Characteristic morphisms of generalized episturmian words

In a recent paper with L.Q. Zamboni, the authors introduced the class of ϑ-episturmian words. An infinite word over A is standard ϑ-episturmian, where ϑ is an involutory antimorphism of A∗, if its set of factors is closed under ϑ and its left special factors are prefixes. When ϑ is the reversal operator, one obtains the usual standard episturmian words. In this paper, we introduce and study ϑ-characteristic morphisms, that is, morphisms which map standard episturmian words into standard ϑ-episturmian words. They are a natural extension of standard episturmian morphisms. The main result of the paper is a characterization of these morphisms when they are injective. In order to prove this result, we also introduce and study a class of biprefix codes which are overlap-free, i.e., any two code words do not overlap properly, and normal, i.e., no proper suffix (prefix) of any code-word is left (right) special in the code. A further result is that any standard ϑ-episturmian word is a morphic image, by an injective ϑ-characteristic morphism, of a standard episturmian word.