Relationship among several types of sensitivity in general semi-flows

  • Xinxing Wu
  • Xu ZhangEmail author
Research Article


In this paper, we show that there exists a monoid, on which neither the syndetic property nor the dual syndetic property holds, and there exists a strongly mixing semi-flow with this monoid action which does not have thick sensitivity, syndetic sensitivity, thickly syndetic sensitivity, or thickly periodical sensitivity. Meanwhile, we show that there exists a thickly sensitive cascade which is not multi-sensitive. The first result answers positively Question 2, and the first and the second results answer negatively Question 3 in (Miller in Appl Gen Topol 19:281–289, 2018).


Semi-flow Topological monoid Sensitivity 



This work was supported by the National Natural Science Foundation of China (Nos. 11601449 and 11701328), the Science and Technology Innovation Team of Education Department of Sichuan for Dynamical System and its Applications (No. 18TD0013), Youth Science and Technology Innovation Team of Southwest Petroleum University for Nonlinear Systems (No. 2017CXTD02), Shandong Provincial Natural Science Foundation, China (Grant ZR2017QA006), Young Scholars Program of Shandong University, Weihai (No. 2017WHWLJH09), and the Fundamental Research Funds for the Central Universities (No. 2019ZRJC005).


  1. 1.
    Banks, J., Brooks, J., Cairns, G., Davis, G., Stacey, P.: On Devaney’s definition of chaos. Am. Math. Mon. 99, 332–334 (1992)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Carvalho, B.: Hyperbolicity, transitivity and the two-sided limit shadowing property. Proc. Am. Math. Soc. 143, 657–666 (2015)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Ceccherini-Silberstein, T., Coornaert, M.: Sensitivity and Devaney’s chaos in uniform spaces. J. Dyn. Control Syst. 19, 349–357 (2013)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Iglesias, J., Portela, A.: Almost open semigroup actions. Semigroup Forum 98, 261–270 (2019)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Kontorovich, E., Megrelishvili, M.: A note on sensitivity of semigroup actions. Semigroup Forum 76, 133–141 (2008)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Li, R.: A note on shadowing with chain transitivity. Commun. Nonlinear Sci. Numer. Simulat. 17, 2815–2823 (2012)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Li, J., Oprocha, P., Wu, X.: Furstenberg families, sensitivity and the space of probability measures. Nonlinearity 30, 987–1005 (2017)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Miller, A.: Envelopes of syndetic subsemigroups of the acting topological semigroup in a semiflows. Topol. Appl. 158, 291–297 (2011)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Miller, A.: Syndetic sensitivity in semiflows. Topol. Appl. 196, 1–7 (2015)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Miller, A.: On various conditions that imply sensitivity of monoid actions. Real Anal. Exch. 42, 9–23 (2017)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Miller, A.: Indecomposability and Devaney’s chaoticity of semiflows with an arbitrary acting abelian topological semigroup. Semigroup Forum 96, 596–599 (2018)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Miller, A.: A note about various types of sensitivity in general semiflows. Appl. Gen. Topol. 19, 281–289 (2018)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Miller, A.: Weak mixing in general semiflows implies multi-sensitivity, but not thick sensitivity. J. Nonlinear Sci. Appl. 12, 120–123 (2019)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Money, C.: Chaos in semiflows. Ph.D. thesis, University of Louisville, Louisville, KY, USA (2015)Google Scholar
  15. 15.
    Moothathu, T.K.S.: Stronger forms of sensitivity for dynamical systems. Nonlinearity 20, 2115–2126 (2007)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Wang, H.: Thickly syndetic sensitivity of semigroup actions. Bull. Korean Math. Soc. 55, 1125–1135 (2018)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Wang, H., Long, X., Fu, H.: Sensitivity and chaos of semigroup actions. Semigroup Forum 84, 81–90 (2012)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Wu, X., Chen, G.: Sensitivity and transitivity of fuzzified dynamical systems. Inform. Sci. 396, 14–23 (2017)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Wu, X., Liang, S., Luo, Y., Xin, M., Zhang, X.: A remark on the limit shadowing property for iterated function systems. U.P.B. Sci. Bull. Ser. A 81, 107–114 (2019)MathSciNetGoogle Scholar
  20. 20.
    Wu, X., Luo, Y., Ma, X., Lu, T.: Rigidity and sensitivity on uniform spaces. Topol. Appl. 252, 145–157 (2019)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Wu, X., Ma, X., Chen, G., Lu, T.: A note on the sensitivity of semiflows. Topol. Appl. (2019, in press)Google Scholar
  22. 22.
    Wu, X., Oprocha, P., Chen, G.: On various definitions of shadowing with average error in tracing. Nonlinearity 29, 1942–1972 (2016)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Wu, X., Wang, J., Chen, G.: \({\mathscr {F}}\)-sensitivity and multi-sensitivity of hyperspatial dynamical systems. J. Math. Anal. Appl. 429, 16–26 (2015)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Wu, X., Zhang, X., Ma, X.: Various shadowing in linear dynamical systems. Int. J. Bifurcat. Chaos 29, 1950042 (2019)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Wu, X., Zhu, P.: Devaney chaos and Li–Yorke sensitivity for product systems. Studia Sci. Math. Hung. 49, 538–548 (2012)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Zhang, X., Wu, X.: Diffeomorphisms with the \({M}_0\)-shadowing property. Acta. Math. Sin. English Ser. 35, 1760–1770 (2019)Google Scholar
  27. 27.
    Zhang, X., Wu, X., Luo, Y.: A remark on limit shadowing for hyperbolic iterated function systems. U.P.B. Sci. Bull. Ser. A 41, 139–146 (2019)Google Scholar

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Authors and Affiliations

  1. 1.School of SciencesSouthwest Petroleum UniversityChengduPeople’s Republic of China
  2. 2.Institute for Artificial Intelligence, Southwest Petroleum UniversityChengduPeople’s Republic of China
  3. 3.Department of MathematicsShandong UniversityWeihaiPeople’s Republic of China

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