A generalized palindromization map in free monoids

The palindromization map $\psi$ in a free monoid $A^*$ was introduced in 1997 by the first author in the case of a binary alphabet $A$, and later extended by other authors to arbitrary alphabets. Acting on infinite words, $\psi$ generates the class of standard episturmian words, including standard Arnoux-Rauzy words. In this paper we generalize the palindromization map, starting with a given code $X$ over $A$. The new map $\psi_X$ maps $X^*$ to the set $PAL$ of palindromes of $A^*$. In this way some properties of $\psi$ are lost and some are saved in a weak form. When $X$ has a finite deciphering delay one can extend $\psi_X$ to $X^{\omega}$, generating a class of infinite words much wider than standard episturmian words. For a finite and maximal code $X$ over $A$, we give a suitable generalization of standard Arnoux-Rauzy words, called $X$-AR words. We prove that any $X$-AR word is a morphic image of a standard Arnoux-Rauzy word and we determine some suitable linear lower and upper bounds to its factor complexity. For any code $X$ we say that $\psi_X$ is conservative when $\psi_X(X^{*})\subseteq X^{*}$. We study conservative maps $\psi_X$ and conditions on $X$ assuring that $\psi_X$ is conservative. We also investigate the special case of morphic-conservative maps $\psi_{X}$, i.e., maps such that $\phi\circ \psi = \psi_X\circ \phi$ for an injective morphism $\phi$. Finally, we generalize $\psi_X$ by replacing palindromic closure with $\theta$-palindromic closure, where $\theta$ is any involutory antimorphism of $A^*$. This yields an extension of the class of $\theta$-standard words introduced by the authors in 2006.Comment: Final version, accepted for publication on Theoret. Comput. Sc

Episturmian words: a survey

In this paper, we survey the rich theory of infinite episturmian words which generalize to any finite alphabet, in a rather resembling way, the well-known family of Sturmian words on two letters. After recalling definitions and basic properties, we consider episturmian morphisms that allow for a deeper study of these words. Some properties of factors are described, including factor complexity, palindromes, fractional powers, frequencies, and return words. We also consider lexicographical properties of episturmian words, as well as their connection to the balance property, and related notions such as finite episturmian words, Arnoux-Rauzy sequences, and "episkew words" that generalize the skew words of Morse and Hedlund.Comment: 36 pages; major revision: improvements + new material + more reference

On some problems related to palindrome closure

In this paper, we solve some open problems related to (pseudo)palindrome closure operators and to the infinite words generated by their iteration, that is, standard episturmian and pseudostandard words. We show that if ϑ is an involutory antimorphism of A*, then the right and left ϑ-palindromic closures of any factor of a ϑ-standard word are also factors of some ϑ-standard word. We also introduce the class of pseudostandard words with “seed”, obtained by iterated pseudopalindrome closure starting from a nonempty word. We show that pseudostandard words with seed are morphic images of standard episturmian words. Moreover, we prove that for any given pseudostandard word s with seed, all sufficiently long left special factors of s are prefixes of it