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Balancedness of epichristoffel words

Ascertain whether epichristoffel words satisfy a balanced property analogous to the balancedness of Christoffel words; if such a property holds, characterize its precise form.

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Background

Christoffel words are characterized as balanced Lyndon words, linking their combinatorial and geometric definitions. For epichristoffel words, which extend Christoffel words to larger alphabets via episturmian morphisms, the balancedness is not established. The paper reiterates Paquin’s open question about whether an appropriate notion of balancedness applies to epichristoffel words and does not resolve it, focusing instead on tree-based structural properties and existence results.

References

Although epichristoffel words share many of the same properties as Christoffel words, Genevieve raised some open problems regarding epichristoffel words. These include the ability to characterize the epichristoffel word of each conjugacy class, whether epichristoffel words satisfy a type of balanced property, and whether there is an epichristoffel word of any length over a k-letter alphabet for a fixed k ≥ 3.

— On a Generalization of the Christoffel Tree: Epichristoffel Trees (2507.15313 - Krishnamoorthy et al., 21 Jul 2025) in Section 1. Introduction