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 Arnoux-Rauzy 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 Xω, 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.