Interval Exchange Words and the Question of Hof, Knill, and Simon

  • Zuzana Masáková
  • Edita Pelantová
  • Štěpán Starosta
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9168)


We consider words coding non-degenerate 3 interval exchange transformation. It is known that such words contain infinitely many palindromic factors. We show that for any morphism \(\xi \) fixing such a word, either \(\xi \) or \(\xi ^2\) is conjugate to a class P morphism. By this, we provide a new family of palindromic infinite words satisfying the conjecture of Hof, Knill and Simon, as formulated by Tan.


Palindrome Morphism Interval exchange 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Arnoux, P., Berthé, V., Masáková, Z., Pelantová, E.: Sturm numbers and substitution invariance of 3iet words. Integers 8 (Article A14) (2008)Google Scholar
  2. 2.
    Blondin Massé, A., Brlek, S., Garon, A., Labbé, S.: Equations on palindromes and circular words. Theoret. Comput. Sci. 412, 2922–2930 (2011)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Blondin Massé, A., Brlek, S., Labbé, S.: Palindromic lacunas of the Thue-Morse word. In: Proc. GASCom 2008, pp. 53–67 (2008)Google Scholar
  4. 4.
    Ferenczi, S., Holton, C., Zamboni, L.: Structure of three-interval exchange transformations II: a combinatorial description of the tranjectories. J. Anal. Math. 89, 239–276 (2003)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Gawrychowski, P., Manea, F., Nowotka, D.: Testing generalised freeness of words. In: Mayr, E.W., Portier, N. (eds.) 31st International Symposium on Theoretical Aspects of Computer Science (STACS 2014). (LIPIcs), vol. 25, pp. 337–349. Dagstuhl, Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, Germany (2014)Google Scholar
  6. 6.
    Hof, A., Knill, O., Simon, B.: Singular continuous spectrum for palindromic Schrödinger operators. Comm. Math. Phys. 174, 149–159 (1995)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Kari, L., Mahalingam, K.: Watson-crick conjugate and commutative words. In: Garzon, M.H., Yan, H. (eds.) DNA 2007. LNCS, vol. 4848, pp. 273–283. Springer, Heidelberg (2008) CrossRefGoogle Scholar
  8. 8.
    Keane, M.: Interval exchange transformations. Math. Z. 141, 25–31 (1975)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Labbé, S., Pelantová, E.: Palindromic sequences generated from marked morphisms. Eur. J. Comb. 51, 200–214 (2016)CrossRefGoogle Scholar
  10. 10.
    Labbé, S.: A counterexample to a question of Hof, Knill and Simon. Electron. J. Comb. 21 (2014)Google Scholar
  11. 11.
    Lothaire, M.: Algebraic combinatorics on words. Encyclopedia of Mathematics and its Applications, vol. 90. Cambridge University Press (2002)Google Scholar
  12. 12.
    Masáková, Z., Pelantová, E., Starosta, Š: Interval exchange words and the question of Hof, Knill, and Simon (2015). (preprint available at)
  13. 13.
    Tan, B.: Mirror substitutions and palindromic sequences. Theoret. Comput. Sci. 389, 118–124 (2007)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Zuzana Masáková
    • 1
  • Edita Pelantová
    • 1
  • Štěpán Starosta
    • 2
  1. 1.Faculty of Nuclear Sciences and Physical EngineeringCzech Technical University in PraguePraha 2Czech Republic
  2. 2.Faculty of Information TechnologyCzech Technical University in PraguePraha 6Czech Republic

Personalised recommendations