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 reversed standard words.
Characteristic Parameters and Special Trapezoidal Words
根据Aldo de Luca等人的早期工作，我们研究了梯形词及其前缀的特征参数$K$和$R$（最短不重复后缀的长度和最短无特殊因子的长度），以及它们的对称形式$H$和$L$。我们认为封闭（即，和非。我们的主要结果特征的权利特殊和严格双特殊梯形的话，所做的德卢卡和Mesquisi Sturmian的话。
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…
We show that the number of length-n words over a k-letter alphabet having no even palindromic prefix is the same as the number of length-n unbordered words, by constructing an explicit bijection between the two sets. A slightly different but analogous result holds for those words having no odd palindromic prefix. Using known results on borders, we get an asymptotic enumeration for the number of words having no even (resp., odd) palindromic prefix . We obtain an analogous result for words having no nontrivial palindromic prefix. Finally, we obtain similar results for words having no square pref…
Mignosi, Restivo, and Salemi (1998) proved that for all $ε> 0$ there exists an integer $N$ such that all prefixes of the Fibonacci word of length $\geq N$ contain a suffix of exponent $α^2-ε$, where $α= (1+\sqrt{5})/2$ is the golden ratio. In this note we show how to prove an explicit version of this theorem with tools from automata theory and logic. Along the way we gain a better understanding of the repetitive structure of the Fibonacci word.
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…
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 $θ$.
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.
We study the prefixes of the Fibonacci word that end with a cube. Using Walnut we obtain an exact description of the positions of the Fibonacci word at which a cube ends.
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…
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 …
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 …
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…
Alternating geometric progressions modulo one and Sturmian words
School of Mathematical Sciences, Beijing Normal University（北京师范大学数学科学学院）；Morningside Center of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院晨兴数学中心）；University of Chinese Academy of Sciences（中国科学院大学）
