Palindromic Prefixes and Episturmian Words

Rote序列的字符串吸引子

String attractors of Rote sequences

本文研究Rote序列伪回文前缀的最小字符串吸引子，证明反回文闭包情形下大小为三，并猜想一般情形至多为四。

关于回文周期性的若干注记

Some Remarks on Palindromic Periodicities

本文研究有限词的回文周期性，分析多个经典无限词中的回文周期结构，并证明相关性质及最少回文周期性数量的结论。

Extensions of rich words

In [X. Droubay et al, Episturmian words and some constructions of de Luca and Rauzy, Theoret. Comput. Sci. 255 (2001)], it was proved that every word w has at most |w|+1 many distinct palindromic factors, including the empty word. The unified study of words which achieve this limit was initiated in [A. Glen et al, Palindromic richness, Eur. Jour. of Comb. 30 (2009)]. They called these words rich (in palindromes). This article contains several results about rich words and especially extending them. We say that a rich word w can be extended richly with a word u if wu is rich. Some notions are al…

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 $abaa$.

On a generalization of Christoffel words: epichristoffel words

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 …

Palindromic closures using multiple antimorphisms

Generalized pseudostandard word $\bf u$, as introduced in 2006 by de Luca and De Luca, is given by a directive sequence of letters from an alphabet ${\cal A}$ and by a directive sequence of involutory antimorphisms acting on ${\cal A}^*$. Prefixes of $\bf u$ with increasing length are constructed using pseudopalindromic closure operator. We show that generalized Thue--Morse words ${\bf t}_{b,m}$, with $b, m \in \N$ and $b, m \geq 2$, are generalized pseudostandard words if and only if ${\bf t}_{b,m}$ is a periodic word or $b \leq m$. This extends the result of de Luca and De Luca obtained for …

On the fixed point of the iterated pseudopalindromic closure

First introduced in the study of the Sturmian words, the iterated palindromic closure was recently generalized to pseudopalindromes. This operator allows one to construct words with an infinity of pseudopalindromic prefixes, called pseudostandard words. We provide here several combinatorial properties of the fixed points under the iterated pseudopalindromic closure.

Palindromic Length of Words with Many Periodic Palindromes

The palindromic length $\text{PL}(v)$ of a finite word $v$ is the minimal number of palindromes whose concatenation is equal to $v$. In 2013, Frid, Puzynina, and Zamboni conjectured that: If $w$ is an infinite word and $k$ is an integer such that $\text{PL}(u)\leq k$ for every factor $u$ of $w$ then $w$ is ultimately periodic. Suppose that $w$ is an infinite word and $k$ is an integer such $\text{PL}(u)\leq k$ for every factor $u$ of $w$. Let $Ω(w,k)$ be the set of all factors $u$ of $w$ that have more than $\sqrt[k]{k^{-1}\vert u\vert}$ palindromic prefixes. We show that $Ω(w,k)$ is an infini…