A generalized palindromization map in free monoids

• Authors: Aldo De Luca;Alessandro De Luca
• Venue: Theoretical Computer Science
• Year: 2012

The palindromization map @j 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, @j 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 @j"X maps X^* to the set PAL of palindromes of A^*. In this way, some properties of @j are lost and some are saved in a weak form. When X has a finite deciphering delay, one can extend @j"X to X^@w, 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 @j"X is conservative when @j"X(X^*)@?X^*. We study conservative maps @j"X and conditions on X assuring that @j"X is conservative. We also investigate the special case of morphic-conservative maps @j"X, i.e., maps such that @f@?@j=@j"X@?@f for an injective morphism @f. Finally, we generalize @j"X by replacing palindromic closure with @q-palindromic closure, where @q is any involutory antimorphism of A^*. This yields an extension of the class of @q-standard words introduced by the authors in 2006.