Abstract
In our paper [A square root map on Sturmian words, Electron. J. Combin. 24.1 (2017)], we introduced a symbolic square root map. Every optimal squareful infinite word s contains exactly six minimal squares and can be written as a product of these squares: \(s = X_1^2 X_2^2 \cdots \). The square root \(\sqrt{s}\) of s is the infinite word \(X_1 X_2 \cdots \) obtained by deleting half of each square. We proved that the square root map preserves the languages of Sturmian words (which are optimal squareful words). The dynamics of the square root map on a Sturmian subshift are well understood. In our earlier work, we introduced another type of subshift of optimal squareful words which together with the square root map form a dynamical system. In this paper, we study these dynamical systems in more detail and compare their properties to the Sturmian case. The main results are characterizations of periodic points and the limit set. The results show that while there is some similarity it is possible for the square root map to exhibit quite different behavior compared to the Sturmian case.
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Notes
- 1.
Without this condition the subshift \(\varOmega \), defined below, does not consist of optimal squareful words; see the remark after [3, Lemma 8.3].
- 2.
If S or L were in \(\varPi (\mathfrak {a},\mathfrak {b})\), then they would be nonprimitive as solutions to (3).
- 3.
In the proof of [3, Theorem 8.7] only the case \(\mathfrak {c}= 1\) was considered. This is of no consequence as the proof given applies to arbitrary product ofs the words S and L.
- 4.
In general, e.g., the word \(\gamma ^2\) can be a factor of \(\overline{\gamma } ^3\).
- 5.
See the proof of [3, Theorem 8.7] for precise details.
References
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Peltomäki, J.: Privileged Words and Sturmian Words. Ph.D. dissertation, Turku Centre for Computer Science, University of Turku (2016). http://www.doria.fi/handle/10024/124473
Peltomäki, J., Whiteland, M.: A square root map on Sturmian words. Electron. J. Comb. 24(1) (2017). Article No. P1.54
Pytheas Fogg, N.: Sturmian Sequences. In: Berthé, V., Ferenczi, S., Mauduit, C., Siegel, A. (eds.) Substitutions in Dynamics, Arithmetics and Combinatorics. Lecture Notes in Mathematics, vol 1794. Springer, Berlin, Heidelberg (2002)
Saari, K.: On the frequency and periodicity of infinite words. Ph.D. dissertation, Turku Centre for Computer Science, University of Turku (2008). http://users.utu.fi/kasaar/pubs/phdth.pdf
Saari, K.: Everywhere \(\alpha \)-repetitive sequences and Sturmian words. Eur. J. Comb. 31, 177–192 (2010)
Acknowledgments
The work of the first author was supported by the Finnish Cultural Foundation by a personal grant. He also thanks the Department of Computer Science at Åbo Akademi for its hospitality. The second author was partially supported by the Vilho, Yrjö and Kalle Väisälä Foundation. Jyrki Lahtonen deserves our thanks for fruitful discussions.
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Peltomäki, J., Whiteland, M. (2017). More on the Dynamics of the Symbolic Square Root Map. In: Brlek, S., Dolce, F., Reutenauer, C., Vandomme, É. (eds) Combinatorics on Words. WORDS 2017. Lecture Notes in Computer Science(), vol 10432. Springer, Cham. https://doi.org/10.1007/978-3-319-66396-8_10
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DOI: https://doi.org/10.1007/978-3-319-66396-8_10
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