A palindromization map on free monoids

This paper is a survey of several results of combinatorial nature which have been obtained starting from a palindromization map on a free monoid $A^*$ introduced by the author in 1997 in the case of a binary alphabet and, successively, generalized by other authors for arbitrary finite alphabets. If one extends the action of the palindromization map to infinite words, one can generate the class of all standard episturmian words, which includes standard Sturmian words and Arnoux–Rauzy words. In this framework, an essential role is played by the class of palindromic prefixes of all standard episturmian words called epicentral words. These words are precisely the images of $A^*$ under the palindromization map. Epicentral words have several different representations and satisfy interesting combinatorial properties. A further extension of the palindromization map to a $\vartheta$-palindromization map, where $\vartheta$ is an arbitrary involutory antimorphism of $A^*$, is also briefly discussed.