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

Palindromic Prefixes and Episturmian Words

Let $w$ be an infinite word on an alphabet $A$. We denote by $(n_i)_{i \geq 1}$ the increasing sequence (assumed to be infinite) of all lengths of palindrome prefixes of $w$. In this text, we give an explicit construction of all words $w$ such that $n_{i+1} \leq 2 n_i + 1$ for any $i$, and study these words. Special examples include characteristic Sturmian words, and more generally standard episturmian words. As an application, we study the values taken by the quantity $\limsup n_{i+1}/n_i$, and prove that it is minimal (among all non-periodic words) for the Fibonacci word.Comment: 27 pages; many minor changes; to appear in J. Comb. Th. Series

Constructions of words rich in palindromes and pseudopalindromes

A narrow connection between infinite binary words rich in classical palindromes and infinite binary words rich simultaneously in palindromes and pseudopalindromes (the so-called $H$-rich words) is demonstrated. The correspondence between rich and $H$-rich words is based on the operation $S$ acting over words over the alphabet $\{0,1\}$ and defined by $S(u_0u_1u_2\ldots) = v_1v_2v_3\ldots$, where $v_i= u_{i-1} + u_i \mod 2$. The operation $S$ enables us to construct a new class of rich words and a new class of $H$-rich words. Finally, the operation $S$ is considered on the multiliteral alphabet $\mathbb{Z}_m$ as well and applied to the generalized Thue--Morse words. As a byproduct, new binary rich and $H$-rich words are obtained by application of $S$ on the generalized Thue--Morse words over the alphabet $\mathbb{Z}_4$.Comment: 26 page

Rich, Sturmian, and trapezoidal words

In this paper we explore various interconnections between rich words, Sturmian words, and trapezoidal words. Rich words, first introduced in arXiv:0801.1656 by the second and third authors together with J. Justin and S. Widmer, constitute a new class of finite and infinite words characterized by having the maximal number of palindromic factors. Every finite Sturmian word is rich, but not conversely. Trapezoidal words were first introduced by the first author in studying the behavior of the subword complexity of finite Sturmian words. Unfortunately this property does not characterize finite Sturmian words. In this note we show that the only trapezoidal palindromes are Sturmian. More generally we show that Sturmian palindromes can be characterized either in terms of their subword complexity (the trapezoidal property) or in terms of their palindromic complexity. We also obtain a similar characterization of rich palindromes in terms of a relation between palindromic complexity and subword complexity.Comment: 7 page

Palindromic complexity of infinite words associated with simple Parry numbers

A simple Parry number is a real number \beta>1 such that the R\'enyi expansion of 1 is finite, of the form d_\beta(1)=t_1...t_m. We study the palindromic structure of infinite aperiodic words u_\beta that are the fixed point of a substitution associated with a simple Parry number \beta. It is shown that the word u_\beta contains infinitely many palindromes if and only if t_1=t_2= ... =t_{m-1} \geq t_m. Numbers \beta satisfying this condition are the so-called confluent Pisot numbers. If t_m=1 then u_\beta is an Arnoux-Rauzy word. We show that if \beta is a confluent Pisot number then P(n+1)+ P(n) = C(n+1) - C(n)+ 2, where P(n) is the number of palindromes and C(n) is the number of factors of length n in u_\beta. We then give a complete description of the set of palindromes, its structure and properties.Comment: 28 pages, to appear in Annales de l'Institut Fourie