Morphic images of episturmian words having finite palindromic defect

Abstract

We study morphisms from certain classes and their action on episturmian words. The first class is $P_{ret}$. In general, a morphism of class $P_{ret}$ can map an infinite word having zero palindromic defect to a word having infinite palindromic defect. We show that the image of an episturmian word, which has zero palindromic defect, under a morphism of class $P_{ret}$ has always its palindromic defect finite. We also focus on letter-to-letter morphisms to binary alphabet: we show that images of ternary episturmian words under such morphisms have zero palindromic defect. These results contribute to the study of an unsolved question of characterization of morphisms that preserve finite (resp. zero) palindromic defect. They also enable us to construct new examples of binary $H$-rich and almost $H$-rich words, where $H = \{\rm{Id}, R, E, RE \}$ is the group generated by both involutory antimorphisms on a binary alphabet.