Advertisement

Infinite Words Rich and Almost Rich in Generalized Palindromes

  • Edita Pelantová
  • Štěpán Starosta
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6795)

Abstract

We focus on Θ-rich and almost Θ-rich words over a finite alphabet \(\mathcal{A}\), where Θ is an involutive antimorphism over \(\mathcal{A}^*\). We show that any recurrent almost Θ-rich word u is an image of a recurrent Θ′-rich word under a suitable morphism, where Θ′ is again an involutive antimorphism. Moreover, if the word u is uniformly recurrent, we show that Θ′ can be set to the reversal mapping. We also treat one special case of almost Θ-rich words. We show that every Θ-standard word with seed is an image of an Arnoux-Rauzy word.

Keywords

palindrome pseudopalindrome palindromic defect richness 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Anne, V., Zamboni, L.Q., Zorca, I.: Palindromes and pseudo-palindromes in episturmian and pseudo-episturmian infinite words. In: Brlek, S., Reutenauer, C. (eds.) Words 2005, vol. (36), pp. 91–100. LACIM (2005)Google Scholar
  2. 2.
    Baláži, P., Masáková, Z., Pelantová, E.: Factor versus palindromic complexity of uniformly recurrent infinite words. Theoret. Comput. Sci. 380(3), 266–275 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Balková, L., Pelantová, E., Starosta, Š.: Infinite words with finite defect. To appear in Adv. Appl. Math., (2011), preprint available at http://arxiv.org/abs/1009.5105
  4. 4.
    Blondin Massé, A., Brlek, S., Garon, A., Labbé, S.: Combinatorial properties of f-palindromes in the Thue-Morse sequence. Pure Math. Appl. 19(2-3), 39–52 (2008)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Brlek, S., Hamel, S., Nivat, M., Reutenauer, C.: On the palindromic complexity of infinite words. Internat. J. Found. Comput. 15(2), 293–306 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bucci, M., De Luca, A.: On a family of morphic images of arnoux-rauzy words. In: Dediu, A.H., Ionescu, A.M., Martín-Vide, C. (eds.) LATA 2009. LNCS, vol. 5457, pp. 259–266. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  7. 7.
    Bucci, M., De Luca, A., Glen, A., Zamboni, L.Q.: A connection between palindromic and factor complexity using return words. Adv. in Appl. Math. 42(1), 60–74 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bucci, M., de Luca, A., De Luca, A.: Characteristic morphisms of generalized episturmian words. Theor. Comput. Sci. 410, 2840–2859 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Bucci, M., de Luca, A., De Luca, A.: Rich and periodic-like words. In: Diekert, V., Nowotka, D. (eds.) DLT 2009. LNCS, vol. 5583, pp. 145–155. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  10. 10.
    Bucci, M., de Luca, A., De Luca, A., Zamboni, L.Q.: On different generalizations of episturmian words. Theoret. Comput. Sci. 393(1-3), 23–36 (2008)Google Scholar
  11. 11.
    Bucci, M., de Luca, A., De Luca, A., Zamboni, L.Q.: On theta-episturmian words. European J. Combin. 30(2), 473–479 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Droubay, X., Justin, J., Pirillo, G.: Episturmian words and some constructions of de Luca and Rauzy. Theoret. Comput. Sci. 255(1-2), 539–553 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Glen, A., Justin, J., Widmer, S., Zamboni, L.Q.: Palindromic richness. European J. Combin. 30(2), 510–531 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Hof, A., Knill, O., Simon, B.: Singular continuous spectrum for palindromic Schrödinger operators. Comm. Math. Phys. 174, 149–159 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Kari, L., Magalingam, K.: Watson-Crick palindromes in DNA computing. Nat. Comput. (9), 297–316 (2010)Google Scholar
  16. 16.
    Pelantová, E., Starosta, Š.: Languages invariant under more symmetries: overlapping factors versus palindromic richness. (2011), preprint available at http://arxiv.org/abs/1103.4051
  17. 17.
    Rauzy, G.: Suites à termes dans un alphabet fini. Séminaire de Théorie des Nombres de Bordeaux Anné 1982–1983(exposé 25) (1983)Google Scholar
  18. 18.
    Starosta, Š.: On theta-palindromic richness. Theoret. Comput. Sci. 412(12-14), 1111–1121 (2011)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Edita Pelantová
    • 1
  • Štěpán Starosta
    • 1
  1. 1.Department of MathematicsFNSPE, Czech Technical University in PraguePraha 2Czech Republic

Personalised recommendations