Morphic images of episturmian words having finite palindromic defect

TL;DR

This paper investigates how certain morphisms affect the palindromic defect of episturmian words, showing that some morphisms preserve finiteness of the defect and enabling the construction of new rich words.

Contribution

It demonstrates that morphisms of class P_{ret} map zero palindromic defect episturmian words to words with finite defect and characterizes binary images of ternary episturmian words.

Findings

Images of episturmian words under P_{ret} morphisms have finite palindromic defect.

Binary images of ternary episturmian words have zero palindromic defect.

Results contribute to understanding morphisms that preserve finite or zero 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.