射标准Lyndon字的凸性与单调性，推导了虚标准Lyndon字的部分结构结果并提出猜想，还提供了可高效计算例外型各阶仿射标准Lyndon字的代码。
Lambda words are sequences obtained by encoding the differences between ordered elements of the form i+jθ, where i and j are non-negative integers and 1 < θ<2. Lambda words are right-infinite words defined over an infinite alphabet that have connections with Sturmian words, Christoffel words, and interspersion arrays. We show that Lambda words are infinite rich words. Furthermore, any Lambda word may be mapped onto a right-infinite word over a three-letter alphabet. Although the mapping preserves palindromes and non-palindromes of the Lambda word, the resulting Gamma word is not rich.
In this paper, we establish a bijection between the infinite reduced words of an affine Weyl group and certain biclosed sets of its positive system and determine all finitely generated biclosed sets in the positive system of an affine Weyl group. Using these results, we show first that the biclosed sets in the standard positive system of rank 3 affine Weyl groups when ordered by inclusion form a complete algebraic ortholattice and secondly that the (generalized) braid graphs of those Coxeter groups are connected, which can be thought of as an infinite version of Tit's solution to the word prob…
We characterize the squares occurring in infinite overlap-free binary words and construct various alpha power-free binary words containing infinitely many overlaps.
Affine Standard Lyndon words: A-type
本文推广 Leclerc 算法以刻画 Lalonde-Ram 双射的仿射推广，并在 $A_n^{(1)}$ 型中计算任意单根次序下的全部仿射标准 Lyndon 词及正根诱导次序性质。
Chains of affine standard Lyndon words
研究仿射标准Lyndon词链的周期性，确定其周期界，推广A型结果。
In a recent paper, Brlek, Jamet and Paquin showed that some extremal infinite smooth words are also infinite Lyndon words. This result raises a natural question: are they the only ones? If no, what do the infinite smooth words that are also Lyndon words look like? In this paper, we give the answer, proving that the only infinite smooth Lyndon words are $m_{\{a<b\}}$, with $a,b$ even, $m_{\{1<b\}}$ and $Δ^{-1}_1(m_{\{1<b\}})$, with $b$ odd, where $m_\A$ is the minimal infinite smooth word with respect to the lexicographic order over a numerical alphabet $\A$ and $Δ$ is the run-length encoding f…
We derive an explicit formula for the Abelian complexity of infinite words associated with quadratic Parry numbers.
In a recent paper, Harju posed three open problems concerning square-free arithmetic progressions in infinite words. In this note we solve two of them.
The study of the structure of infinite words having bounded abelian complexity was initiated by G. Richomme, K. Saari, and L. Q. Zamboni. In this note we define bounded additive complexity for infinite words over a finite subset of Z^m. We provide an alternative proof of one of the results of Richomme, Saari, and Zamboni.
In this paper, we consider infinite words that arise as fixed points of primitive substitutions on a finite alphabet and finite colorings of their factors. Any such infinite word exhibits a "hierarchal structure" that will allow us to define, under the additional condition of strong recognizability, certain remarkable finite colorings of its factors. In particular, we generalize two combinatorial results by Justin and Pirillo concerning arbitrarily large monochromatic $k$-powers occurring in infinite words, in view of a recent paper by de Luca, Pribavkina and Zamboni, we will give new examples…
Given two finite words $u$ and $v$ of equal length, define the \emph{overlap gap between $u$ and $v$}, denoted $og(u,v)$, as the least integer $m$ for which there exist words $x$ and $x'$ of length $m$ such that $xu=vx'$ or $ux=x'v$. Informally, the overlap gap measures the outside parts of the greatest overlap of the given words. For a left-infinite word $λ$ and a right-infinite word $ρ$, let $og_{λ,ρ}$ be the function defined, for each non-negative integer $n$, by $og_{λ,ρ}(n)=og(λ_n,ρ_n)$, where $λ_n$ and $ρ_n$ are, respectively, the suffix of $λ$ and the prefix of $ρ$ of length $n$. Also, …
Consider a graph with vertex set S. A word in the alphabet S has the intervening neighbours property if any two occurrences of the same letter are separated by all its graph neighbours. For a Coxeter graph, words represent group elements. Speyer recently proved that words with the intervening neighbours property are irreducible if the group is infinite and irreducible. We present a new and shorter proof using the root automaton for recognition of irreducible words.
We describe an algorithm to identify a minimal set of "braid relations" which span and preserve all sets of involution words for twisted Coxeter systems of finite or affine type. We classify the cases in which adding the smallest possible set of "half-braid" relations to the ordinary braid relations produces a spanning set: in the untwisted case, this occurs for the Coxeter systems which are finite with rank two or type $A_n$, or affine with rank three or type $\tilde A_n$. These results generalize recent work of Hu and Zhang on the finite classical cases.
A complete partition theory is presented for omega-located words (and omega-words), namely for located words over an infinite alphabet dominated by a fixed increasing sequence. This theory strengthens in an essential way the classical Carlson, Furstenberg-Katznelson, and Bergelson-Blass-Hindman partition theory for words over a finite alphabet. Consequences of this theory are strong simultaneous extensions of the classical Hindman, Milliken-Taylor partition theorem, and of a van der Waerden theorem for general semigroups, extending results of Hindman-Strauss and Beiglbock.
In this paper, we study the critical exponent of infinite words $\ubeta$ coding $β$-integers for $β$ being a~non-simple Parry number. In other words, we investigate the maximal consecutive repetitions of factors that occur in the infinite word in question. We calculate also the ultimate critical exponent that expresses how long repetitions occur in the infinite word $\ubeta$ when the factors of length growing ad infinitum are considered. The basic ingredients of our method are the description of all bispecial factors of $\ubeta$ and the notion of return words. This method can be applied to any…
Let $A_q$ be a $q$-letter alphabet and $w$ be a right infinite word on this alphabet. A subword of $w$ is a block of consecutive letters of $w$. The subword complexity function of $w$ assigns to each positive integer $n$ the number $f_w(n)$ of distinct subwords of length $n$ of $w$. The gap function of an infinite word over the binary alphabet $\{0,1 \}$ gives the distances between consecutive 1's in this word. In this paper we study infinite binary words whose gap function is injective or "almost injective". A method for computing the subword complexity of such words is given. A necessary and…
We study the palindromic complexity of infinite words $u_β$, the fixed points of the substitution over a binary alphabet, $ϕ(0)=0^a1$, $ϕ(1)=0^b1$, with $a-1\geq b\geq 1$, which are canonically associated with quadratic non-simple Parry numbers $β$.
We consider questions related to the structure of infinite words (over an integer alphabet) with bounded additive complexity, i.e., words with the property that the number of distinct sums exhibited by factors of the same length is bounded by a constant that depends only on the word. We describe how bounded additive complexity impacts the slope of the word and how a non-erasing morphism may affect the boundedness of a given word's additive complexity. We prove the existence of recurrent words with constant additive complexity equal to any given odd positive integer. In the last section, we dis…
We study the following problem, first introduced by Dekking. Consider an infinite word x over an alphabet {0,1,...,k-1} and a semigroup homomorphism S:{0,1,...,k-1}* -> N. Let L_x denote the set of factors of x. What conditions on S and the abelian complexity of x guarantee that S(L_x) contains all but finitely many elements of N? We examine this question for some specific infinite words x having different abelian complexity functions.
We study infinite words u over an alphabet A satisfying the property P : P(n)+ P(n+1) = 1+ #A for any n in N, where P(n) denotes the number of palindromic factors of length n occurring in the language of u. We study also infinite words satisfying a stronger property PE: every palindrome of u has exactly one palindromic extension in u. For binary words, the properties P and PE coincide and these properties characterize Sturmian words, i.e., words with the complexity C(n)=n+1 for any n in N. In this paper, we focus on ternary infinite words with the language closed under reversal. For such words…
We show that every infinite word $ω$ on a finite subset of $\mathbb{Z}$ must contain arbitrarily large factors $B_1B_2$ which are "close" to being \textit{additive squares}. We also show that for all $k>1, \ ω$ must contain a factor $U_1U_2 ... U_k$ where $U_1,U_2,..., U_k$ all have the same \textit{average.}
We study the relation between the palindromic and factor complexity of infinite words. We show that for uniformly recurrent words one has P(n)+P(n+1) \leq ΔC(n) + 2, for all n \in N. For a large class of words it is a better estimate of the palindromic complexity in terms of the factor complexity then the one presented by Allouche et al. We provide several examples of infinite words for which our estimate reaches its upper bound. In particular, we derive an explicit prescription for the palindromic complexity of infinite words coding r-interval exchange transformations. If the permutation πcon…
In this paper we introduce a new notion of a sequence of symmetry groups of an infinite word. Given a subgroup $G_n$ of the symmetric group $S_n$, it acts on the set of finite words of length $n$ by permutation. We associate to an infinite word $w$ a sequence $(G_n(w))_{n\geq 1}$ of its symmetry groups: For each $n$, a symmetry group of $w$ is a subgroup $G_n(w)$ of the symmetric group $S_n$ such that $g(v)$ is a factor of $w$ for each permutation $g \in G_n(w)$ and each factor $v$ of length $n$ of $w$. We study general properties of the symmetry groups of infinite words and characterize the s…
In this paper we study the asymptotic behaviour of two relatively new complexity functions defined on infinite words and their relationship to periodicity. Given a factor $u$ of an infinite word $x$, we say $u$ is closed if it is a letter or if it is a complete first return to some factor $v$ of $x$; otherwise $u$ is said to be open. We show that for an aperiodic word $x$ over a finite alphabet, the complexity functions that count the number of closed and the number of open factors of $x$ of each given length are both unbounded. More precisely, we show that if $x$ is aperiodic then the limit i…
We introduce two classes of morphisms over the alphabet $A=\{0,1\}$ whose fixed points contain infinitely many antipalindromic factors. An antipalindrome is a finite word invariant under the action of the antimorphism $\mathrm{E}:\{0,1\}^*\to\{0,1\}^*$, defined by $\mathrm{E}(w_1\cdots w_n)=(1-w_{n})\cdots(1-w_1)$. We conjecture that these two classes contain all morphisms (up to conjugation) which generate infinite words with infinitely many antipalindromes. This is an analogue to the famous HKS conjecture concerning infinite words containing infinitely many palindromes. We prove our conjectu…
Improvement Ergodic Theory For The Infinite Word $\mathfrak{F}=\mathfrak{F}_{b}:=\left({ }_{b} f_{n}\right)_{n \geqslant 0}$ on Fibonacci Density
本文研究斐波那契密度上无限词的遍历理论，分析其子词多样性与广义模式分布，并揭示替换规则、代数数论与组合复杂性之间的深层联系。
Bounds on the closed-rich constant of infinite words
本文研究无限闭合丰富词的闭合丰富常数可能取值，证明了闭合丰富常数的上界不超过0.165952，并对斐波那契词的闭合丰富常数给出了范围，从而为闭合丰富常数的上界提供了下界。
We focus on $Θ$-rich and almost $Θ$-rich words over a finite alphabet $\mathcal{A}$, where $Θ$ is an involutive antimorphism over $\mathcal{A}^*$. We show that any recurrent almost $Θ$-rich word $\uu$ is an image of a recurrent $Θ'$-rich word under a suitable morphism, where $Θ'$ is again an involutive antimorphism. Moreover, if the word $\uu$ is uniformly recurrent, we show that $Θ'$ can be set to the reversal mapping. We also treat one special case of almost $Θ$-rich words. We show that every $Θ$-standard words with seed is an image of an Arnoux-Rauzy word.
Infinite friezes of affine type D
该研究针对仿射型$D$丛范畴衍生的无限frieze展开，确定其增长系数，并证明所有由tube给出的仿射型$D$ frieze均具有一致的增长行为。
In this paper, we provide a new characterization of uniformly recurrent words with finite defect based on a relation between the palindromic and factor complexity. Furthermore, we introduce a class of morphisms P_ret closed under composition and we show that a uniformly recurrent word with finite defect is an image of a rich (also called full) word under a morphism of class P_ret. This class is closely related to the well-known class P defined by Hof, Knill, and Simon; every morphism from P_ret is conjugate to a morphism of class P.
On modulo-recurrence and window complexity in infinite words
引入均匀模递归和强模递归概念，证明Sturmian词和最大复杂度词是强模递归的，并建立Thue-Morse词窗口复杂度与经典复杂度的关系，构造窗口复杂度有界或呈次线性增长的词，证明一致递归非周期词的窗口复杂度无界。
In this article, we count the number of return words in some infinite words with complexity 2n+1. We also consider some infinite words given by codings of rotation and interval exchange transformations on k intervals. We prove that the number of return words over a given word w for these infinite words is exactly k.
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.
A generalized lexicographic order on words is a lexicographic order where the total order of the alphabet depends on the position of the comparison. A generalized Lyndon word is a finite word which is strictly smallest among its class of rotations with respect to a generalized lexicographic order. This notion can be extended to infinite words: an infinite generalized Lyndon word is an infinite word which is strictly smallest among its class of suffixes. We prove a conjecture of Dolce, Restivo, and Reutenauer: every infinite word has a unique nonincreasing factorization into finite and infinite…
The $k$-Young lattice $Y^k$ is a partial order on partitions with no part larger than $k$. This weak subposet of the Young lattice originated from the study of the $k$-Schur functions(atoms) $s_λ^{(k)}$, symmetric functions that form a natural basis of the space spanned by homogeneous functions indexed by $k$-bounded partitions. The chains in the $k$-Young lattice are induced by a Pieri-type rule experimentally satisfied by the $k$-Schur functions. Here, using a natural bijection between $k$-bounded partitions and $k+1$-cores, we establish an algorithm for identifying chains in the $k$-Young l…
The $\ell$-Rauzy graph of order $k$ for any infinite word is a directed graph in which an arc $(v_1,v_2)$ is formed if the concatenation of the word $v_1$ and the suffix of $v_2$ of length $k-\ell$ is a subword of the infinite word. In this paper, we consider one of the important aperiodic recurrent words, the infinite Fibonacci word for discussion. We prove a few basic properties of the $\ell$-Rauzy graph of the infinite Fibonacci word. We also prove that the $\ell$-Rauzy graphs for the infinite Fibonacci word are strongly connected.
We answer a question of Harju: An infinite square-free ternary word with an $n$-stem factorization exists for any $n\ge 13$. We show that there are uniform ternary morphisms of length $k$ for every $k\ge 23$. This resolves almost completely a problem of the author and Rampersad.
Growth of infinite frieze patterns of affine type
本文研究了通过聚类代数产生的无限条纹图案的增长系数，证明了固定affine类型的聚类类别中，同构和非同构稳定管生成的无限条纹图案具有相同增长系数，并推导了第k个增长系数的显式公式。
For each $α> 2$ there is a binary word with critical exponent $α$.
Over an alphabet of size 3 we construct an infinite balanced word with critical exponent 2+sqrt(2)/2. Over an alphabet of size 4 we construct an infinite balanced word with critical exponent (5+sqrt(5))/4. Over larger alphabets, we give some candidates for balanced words (found computationally) having small critical exponents. We also explore a method for proving these results using the automated theorem prover Walnut.
Rich words are characterized by containing the maximum possible number of distinct palindromes. Several characteristic properties of rich words have been studied; yet the analysis of repetitions in rich words still involves some interesting open problems. We address lower bounds on the repetition threshold of infinite rich words over 2 and 3-letter alphabets, and construct a candidate infinite rich word over the alphabet $Σ_2=\{0,1\}$ with a small critical exponent of $2+\sqrt{2}/2$. This represents the first progress on an open problem of Vesti from 2017.
A word is called closed if it has a prefix which is also its suffix and there is no internal occurrences of this prefix in the word. In this paper we study words that are rich in closed factors, i.e., which contain the maximal possible number of distinct closed factors. As the main result, we show that for finite words the asymptotics of the maximal number of distinct closed factors in a word of length $n$ is $\frac{n^2}{6}$. For infinite words, we show there exist words such that each their factor of length $n$ contains a quadratic number of distinct closed factors, with uniformly bounded con…
String attractors and bi-infinite words
本文刻画了具有有限字符串吸引子的双向无限词为特征Sturmian词及其态射像，并研究了无限字符串吸引子的结构性质。
An infinite word x is said to be quasiperiodic if there exists a finite word q such that x is covered by occurrences of q (such a q is called a quasiperiod of x). Using the notion of derivation, we show that this definition is not sufficient to imply any symmetry in an infinite word. Therefore we introduce multi-scale quasiperiodic words, i.e. quasiperiodic words that admit an infinite number of quasiperiods. Such words are uniformly recurrent, this allows us to study the subshift they generate. We prove that multi-scale quasiperiodic subshifts are uniquely ergodic and have zero topological en…
In a recent paper, one of us posed three open problems concerning squarefree arithmetic progressions in infinite words. In this note we solve these problems and prove some additional results.
We consider three aspects of avoiding large squares in infinite binary words. First, we construct an infinite binary word avoiding both cubes xxx and squares yy with |y| >= 4; our construction is somewhat simpler than the original construction of Dekking. Second, we construct an infinite binary word avoiding all squares except 00, 11, and 0101; our construction is somewhat simpler than the original construction of Fraenkel and Simpson. In both cases, we also show how to modify our construction to obtain exponentially many words of length n with the given avoidance properties. Finally, we answe…
The limit weak order on an affine Weyl group was introduced by Lam and Pylyavskyy in their study of total positivity for loop groups. They showed that in the case of the affine symmetric group the minimal elements of this poset coincide with the infinite fully commutative reduced words and with infinite powers of Coxeter elements. We answer several open problems raised there by classifying minimal elements in all affine types and relating these elements to the classes of fully commutative and Coxeter elements. Interestingly, the infinite fully commutative elements correspond to the minuscule a…
Prefix normal words are binary words that have no factor with more $1$s than the prefix of the same length. Finite prefix normal words were introduced in [Fici and Lipták, DLT 2011]. In this paper, we study infinite prefix normal words and explore their relationship to some known classes of infinite binary words. In particular, we establish a connection between prefix normal words and Sturmian words, between prefix normal words and abelian complexity, and between prefix normality and lexicographic order.
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 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 uniformly recurrent binary words, periodic and aperiodic Sturmian words are exactly those words for which $\mathcal{A}(\mathbf{x})$ equals the shift …
Palindromic length of infinite aperiodic words
本文证明了2013年提出的猜想：每个无限非周期字的因子的回文长度是无界的。
We prove that semi-infinite Bruhat order on an affine Weyl group is completely determined from those on the quotients by affine Weyl subgroups associated with various maximal (standard) parabolic subgroups of finite type. Furthermore, for an affine Weyl group of classical type, we give a complete classification of all cover relations of semi-infinite Bruhat order (or equivalently, all edges of the quantum Bruhat graphs) on the quotients in terms of tableaux. Combining these we obtain a tableau criterion for semi-infinite Bruhat order on an affine Weyl group of classical type. As an application…
Recently the Fibonacci word $W$ on an infinite alphabet was introduced by [Zhang et al., Electronic J. Combinatorics 24-2 (2017) #P2.52] as a fixed point of the morphism $ϕ: (2i) \mapsto (2i)(2i+ 1),\ (2i+ 1) \mapsto (2i+ 2)$ over all $i \in \mathbb{N}$. In this paper we investigate the occurrence of squares, palindromes, and Lyndon factors in this infinite word.
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 …
We construct an infinite binary word with critical exponent 3 that avoids abelian 4-powers. Our method gives an algorithm to determine if certain types of morphic sequences avoid additive powers. We also show that there are $Ω(1.172^n)$ binary words of length $n$ that avoid abelian 4-powers, which improves on previous estimates.
Let $β=\frac{1+\sqrt{5}}{2}$, $(a_n)_{n \in \mathbb{N}^+}$ be a non-uniform morphic sequence involving the infinite Fibonacci word and $(δ(n))_{n \in \mathbb{N}^+}$ be a positive sequence such that for all positive integers $n$, $δ(n)=\frac{1}{\sqrt{5}}\sum_{j \geq 0}ε_jβ^{j+2}$ if the unique Zeckendorf expansion of $n$ is $n=\sum_{j \geq 0}ε_jF_{j+2}$ with Fibonacci numbers $F_0,F_1,F_2...$. We define and study some Dirichlet series in the form of $\sum_{n\geq 1}\frac{a_n}{(δ(n))^s}$ and relations between them. Moreover, we compute the values of some infinite series involving the infinite Fib…
The factor complexity of the infinite word $\ubeta$ canonically associated to a non-simple Parry number $β$ is studied. Our approach is based on the notion of special factors introduced by Berstel and Cassaigne. At first, we give a handy method for determining infinite left special branches; this method is applicable to a broad class of infinite words which are fixed points of a primitive substitution. In the second part of the article, we focus on infinite words $\ubeta$ only. To complete the description of its special factors, we define and study $(a,b)$-maximal left special factors. This en…
The initial non-repetitive complexity function of an infinite word x (first defined by Moothathu) is the function of n that counts the number of distinct factors of length n that appear at the beginning of x prior to the first repetition of a length-n factor. We examine general properties of the initial non-repetitive complexity function, as well as obtain formulas for the initial non-repetitive complexity of the Thue-Morse word, the Fibonacci word and the Tribonacci word.
In 1976, Dekking showed that there exists an infinite binary word that contains neither squares yy with y >= 4 nor cubes xxx. We show that `cube' can be replaced by any fractional power > 5/2. We also consider the analogous problem where `4' is replaced by any integer. This results in an interesting and subtle hierarchy.
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