A characterization of substitutive sequences using return words
Fabien Durand
我们证明了一个序列是本原置换的当且仅当它的导出序列的集合是有限的;我们在这里定义了这些序列。
We prove that a sequence is primitive substitutive if and only if the set of its derived sequences is finite; we defined these sequences here.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
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…
We consider the complexities of substitutive sequences over a binary alphabet. By studying various types of special words, we show that, knowing some initial values, its complexity can be completely formulated via a recurrence formula determined by the characteristic polynomial.
An S-adic expansion of an infinite word is a way of writing it as the limit of an infinite product of substitutions (i.e., morphisms of a free monoid). Such a description is related to continued fraction expansions of numbers and vectors. A fundamental example of this relation is between Sturmian sequences and regular continued fractions. We study S-adic words from different perspectives, namely word combinatorics, ergodic theory, and Diophantine approximation, by stressing the parallel with continued fraction expansions.
Automaticity of uniformly recurrent substitutive sequences
本文研究了一致递归替换序列的自动性，通过 incidence 矩阵刻画其自动性，并构造了一个与 2 进制 odometer 相关的最小替换系统，提出关于 k 自动替换系统的猜想。
In a 1995 paper, Hof, Knill and Simon obtain a sufficient combinatorial criterion on the hull $Ω$ of the potential of a discrete Schrödinger operator which guarantees purely singular continuous spectrum on a generic subset of $Ω.$ In part, this condition requires the existence of infinitely many palindromic factors. In this same paper, they introduce the class P of morphisms $f:A^*\rightarrow B^*$ of the form $a\mapsto pq_a$ and ask whether every palindromic subshift generated by a primitive substitution arises from morphisms of class P or by morphisms of the form $a\mapsto q_ap$ where again $…
Complementary symmetric Rote sequences are binary sequences which have factor complexity $\mathcal{C}(n) = 2n$ for all integers $n \geq 1$ and whose languages are closed under the exchange of letters. These sequences are intimately linked to Sturmian sequences. Using this connection we investigate the return words and the derivated sequences to the prefixes of any complementary symmetric Rote sequence $\mathbf{v}$ which is associated with a standard Sturmian sequence $\mathbf{u}$. We show that any non-empty prefix of $\mathbf{v}$ has three return words. We prove that any derivated sequence of …
Effective Computation of Generalized Abelian Complexity for Pisot Type Substitutive Sequences
该研究针对 Pisot 型替换序列，提出有效计算其广义阿贝尔复杂度的方法，利用 Walnut 定理证明器，并揭示了 Tribonacci 序列的平衡性统一界及复杂度函数的线性表示。
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. …
On the number of return words in infinite words with complexity 2n+1
本文对复杂度为2n+1的无穷词中返回词的个数进行了统计。我们还考虑了由旋转编码和k区间上的区间交换变换所给出的一些无限字。我们证明了对于这些无限单词，在给定单词w上的返回单词的数量正好是k。
We investigate the natural codings of linear involutions. We deduce from the geometric representation of linear involutions as Poincaré maps of measured foliations a suitable definition of return words which yields that the set of first return words to a given word is a symmetric basis of the free group on the underlying alphabet $A$. The set of first return words with respect to a subgroup of finite index $G$ of the free group on $A$ is also proved to be a symmetric basis of $G$.
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。
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.
Sequences with constant number of return words
如果每个因子都恰好有$m$返回字，则无限字具有属性$R_m$。Vuillon证明了$R_2$刻画了Sturmian词。证明了一个词满足$R_m$，如果它的复杂度函数是$(m-1)n+1$，并且它不包含弱双特殊因子.这些条件对于$m=3$是必要的，而对于$m=4$，复杂度函数不必是$3n+1$。满足$R_m$的词的新的例子给出了与实数基中的数字扩展有关的词。
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.
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 $ψ$ …
We prove in this note that, for an alphabet with three letters, the set of first return to a given word in a set satisfying the tree condition is a basis of the free group.
A connection between palindromic and factor complexity using return words
本文证明了：对于任意因子集在反转下闭的无限字W，下列条件等价：（I）所有完全返回到回文的都是回文的;（II）P（n）+ P（n+1）= C（n+1）- C（n）+ 2，对所有n，其中P（分别为P（n）+ P（n+1））= C（n+1）- C（n）+ 2. C）表示回文复杂度（resp.因子复杂度）函数，它计算不同回文因子的数量（分别为每个长度W的系数）。
A double occurrence word (DOW) is a word in which every symbol appears exactly twice; two DOWs are equivalent if one is a symbol-to-symbol image of the other. We consider the so called repeat pattern ($αα$) and the return pattern ($αα^R$), with gaps allowed between the $α$'s. These patterns generalize square and palindromic factors of DOWs, respectively. We introduce a notion of inserting repeat/return words into DOWs and study how two distinct insertions into the same word can produce equivalent DOWs. Given a DOW $w$, we characterize the structure of $w$ which allows two distinct insertions t…
Stability properties for subgroups generated by return words
本文引入返回词生成子群的稳定性概念，统一并推广了现有结果，并探讨了可判定性及运算封闭性等一般性质。
Certified Return-Word Induction for a Perturbed Hofstadter Recursion
本文研究了一种扰动的霍夫达特型递归，并通过计算机认证的有限状态归纳法证明了该递归对所有n≥1都是良定义的。
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.
Episturmian words: a survey
在本文中，我们调查了丰富的理论无限episturmian词推广到任何有限的字母表，在一个相当类似的方式，著名的家庭Sturmian词的两个字母。在回顾定义和基本属性后，我们考虑episturmian态射，允许更深入地研究这些词。因子的一些属性，包括因子复杂度，回文，分数幂，频率和返回字。我们还考虑episturmian词的词典编纂性质，以及它们与平衡性质的联系，以及相关的概念，如有限episturmian词，Arnoux-Rauzy序列，和“episkew词”，概括了莫尔斯和Hedlund的歪斜词。
We study infinite words fixed by a morphism and their derived words. A derived word is a coding of return words to a factor. We exhibit two examples of sets of morphisms which are closed under derivation --- any derived word with respect to any factor of the fixed point is again fixed by a morphism from this set. The first example involves standard episturmian morphisms, and the second concerns the period doubling morphism.
Given a finitely generated group with generating set $S$, we study the \emph{cogrowth} sequence, which is the number of words of length $n$ over the alphabet $S$ that are equal to one. This is related to the probability of return for walks in a Cayley graph with steps from $S$. We prove that the cogrowth sequence is not $P$-recursive when~$G$ is an amenable group of superpolynomial growth, answering a question of Garrabant and Pak.
Algebraic characterization of dendricity
本文证明了树状移位空间中返回定理的逆命题，从而以返回词构成自由群基这一代数条件完整刻画了树状性。
Given a set $F$ of words, one associates to each word $w$ in $F$ an undirected graph, called its extension graph, and which describes the possible extensions of $w$ on the left and on the right. We investigate the family of sets of words defined by the property of the extension graph of each word in the set to be acyclic or connected or a tree. We prove that in a uniformly recurrent tree set, the sets of first return words are bases of the free group on the alphabet. Concerning acyclic sets, we prove as a main result that a set $F$ is acyclic if and only if any bifix code included in $F$ is a …
Quadratic bounds for uncompletable words and matrix mortality
该研究为不可补全词和矩阵死亡性建立了二次长度上界，并给出多项式时间构造算法，在 Lean 中完成形式化证明。
Shuffle squares in smooth binary words and the Kolakoski sequence
本文证明长度至少18且两字母出现偶数次的光滑二元词必为洗牌平方，并应用于Kolakoski序列，刻画其洗牌平方前缀的条件。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。