Abstract
We introduce a square root map on Sturmian words and study its properties. Given a Sturmian word of slope \(\alpha \), there exists exactly six minimal squares in its language. A minimal square does not have a square as a proper prefix. A Sturmian word s of slope \(\alpha \) can be written as a product of these six minimal squares: \(s = X_1^2 X_2^2 X_3^2 \cdots \). The square root of s is defined to be the word \(\sqrt{s} = X_1 X_2 X_3 \cdots \). We prove that \(\sqrt{s}\) is also a Sturmian word of slope \(\alpha \). Moreover, we describe how to find the intercept of \(\sqrt{s}\) and an occurrence of any prefix of \(\sqrt{s}\) in s. Related to the square root map, we characterize the solutions of the word equation \(X_1^2 X_2^2 \cdots X_m^2 = (X_1 X_2 \cdots X_m)^2\) in the language of Sturmian words of slope \(\alpha \) where the words \(X_i^2\) are minimal squares of slope \(\alpha \).
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Acknowledgments
We thank our supervisors Juhani Karhumäki and Luca Zamboni for suggesting that the square root map might preserve the language of a Sturmian word.
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Peltomäki, J., Whiteland, M. (2015). A Square Root Map on Sturmian Words. In: Manea, F., Nowotka, D. (eds) Combinatorics on Words. WORDS 2015. Lecture Notes in Computer Science(), vol 9304. Springer, Cham. https://doi.org/10.1007/978-3-319-23660-5_17
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DOI: https://doi.org/10.1007/978-3-319-23660-5_17
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