Advertisement

More on the Dynamics of the Symbolic Square Root Map

(Extended Abstract)
  • Jarkko PeltomäkiEmail author
  • Markus Whiteland
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10432)

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.

Notes

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.

References

  1. 1.
    Lothaire, M.: Algebraic Combinatorics on Words. No. 90 in Encyclopedia of Mathematics and Its Applications. Cambridge University Press, Cambridge (2002)Google Scholar
  2. 2.
    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
  3. 3.
    Peltomäki, J., Whiteland, M.: A square root map on Sturmian words. Electron. J. Comb. 24(1) (2017). Article No. P1.54Google Scholar
  4. 4.
    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)Google Scholar
  5. 5.
    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
  6. 6.
    Saari, K.: Everywhere \(\alpha \)-repetitive sequences and Sturmian words. Eur. J. Comb. 31, 177–192 (2010)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Turku Centre for Computer Science TUCSTurkuFinland
  2. 2.Department of Mathematics and StatisticsUniversity of TurkuTurkuFinland

Personalised recommendations