On the conjugates of Christoffel words

TL;DR

This paper develops a finitary, Ostrowski-based parametrization of the conjugates of Christoffel words, enabling an explicit, representation-independent construction of the entire conjugation class. By combining continuant polynomials, Rauzy-like standard-word techniques, and endomorphisms, the authors provide a precise description of the borders of conjugates and establish that these borders are themselves conjugates or powers of them. The work further connects these finite Christoffel word structures to Sturmian graphs, compact graphs, central words, and the Stern–Brocot tree, embedding finite conjugates into a broader combinatorial and automata-theoretic framework. Overall, it offers a noncommutative lifting of Ostrowski numeration and extends Frid’s prefix parametrization to the realm of Christoffel word conjugates, with implications for Sturmian word theory and automaton representations of suffix languages.

Abstract

We introduce a parametrization of the conjugates of Christoffel words based on the integer Ostrowski numeration system. We use it to give a precise description of the borders (prefixes which are also suffixes) of the conjugates of Christoffel words and to revisit the notion of Sturmian graph introduced by Epifanio et al.

Figures (2)

• Figure 1: The compact graph $(V,E)$
• Figure 2: A path in the tree of central words, a compact graph and a Sturmian graph

Theorems & Definitions (74)

• Proposition 3.1
• Lemma 3.2
• proof
• Lemma 4.1
• proof
• Lemma 5.1
• Lemma 7.1
• proof
• Lemma 7.2
• proof