Skip to main content
Log in

Special Flow, Weak Mixing and Complexity

Dedicated to celebrate the Sixtieth anniversary of USTC

  • Published:
Communications in Mathematics and Statistics Aims and scope Submit manuscript

Abstract

We focus on the complexity of a special flow built over an irrational rotation of the unit circle and under a roof function on the unit circle. We construct a weak mixing minimal special flow with bounded topological complexity. We also prove that if the roof function is \(C^\infty \), then the special flow has sub-polynomial topological complexity and the time one map meets the condition of Sarnak’s conjecture.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Blanchard, F., Host, B., Maass, A.: Topological complexity. Ergod. Theory Dyn. Syst. 20(3), 641–662 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bowen, R., Walters, P.: Expansive one-parameter flows. J. Differ. Equ. 12, 180–193 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  3. Davenport, H.: On some infinite series involving arithmetical functions II. Quat. J. Math. 8, 313–320 (1937)

    Article  MATH  Google Scholar 

  4. Fayad, B.: Polynomial decay of correlations for a class of smooth flows on the two torus. Bull. Soc. Math. France 129(4), 487–503 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  5. Fayad, B.: Weak mixing for reparameterized linear flows on the torus. Ergod. Theory Dyn. Syst. 22(1), 187–201 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  6. Fayad, B., Katok, A., Windsor, A.: Mixed spectrum reparameterizations of linear flows on \({\mathbb{T}}^2\). Dedicated to the memory of I. G. Petrovskii on the occasion of his 100th anniversary. Mosc. Math. J. 1(4), 521–537, 644(2001)

  7. Fayad, B., Windsor, A.: A dichotomy between discrete and continuous spectrum for a class of special flows over rotations. J. Mod. Dyn. 1(1), 107–122 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  8. Ferenczi, S.: Measure-theoretic complexity of ergodic systems. Israel J. Math. 100(1), 189–207 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  9. Fraczek, K., Lemańczyk, M.: A class of special flows over irrational rotations which is disjoint from mixing flows. Ergod. Theory Dyn. Syst. 24(4), 1083–1095 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  10. Fraczek, K., Lemańczyk, M.: On mild mixing of special flows over irrational rotations under piecewise smooth functions. Ergod. Theory Dyn. Syst. 26(3), 719–738 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  11. Fraczek, K., Lemańczyk, M., Lesigne, E.: Mild mixing property for special flows under piecewise constant functions. Dis. Contin. Dyn. Syst. 19(4), 691–710 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  12. Furstenberg, H.: Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation. Math. Syst. Theory 1, 1–49 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  13. Furstenberg, H., Keynes, H., Shapiro, L.: Prime flows in topological dynamics. Israel J. Math. 14, 26–38 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  14. Gottschalk, W., Hedlund, G.: Topological Dynamics. Amer. Math. Soc. Colloq., vol. 36, Providence, R.I.(1955)

  15. Huang, W., Wang, Z., Ye, X.: Measure complexity and Möbius disjointness, preprint. arXiv:1707.06345

  16. Huang, W., Li, J., Thouvent, J. P., Xu, L., Ye, X.: Bounded complexity, mean equicontinuity and discrete spectrum, preprint. arXiv:1806.02980

  17. Katok, A.: Lyapunov exponents, entropy and the periodic orbits for diffeomorphisms. Inst. Hautes tudes Sci. Publ. Math. 51, 137–173 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  18. Khanin, K.M.: Mixing for area-preserving flows on the two-dimensional torus. J. Mod. Phys. B 10(18–19), 2167–2188 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  19. Khinchin, A. Y.: Continued Fractions, Translated from the third (1961) Russian edition, Dover Publications, Inc., Mineola, NY. With a preface by B. V. Gnedenko; Reprint of the 1964 translation (1997)

  20. Kochergin, A.V.: On the absence of mixing in special flows over the rotation of a circle and in flows on a two-dimensional torus. Dokl. Akad. Nauk SSSR 205, 949–952 (1972)

    MathSciNet  Google Scholar 

  21. Kochergin ,A.V.: A mixing special flow over a rotation of the circle with an almost Lipschitz function. (Russian) Mat. Sb. 193(3), 51–78(2002); translation in Sb. Math. 193(3–4), 359–385(2002)

  22. Kolmogorov, A.N.: On dynamical systems with an integral invariant on the torus. Doklady Akad. Nauk SSSR (N.S.)) 93, 763–766 (1953)

    MathSciNet  MATH  Google Scholar 

  23. Lemańczyk, M.: Sur l’absence de mlange pour des flots spciaux au dessus d’une rotation irrationelle. Colloq. Math. 84(85), 29–41 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  24. Liu, J., Sarnak, P.: The Möbius function and distal flows. Duke Math. J. 164(7), 1353–1399 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  25. Qiao, Y.: Topological complexity, minimality and systems of order two on torus. Sci. China Math. 59, 503–514 (2016)

    Article  MathSciNet  MATH  Google Scholar 

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

  27. Sinai, Y.G., Khanin, K.M.: Mixing of some classes of special flows over rotations of the circle. (Russian) Funktsional. Anal. i Prilozhen. 26(3), 1–21(1992); translation in Funct. Anal. Appl. 26(3), 155–169(1992)

  28. Šklover, M.D.: Classical dynamical systems on the torus with continuous spectrum. Izv. Vysš. Učebn. Zaved. Matematika 65(10), 113–124 (1967)

    MathSciNet  Google Scholar 

  29. Vershik, A.: Scaling entropy and automorphisms with pure point spectrum. St. Petersburg Math. J. 23(1), 75–91 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  30. 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 

  31. Von Neumann, J.: Zur Operatorenmethode in der klassischen Mechanik. Ann. Math. 33(3), 587–642 (1932)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Wen Huang.

Additional information

Huang is partially supported by NNSF of China (11431012,11731003) and Xu is partially supported by NNSF of China (11801538, 11871188).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Huang, W., Xu, L. Special Flow, Weak Mixing and Complexity. Commun. Math. Stat. 7, 85–122 (2019). https://doi.org/10.1007/s40304-018-0166-5

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40304-018-0166-5

Keywords

Mathematics Subject Classification

Navigation