Measure Complexity and Rigid Systems

Abstract

In this paper we introduce two metrics: the max metric dn,q and the mean metric \({\bar d_{n,q}}\). We give an equivalent characterization of rigid measure preserving systems by the two metrics. It turns out that an invariant measure μ on a topological dynamical system (X, T) has bounded complexity with respect to dn,q if and only if μ has bounded complexity with respect to \({\bar d_{n,q}}\) if and only if (X, \({\cal B}x\), μ, T) is rigid. We also obtain computation formulas of the measure-theoretic entropy of an ergodic measure preserving system (resp. the topological entropy of a topological dynamical system) by the two metrics dn,q and \({\bar d_{n,q}}\).

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References

  1. [1]

    Adler, R. L., Konheim, A. G., McAndrew, M. H.: Topological entropy. Trans. Amer. Math. Soc., 114, 309–319 (1965)

    MathSciNet  Article  Google Scholar 

  2. [2]

    Blachard, F., Host, B., Maass, A.: Topological complexity. Ergodic Theory Dynam. Systems, 20(3), 641–662 (2000)

    MathSciNet  Article  Google Scholar 

  3. [3]

    Bowen, R.: Entropy for group endomorphisms and homogeneous spaces. Trans. Amer. Math. Soc., 153, 401–414 (1971)

    MathSciNet  Article  Google Scholar 

  4. [4]

    Chaika, J., Eskin, A.: Möbius disjointness for intervl exchange transformations on three intervals. J. Mod. Dyn., 14, 55–86 (2019)

    MathSciNet  Article  Google Scholar 

  5. [5]

    Dinaburg, E. I.: A connection between various entropy characterizations of dynamical systems. (Russian) Izv. Akad. Nauk SSSR Ser. Mat., 35, 324–366 (1971)

    MathSciNet  Google Scholar 

  6. [6]

    Ferenczi, S.: Measure-theoretic complexity of ergodic systems. Israel J. Math., 100, 189–207 (1997)

    MathSciNet  Article  Google Scholar 

  7. [7]

    Huang, W., Li, J., Thouvenot, J., et al.: Bounded complexity, mean equicontinuity and discrete spectrum. Ergodic Theory Dynam. Systems, 41(2), 494–533 (2021)

    MathSciNet  Article  Google Scholar 

  8. [8]

    Huang, W., Wang, Z., Ye, X.: Measure complexity and Möbius disjointness. Adv. Math., 347, 827–858 (2019)

    MathSciNet  Article  Google Scholar 

  9. [9]

    Huang, W., Xu, L.: Special flow, weak mixing and complexity. Commun. Math. Stat., 7(1), 85–122 (2019)

    MathSciNet  Article  Google Scholar 

  10. [10]

    Huang, W., Ye, X.: Combinatorial lemmas and application to dynamics. Adv. Math., 220(6), 1689–1716 (2009)

    MathSciNet  Article  Google Scholar 

  11. [11]

    Huang, W., Ye, X., Zhang, G.: Lowing topological entropy over subsets. Ergodic Theory Dynam. Systems, 30(1), 181–209 (2010)

    MathSciNet  Article  Google Scholar 

  12. [12]

    Kamae, T., Zamboni, L.: Sequence entropy and maximal pattern complexity of infinite words. Ergodic Theory Dynam. Systems, 22(4), 1191–1199 (2002)

    MathSciNet  MATH  Google Scholar 

  13. [13]

    Kamae T., Zamboni, L.: Maximal pattern complexity for discrete systems. Ergodic Theory Dynam. Systems, 22(4), 1201–1214 (2002)

    MathSciNet  MATH  Google Scholar 

  14. [14]

    Kanigowski, A., Lemańczyk, M., Radziwiłł, M.: Rigidity in dynamics and möbius disjointness. arXiv:1905.13256v2

  15. [15]

    Katok, A.: Lyapunov exponents, entropy and periodic orbits for diffeomorphisms. Inst. Hautes Études Sci. Publ. Math. No., 51, 137–173 (1980)

    MathSciNet  Article  Google Scholar 

  16. [16]

    Kolmogorov, A. N.: A new metric invariant of transient dynamical systems and automorphisms in Lebesgue spaces. (Russian) Dokl. Akad. Nauk SSSR (N.S.), 119, 861–864 (1958)

    MathSciNet  MATH  Google Scholar 

  17. [17]

    Kolmogorov, A. N.: Entropy per unit time as a metric invariant of automorphisms. (Russian) Dokl. Akad. Nauk SSSR, 124, 754–755 (1959)

    MathSciNet  MATH  Google Scholar 

  18. [18]

    Lindenstrauss, E., Tsukamoto, M.: From rate distortion theory to metric mean dimension: variational principle. IEEE Trans. Inform. Theory, 64(5), 3590–3609 (2018)

    MathSciNet  Article  Google Scholar 

  19. [19]

    Li, J., Ye, X., Yu, T.: Mean equicontinuity, complexity and applications. Discrete Contin. Dyn. Syst., 41(1), 359–393 (2021)

    MathSciNet  Article  Google Scholar 

  20. [20]

    Sarnak, P.: Three lectures on the Möbius function, randomness and dynamics. Lecture Notes, IAS (2009)

  21. [21]

    Sinai, Ja.: On the concept of entropy for a dynamic system. (Russian) Dokl. Akad. Nauk SSSR, 124, 768–771 (1959)

    MathSciNet  MATH  Google Scholar 

  22. [22]

    Velozo, A., Velozo, R.: Rate distortion theory, metric mean dimension and measure theoretic entropy. arXiv:1707.05762

  23. [23]

    Vershik, A., Zatitskiy, P., Petrov, F.: Geometry and dynamics of admissible metrics in measure spaces. Cent. Eur. J. Math., 11(3), 379–400 (2013)

    MathSciNet  MATH  Google Scholar 

  24. [24]

    Walters, P.: An Introduction to Ergodic Theory, Graduate Texts in Mathematics, Vol. 79, ix+250 pp., Springer-Verlag, New York-Berlin, 1982

    Book  Google Scholar 

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Acknowledgements

We would like to thank Leiye Xu for valuable remarks and discussions. We also thank the anonymous referees for their careful reading and useful suggestions that greatly improved the manuscript.

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Correspondence to Run Ju Wei.

Additional information

Supported by NNSF of China (Grant Nos. 11971455, 11801538, 11801193, 11871188, 11731003 and 12090012); Tao Yu is supported by STU Scientific Research Foundation for Talents (Grant No. NTF19047)

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Huang, W., Wei, R.J., Yu, T. et al. Measure Complexity and Rigid Systems. Acta. Math. Sin.-English Ser. (2021). https://doi.org/10.1007/s10114-021-0179-y

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Keywords

  • Rigid system
  • entropy
  • bounded complexity

MR(2010) Subject Classification

  • 37B05
  • 54H20