On Christoffel and standard words and their derivatives

TL;DR

This paper introduces derivatives for Christoffel, standard, and Sturmian words, exploring their properties, defining depth, and providing combinatorial and arithmetic bounds, advancing understanding of their structural characteristics.

Contribution

It presents a novel framework for derivatives of Christoffel and standard words, including definitions, properties, and bounds for their depth, enriching combinatorial word theory.

Findings

Derivatives preserve word classes and are realized via specific morphisms.

Depth of words is characterized and bounded combinatorially and arithmetically.

Higher-order derivatives lead to a hierarchy of word structures.

Abstract

We introduce and study natural derivatives for Christoffel and finite standard words, as well as for characteristic Sturmian words. These derivatives, which are realized as inverse images under suitable morphisms, preserve the aforementioned classes of words. In the case of Christoffel words, the morphisms involved map $a$ to $a^{k+1}b$ (resp.,~$ab^{k}$) and $b$ to $a^{k}b$ (resp.,~$ab^{k+1}$) for a suitable $k>0$. As long as derivatives are longer than one letter, higher-order derivatives are naturally obtained. We define the depth of a Christoffel or standard word as the smallest order for which the derivative is a single letter. We give several combinatorial and arithmetic descriptions of the depth, and (tight) lower and upper bounds for it.