On Theta-palindromic Richness

In this paper we study generalization of the reversal mapping realized by an arbitrary involutory antimorphism $\Theta$. It generalizes the notion of a palindrome into a $\Theta$-palindrome -- a word invariant under $\Theta$. For languages closed under $\Theta$ we give the relation between $\Theta$-palindromic complexity and factor complexity. We generalize the notion of richness to $\Theta$-richness and we prove analogous characterizations of words that are $\Theta$-rich, especially in the case of set of factors invariant under $\Theta$. A criterion for $\Theta$-richness of $\Theta$-episturmian words is given together with other examples of $\Theta$-rich words.Comment: 14 page

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 $\Theta \in D_m$, we show that $\mathbf{t}_{b,m}$ is saturated by $\Theta$-palindromes up to the highest possible level. Using the terminology generalizing the notion of palindromic richness for more antimorphisms recently introduced by the author and E. Pelantov\'a, we show that $\mathbf{t}_{b,m}$ is $D_m$-rich. We also calculate the factor complexity of $\mathbf{t}_{b,m}$.Comment: 11 page

Languages invariant under more symmetries: overlapping factors versus palindromic richness

Factor complexity $\mathcal{C}$ and palindromic complexity $\mathcal{P}$ of infinite words with language closed under reversal are known to be related by the inequality $\mathcal{P}(n) + \mathcal{P}(n+1) \leq 2 + \mathcal{C}(n+1)-\mathcal{C}(n)$ for any $n\in \mathbb{N}$\,. Word for which the equality is attained for any $n$ is usually called rich in palindromes. In this article we study words whose languages are invariant under a finite group $G$ of symmetries. For such words we prove a stronger version of the above inequality. We introduce notion of $G$-palindromic richness and give several examples of $G$-rich words, including the Thue-Morse sequence as well.Comment: 22 pages, 1 figur

On morphisms preserving palindromic richness

It is known that each word of length $n$ contains at most $n+1$ distinct palindromes. A finite rich word is a word with maximal number of palindromic factors. The definition of palindromic richness can be naturally extended to infinite words. Sturmian words and Rote complementary symmetric sequences form two classes of binary rich words, while episturmian words and words coding symmetric $d$-interval exchange transformations give us other examples on larger alphabets. In this paper we look for morphisms of the free monoid, which allow to construct new rich words from already known rich words. We focus on morphisms in Class $P_{ret}$. This class contains morphisms injective on the alphabet and satisfying a particular palindromicity property: for every morphism $\varphi$ in the class there exists a palindrome $w$ such that $\varphi(a)w$ is a first complete return word to $w$ for each letter $a$. We characterize $P_{ret}$ morphisms which preserve richness over a binary alphabet. We also study marked $P_{ret}$ morphisms acting on alphabets with more letters. In particular we show that every Arnoux-Rauzy morphism is conjugated to a morphism in Class $P_{ret}$ and that it preserves richness

On Words with the Zero Palindromic Defect

We study the set of finite words with zero palindromic defect, i.e., words rich in palindromes. This set is factorial, but not recurrent. We focus on description of pairs of rich words which cannot occur simultaneously as factors of a longer rich word

Repetitions in infinite palindrome-rich words

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 $\Sigma_2=\{0,1\}$ with a small critical exponent of $2+\sqrt{2}/2$. This represents the first progress on an open problem of Vesti from 2017.Comment: 12 page

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 also made about the infinite defect of a word, the number of rich words of length n and two-dimensional rich words.Comment: 19 pages, 3 figure