A generalized palindromization map in free monoids

The palindromization map ψ 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, ψ generates the class of standard episturmian words, including standard ArnouxRauzy words. In this paper we generalize the palindromization map, starting with a given code X over A. The new map ψX maps X to the set PAL of palindromes of A. In this way some properties of ψ are lost and some are saved in a weak form. When X has a finite deciphering delay one can extend ψX to X, 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 ψX is conservative when ψX(X) ⊆ X. We study conservative maps ψX and conditions on X assuring that ψX is conservative. We also investigate the special case of morphic-conservative maps ψX , i.e., maps such that φ ◦ ψ = ψX ◦ φ for an injective morphism φ. Finally, we generalize ψX by replacing palindromic closure with θ-palindromic closure, where θ is any involutory antimorphism of A. This yields an extension of the class of θ-standard words introduced by the authors in 2006.