## Francesco Dolce ; Edita Pelantová - On morphisms preserving palindromic richness

fi:7243 - Fundamenta Informaticae, March 10, 2022, Volume 185, Issue 1
On morphisms preserving palindromic richness

Authors: Francesco Dolce ; Edita Pelantová

It is known that each word of length $n$ contains at most $n+1$ distinct palindromes. A finite rich word is a word with maximal number of palindromic factors. The definition of palindromic richness can be naturally extended to infinite words. Sturmian words and Rote complementary symmetric sequences form two classes of binary rich words, while episturmian words and words coding symmetric $d$-interval exchange transformations give us other examples on larger alphabets. In this paper we look for morphisms of the free monoid, which allow us to construct new rich words from already known rich words. We focus on morphisms in Class $P_{ret}$. This class contains morphisms injective on the alphabet and satisfying a particular palindromicity property: for every morphism $\varphi$ in the class there exists a palindrome $w$ such that $\varphi(a)w$ is a first complete return word to $w$ for each letter $a$. We characterize $P_{ret}$ morphisms which preserve richness over a binary alphabet. We also study marked $P_{ret}$ morphisms acting on alphabets with more letters. In particular we show that every Arnoux-Rauzy morphism is conjugated to a morphism in Class $P_{ret}$ and that it preserves richness.

Volume: Volume 185, Issue 1
Published on: March 10, 2022
Accepted on: February 24, 2022
Submitted on: March 4, 2021
Keywords: Computer Science - Formal Languages and Automata Theory,Computer Science - Discrete Mathematics,68R15