Advertisement

Ergodic Infinite Permutations of Minimal Complexity

  • Sergey V. Avgustinovich
  • Anna E. Frid
  • Svetlana PuzyninaEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9168)

Abstract

An infinite permutation can be defined as a linear ordering of the set of natural numbers. Similarly to infinite words, a complexity p(n) of an infinite permutation is defined as a function counting the number of its factors of length n. For infinite words, a classical result of Morse and Hedlund, 1940, states that if the complexity of an infinite word satisfies \(p(n)\le n\) for some n, then the word is ultimately periodic. Hence minimal complexity of aperiodic words is equal to \(n+1\), and words with such complexity are called Sturmian. For infinite permutations this does not hold: There exist aperiodic permutations with complexity functions of arbitrarily slow growth, and hence there are no permutations of minimal complexity.

In the paper we introduce a new notion of ergodic permutation, i.e., a permutation which can be defined by a sequence of numbers from [0, 1], such that the frequency of its elements in any interval is equal to the length of that interval. We show that the minimal complexity of an ergodic permutation is \(p(n)=n\), and that the class of ergodic permutations of minimal complexity coincides with the class of so-called Sturmian permutations, directly related to Sturmian words.

Keywords

Conjugate Class Permutation Entropy Irrational Rotation Minimal Complexity Canonical Sequence 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Allouche, J.-P., Shallit, J.: Automatic sequences – theory, applications, generalizations. Cambridge University Press (2003)Google Scholar
  2. 2.
    Allouche, J.-P., Shallit, J.: The ubiquitous Prouhet-Thue-Morse sequence. In: Sequences and Their Applications, Discrete Mathematics and Theoretical Computer Science, pp. 1–16. Springer, London (1999)Google Scholar
  3. 3.
    Amigó, J.: Permutation Complexity in Dynamical Systems - Ordinal Patterns. Permutation Entropy and All That, Springer Series in Synergetics (2010)Google Scholar
  4. 4.
    Avgustinovich, S.V., Frid, A., Kamae, T., Salimov, P.: Infinite permutations of lowest maximal pattern complexity. Theoretical Computer Science 412, 2911–2921 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Avgustinovich, S.V., Kitaev, S., Pyatkin, A., Valyuzhenich, A.: On square-free permutations. J. Autom. Lang. Comb. 16(1), 3–10 (2011)Google Scholar
  6. 6.
    Bandt, C., Keller, G., Pompe, B.: Entropy of interval maps via permutations. Nonlinearity 15, 1595–1602 (2002)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Cassaigne, J., Nicolas, F.: Factor complexity. Combinatorics, automata and number theory, Encyclopedia Math. Appl. 135, 163–247 (2010). Cambridge Univ. PressMathSciNetCrossRefGoogle Scholar
  8. 8.
    Elizalde, S.: The number of permutations realized by a shift. SIAM J. Discrete Math. 23, 765–786 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Ferenczi, S., Monteil, T.: Infinite words with uniform frequencies, and invariant measures. Combinatorics, automata and number theory. Encyclopedia Math. Appl. 135, 373–409 (2010). Cambridge Univ. PressMathSciNetGoogle Scholar
  10. 10.
    Fon-Der-Flaass, D.G., Frid, A.E.: On periodicity and low complexity of infinite permutations. European J. Combin. 28, 2106–2114 (2007)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Frid, A.: Fine and Wilf’s theorem for permutations. Sib. Elektron. Mat. Izv. 9, 377–381 (2012)MathSciNetGoogle Scholar
  12. 12.
    Frid, A., Zamboni, L.: On automatic infinite permutations. Theoret. Inf. Appl. 46, 77–85 (2012)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Kamae, T., Zamboni, L.: Sequence entropy and the maximal pattern complexity of infinite words. Ergodic Theory and Dynamical Systems 22, 1191–1199 (2002)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Kamae, T., Zamboni, L.: Maximal pattern complexity for discrete systems. Ergodic Theory and Dynamical Systems 22, 1201–1214 (2002)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Lothaire, M.: Algebraic combinatorics on words. Cambridge University Press (2002)Google Scholar
  16. 16.
    Makarov, M.: On permutations generated by infinite binary words. Sib. Elektron. Mat. Izv. 3, 304–311 (2006)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Makarov, M.: On an infinite permutation similar to the Thue-Morse word. Discrete Math. 309, 6641–6643 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Makarov, M.: On the permutations generated by Sturmian words. Sib. Math. J. 50, 674–680 (2009)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Morse, M., Hedlund, G.: Symbolic dynamics II: Sturmian sequences. Amer. J. Math. 62, 1–42 (1940)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Valyuzhenich, A.: On permutation complexity of fixed points of uniform binary morphisms. Discr. Math. Theoret. Comput. Sci. 16, 95–128 (2014)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Widmer, S.: Permutation complexity of the Thue-Morse word. Adv. Appl. Math. 47, 309–329 (2011)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Widmer, S.: Permutation complexity related to the letter doubling map, WORDS (2011)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Sergey V. Avgustinovich
    • 1
  • Anna E. Frid
    • 2
  • Svetlana Puzynina
    • 1
    • 3
    Email author
  1. 1.Sobolev Institute of MathematicsNovosibirskRussia
  2. 2.Aix-Marseille UniversitéMarseilleFrance
  3. 3.LIP, ENS de LyonUniversité de LyonLyonFrance

Personalised recommendations