Palindromic closures using multiple antimorphisms

Palindromic richness for languages invariant under more symmetries

For a given finite group $G$ consisting of morphisms and antimorphisms of a free monoid $\mathcal{A}^*$, we study infinite words with language closed under the group $G$. We focus on the notion of $G$-richness which describes words rich in generalized palindromic factors, i.e., in factors $w$ satisfying $Θ(w) = w$ for some antimorphism $Θ\in G$. We give several equivalent descriptions which are generalizations of know characterizations of rich words (in the terms of classical palindromes) and show two examples of $G$-rich words.

Generalized Thue-Morse words and palindromic richness

We prove that the generalized Thue-Morse word $\mathbf{t}_{b,m}$ defined for $b \geq 2$ and $m \geq 1$ as $\mathbf{t}_{b,m} = (s_b(n) \mod m)_{n=0}^{+\infty}$, where $s_b(n)$ denotes the sum of digits in the base-$b$ representation of the integer $n$, has its language closed under all elements of a group $D_m$ isomorphic to the dihedral group of order $2m$ consisting of morphisms and antimorphisms. Considering simultaneously antimorphisms $Θ\in D_m$, we show that $\mathbf{t}_{b,m}$ is saturated by $Θ$-palindromes up to the highest possible level. Using the terminology generalizing the notion o…

自由群中较小的回文长度及带有反演运算的词方程

Small palindromic lengths in free groups and word equations with antimorphisms

本文研究了自由群中回文长度为2和3的词特性，并探讨了自由群中回文长度计算的复杂性。

On Theta-palindromic Richness

In this paper we study generalization of the reversal mapping realized by an arbitrary involutory antimorphism $Θ$. It generalizes the notion of a palindrome into a $Θ$-palindrome -- a word invariant under $Θ$. For languages closed under $Θ$ we give the relation between $Θ$-palindromic complexity and factor complexity. We generalize the notion of richness to $Θ$-richness and we prove analogous characterizations of words that are $Θ$-rich, especially in the case of set of factors invariant under $Θ$. A criterion for $Θ$-richness of $Θ$-episturmian words is given together with other examples of …

Morphic images of episturmian words having finite palindromic defect

We study morphisms from certain classes and their action on episturmian words. The first class is $P_{ret}$. In general, a morphism of class $P_{ret}$ can map an infinite word having zero palindromic defect to a word having infinite palindromic defect. We show that the image of an episturmian word, which has zero palindromic defect, under a morphism of class $P_{ret}$ has always its palindromic defect finite. We also focus on letter-to-letter morphisms to binary alphabet: we show that images of ternary episturmian words under such morphisms have zero palindromic defect. These results contribut…

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。