On morphisms preserving palindromic richness

TL;DR

This paper investigates specific morphisms in Class P_{ret} that generate new rich words from known ones, characterizing their properties and showing their preservation of palindromic richness over binary and larger alphabets.

Contribution

It characterizes P_{ret} morphisms that preserve palindromic richness and demonstrates their relation to Arnoux-Rauzy morphisms on larger alphabets.

Findings

P_{ret} morphisms preserving richness are characterized over binary alphabets.

Every Arnoux-Rauzy morphism is conjugated to a P_{ret} morphism and preserves richness.

The study extends the understanding of morphisms that generate rich words in combinatorics on words.

Abstract

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.