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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2507.15313 |
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| _version_ | 1866915401782263808 |
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| author | Krishnamoorthy, Abhishek Thamburaj, Robinson Thomas, Durairaj Gnanaraj |
| author_facet | Krishnamoorthy, Abhishek Thamburaj, Robinson Thomas, Durairaj Gnanaraj |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_15313 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On a Generalization of the Christoffel Tree: Epichristoffel Trees Krishnamoorthy, Abhishek Thamburaj, Robinson Thomas, Durairaj Gnanaraj Formal Languages and Automata Theory G.2.1:F.2.2 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. |
| title | On a Generalization of the Christoffel Tree: Epichristoffel Trees |
| topic | Formal Languages and Automata Theory G.2.1:F.2.2 |
| url | https://arxiv.org/abs/2507.15313 |