On a Generalization of the Christoffel Tree: Epichristoffel Trees

TL;DR

This paper introduces epichristoffel trees, a new combinatorial structure that helps analyze and factorize epichristoffel words, generalizing properties of Christoffel words to alphabets with three or more letters.

Contribution

It defines epichristoffel trees and demonstrates their usefulness in factorizing epichristoffel words, extending the understanding of these words beyond previous properties.

Findings

Epichristoffel trees facilitate factorization of epichristoffel words.

The structure helps identify subclasses sharing Christoffel word properties.

New insights into the combinatorial structure of epichristoffel words.

Abstract

Sturmian words form a family of one-sided infinite words over a binary alphabet that are obtained as a discretization of a line with an irrational slope starting from the origin. A finite version of this class of words called Christoffel words has been extensively studied for their interesting properties. It is a class of words that has a geometric and an algebraic definition, making it an intriguing topic of study for many mathematicians. Recently, a generalization of Christoffel words for an alphabet with 3 letters or more, called epichristoffel words, using episturmian morphisms has been studied, and many of the properties of Christoffel words have been shown to carry over to epichristoffel words; however, many properties are not shared by them as well. In this paper, we introduce the notion of an epichristoffel tree, which proves to be a useful tool in determining a subclass of epichristoffel words that share an important property of Christoffel words, which is the ability to factorize an epichristoffel word as a product of smaller epichristoffel words. We also use the epichristoffel tree to present some interesting results that help to better understand epichristoffel words.