On a generalization of Christoffel words: epichristoffel words

TL;DR

This paper introduces epichristoffel words, a new class of finite words over three or more letters, generalizing Christoffel words using episturmian morphisms, and explores which properties carry over.

Contribution

The paper defines epichristoffel words as a generalization of Christoffel words for larger alphabets using episturmian morphisms and analyzes their properties.

Findings

Epichristoffel words are successfully defined and constructed.

Some properties of Christoffel words are extended to epichristoffel words.

The paper discusses which properties do not generalize naturally.

Abstract

Sturmian sequences are well-known as the ones having minimal complexity over a 2-letter alphabet. They are also the balanced sequences over a 2-letter alphabet and the sequences describing discrete lines. They are famous and have been extensively studied since the 18th century. One of the {extensions} of these sequences over a $k$-letter alphabet, with $k\geq 3$, are the episturmian sequences, which generalizes a construction of Sturmian sequences using the palindromic closure operation. There exists a finite version of the Sturmian sequences called the Christoffel words. They are known since the works of Christoffel and have interested many mathematicians. In this paper, we introduce a generalization of Christoffel words for an alphabet with 3 letters or more, using the episturmian morphisms. We call them the {\it epichristoffel words}. We define this new class of finite words and show how some of the properties of the Christoffel words can be generalized naturally or not for this class.