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Ergodic Infinite Permutations of Minimal Complexity

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Book cover Developments in Language Theory (DLT 2015)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9168))

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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.

S. Puzynina—Supported by the LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program “Investissements d’Avenir” (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR).

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References

  1. Allouche, J.-P., Shallit, J.: Automatic sequences – theory, applications, generalizations. Cambridge University Press (2003)

    Google Scholar 

  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. Amigó, J.: Permutation Complexity in Dynamical Systems - Ordinal Patterns. Permutation Entropy and All That, Springer Series in Synergetics (2010)

    Google Scholar 

  4. Avgustinovich, S.V., Frid, A., Kamae, T., Salimov, P.: Infinite permutations of lowest maximal pattern complexity. Theoretical Computer Science 412, 2911–2921 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  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. Bandt, C., Keller, G., Pompe, B.: Entropy of interval maps via permutations. Nonlinearity 15, 1595–1602 (2002)

    Article  MathSciNet  Google Scholar 

  7. Cassaigne, J., Nicolas, F.: Factor complexity. Combinatorics, automata and number theory, Encyclopedia Math. Appl. 135, 163–247 (2010). Cambridge Univ. Press

    Article  MathSciNet  Google Scholar 

  8. Elizalde, S.: The number of permutations realized by a shift. SIAM J. Discrete Math. 23, 765–786 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  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. Press

    MathSciNet  Google Scholar 

  10. Fon-Der-Flaass, D.G., Frid, A.E.: On periodicity and low complexity of infinite permutations. European J. Combin. 28, 2106–2114 (2007)

    Article  MathSciNet  Google Scholar 

  11. Frid, A.: Fine and Wilf’s theorem for permutations. Sib. Elektron. Mat. Izv. 9, 377–381 (2012)

    MathSciNet  Google Scholar 

  12. Frid, A., Zamboni, L.: On automatic infinite permutations. Theoret. Inf. Appl. 46, 77–85 (2012)

    Article  MathSciNet  Google Scholar 

  13. Kamae, T., Zamboni, L.: Sequence entropy and the maximal pattern complexity of infinite words. Ergodic Theory and Dynamical Systems 22, 1191–1199 (2002)

    MathSciNet  MATH  Google Scholar 

  14. Kamae, T., Zamboni, L.: Maximal pattern complexity for discrete systems. Ergodic Theory and Dynamical Systems 22, 1201–1214 (2002)

    MathSciNet  MATH  Google Scholar 

  15. Lothaire, M.: Algebraic combinatorics on words. Cambridge University Press (2002)

    Google Scholar 

  16. Makarov, M.: On permutations generated by infinite binary words. Sib. Elektron. Mat. Izv. 3, 304–311 (2006)

    MathSciNet  MATH  Google Scholar 

  17. Makarov, M.: On an infinite permutation similar to the Thue-Morse word. Discrete Math. 309, 6641–6643 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  18. Makarov, M.: On the permutations generated by Sturmian words. Sib. Math. J. 50, 674–680 (2009)

    Article  MathSciNet  Google Scholar 

  19. Morse, M., Hedlund, G.: Symbolic dynamics II: Sturmian sequences. Amer. J. Math. 62, 1–42 (1940)

    Article  MathSciNet  Google Scholar 

  20. Valyuzhenich, A.: On permutation complexity of fixed points of uniform binary morphisms. Discr. Math. Theoret. Comput. Sci. 16, 95–128 (2014)

    MathSciNet  MATH  Google Scholar 

  21. Widmer, S.: Permutation complexity of the Thue-Morse word. Adv. Appl. Math. 47, 309–329 (2011)

    Article  MathSciNet  Google Scholar 

  22. Widmer, S.: Permutation complexity related to the letter doubling map, WORDS (2011)

    Google Scholar 

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Authors and Affiliations

  1. Sobolev Institute of Mathematics, Novosibirsk, Russia

    Sergey V. Avgustinovich & Svetlana Puzynina

  2. Aix-Marseille Université, Marseille, France

    Anna E. Frid

  3. LIP, ENS de Lyon, Université de Lyon, Lyon, France

    Svetlana Puzynina

Authors
  1. Sergey V. Avgustinovich
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  2. Anna E. Frid
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  3. Svetlana Puzynina
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Corresponding author

Correspondence to Svetlana Puzynina .

Editor information

Editors and Affiliations

  1. University of Liverpool, Liverpool, United Kingdom

    Igor Potapov

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Avgustinovich, S.V., Frid, A.E., Puzynina, S. (2015). Ergodic Infinite Permutations of Minimal Complexity. In: Potapov, I. (eds) Developments in Language Theory. DLT 2015. Lecture Notes in Computer Science(), vol 9168. Springer, Cham. https://doi.org/10.1007/978-3-319-21500-6_5

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  • DOI: https://doi.org/10.1007/978-3-319-21500-6_5

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-21499-3

  • Online ISBN: 978-3-319-21500-6

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