Theory of Computing Systems

, Volume 59, Issue 2, pp 161–179 | Cite as

Weak Abelian Periodicity of Infinite Words

  • Sergey Avgustinovich
  • Svetlana Puzynina


An infinite word is called weak abelian periodic if it can be represented as an infinite concatenation of finite words with identical frequencies of letters. In the paper we undertake a general study of the weak abelian periodicity property. We consider its relation with the notions of balance and letter frequency, and study operations preserving weak abelian periodicity. We establish necessary and sufficient conditions for the weak abelian periodicity of fixed points of uniform binary morphisms. Finally, we discuss weak abelian periodicity in minimal subshifts.


Infinite word Abelian equivalence Periodicity Letter frequency Fixed points of morphisms Subshifts 



Supported in part by the Academy of Finland under grant 251371 and by the LABEX MILYON (ANR-10-LABX-0070) of Universit´e de Lyon, within the program “Investissements d’Avenir” (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR).


  1. 1.
    Adamczewski, B.: Balances for fixed points of primitive substitutions. Theor. Comput. Sci. 307, 47–75 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Avgustinovich, S., Karhumäki, J., Puzynina, S.: On abelian versions of critical factorization theorem. RAIRO - Theor. Inform. Appl. 46, 3–15 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Berthé, V., Rigo, M.: Combinatorics, Automata and Number Theory. Cambridge University Press, Cambridge (2010)CrossRefzbMATHGoogle Scholar
  4. 4.
    Cassaigne, J., Karhumäki, J.: Toeplitz words, generalized periodicity and periodically iterated morphisms. Eur. J. Comb. 18(5), 497–510 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cassaigne, J., Richomme, G., Saari, K., Zamboni, L.Q.: Avoiding Abelian powers in binary words with bounded Abelian complexity. Int. J. Found. Comput. Sci. 22(4), 905–920 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Constantinescu, S., Ilie, L.: Fine and Wilf’s theorem for abelian periods. Bull. Eur. Assoc. Theor. Comput. Sci. EATCS 89, 167–170 (2006)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Christou, M., Crochemore, M., Iliopoulos, C.S.: Identifying all abelian periods of a string in quadratic time and relevant problems. Int. J. Found. Comput. Sci. 23(6), 1371–1384 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Crochemore, M., Iliopoulos, C.S., Kociumaka, T., Kubica, M., Pachocki, J., Radoszewski, J., Rytter, W., Tyczynski, W., Walen, T.: A note on efficient computation of all abelian periods in a string. Inf. Process. Lett. 113(3), 74–77 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Durand, F.: A characterization of substitutive sequences using return words. Discret. Math. 179(1–3), 89–101 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Erdös, P.: Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. 6, 221–254 (1961)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Evdokimov, A.A.: Strongly asymmetric sequences generated by a finite number of symbols. Dokl. Akad. Nauk. SSSR 179, 1268–1271 (1968); Soviet Math. Dokl. 9, 536–539 (1968)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Frid, A.: On the frequency of factors in D0L words. J. Autom. Lang. Comb. 3(1), 29–41 (1998)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Gerver, J.L., Ramsey, L.T.: On certain sequences of lattice points. Pac. J. Math. 83(2), 357–363 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Holton, C., Zamboni, L.Q.: Geometric realizations of substitutions. Bull. Soc. Math. France 126, 149–179 (1998)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Keränen, V.: Abelian squares are avoidable on 4 letters. In: Automata, Languages and Programming (Vienna, 1992), volume 623 of Lecture Notes in Comput. Sci., pp. 41–52. Springer, Berlin (1992)Google Scholar
  16. 16.
    Kociumaka, T., Radoszewski, J., Rytter, W.: Fast algorithms for abelian periods in words and greatest common divisor queries. In: STACS 2013. LIPIcs 20, pp. 245–256. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2013)Google Scholar
  17. 17.
    Krajnev, V.A.: Words that do not contain consecutive factors with equal frequencies of letters. Metody Discretnogo Analiza v Reshenii Kombinatornyh Zadach 34, 27–37 (1980). [in Russian]zbMATHGoogle Scholar
  18. 18.
    Lothaire, M.: Algebraic Combinatorics on Words. Cambridge University Press, Cambridge (2002)CrossRefzbMATHGoogle Scholar
  19. 19.
    Pólya, G.: Über eine Aufgabe der Wahrscheinlichkeitsrechnung betreffend die Irrfahrt im Strassennetz. Math. Ann. 84(1–2), 149–160 (1921)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Puzynina, S., Zamboni, L.Q.: Abelian returns in Sturmian words. J. Combin. Theory, Ser. A 120(2), 390–408 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Richomme, G., Saari, K., Zamboni, L.Q.: Abelian complexity of minimal subshifts. J. Lond. Math. Soc. 83(1), 79–95 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Saari, K.: On the Frequency and Periodicity of Infinite Words, PhD thesis, Turku (2008)Google Scholar
  23. 23.
    Samsonov, A.V., Shur, A.M.: On Abelian repetition threshold. RAIRO Inform. Theor. Appl. 46, 147–163 (2012)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Sobolev Institute of MathematicsNovosibirskRussia
  2. 2.LIP, École Normale Supérieure de LyonUniversité de LyonLyonFrance

Personalised recommendations