Palindromic Richness

TL;DR

This paper investigates the combinatorial and structural properties of rich words, a class of words with maximal palindromic content, providing classifications, characterizations, and insights into their behavior under morphisms.

Contribution

It offers a unified framework for understanding rich words, characterizes periodic and balanced cases, and explores their relation to Fraenkel's conjecture and morphism actions.

Findings

Recurrent balanced rich infinite words are episturmian.

Weakly rich words on three or more letters obey Fraenkel's conjecture.

Certain almost rich words are either episturmian or have repeated letter frequencies.

Abstract

In this paper, we study combinatorial and structural properties of a new class of finite and infinite words that are 'rich' in palindromes in the utmost sense. A characteristic property of so-called "rich words" is that all complete returns to any palindromic factor are themselves palindromes. These words encompass the well-known episturmian words, originally introduced by the second author together with X. Droubay and G. Pirillo in 2001. Other examples of rich words have appeared in many different contexts. Here we present the first unified approach to the study of this intriguing family of words. Amongst our main results, we give an explicit description of the periodic rich infinite words and show that the recurrent balanced rich infinite words coincide with the balanced episturmian words. We also consider two wider classes of infinite words, namely "weakly rich words" and almost rich words (both strictly contain all rich words, but neither one is contained in the other). In particular, we classify all recurrent balanced weakly rich words. As a consequence, we show that any such word on at least three letters is necessarily episturmian; hence weakly rich words obey Fraenkel's conjecture. Likewise, we prove that a certain class of almost rich words obeys Fraenkel's conjecture by showing that the recurrent balanced ones are episturmian or contain at least two distinct letters with the same frequency. Lastly, we study the action of morphisms on (almost) rich words with particular interest in morphisms that preserve (almost) richness. Such morphisms belong to the class of "P-morphisms" that was introduced by A. Hof, O. Knill, and B. Simon in 1995.