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Stronger Forms of Sensitivity in the Dynamical System of Abelian Semigroup Actions

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Abstract

In this paper, we continue the study of abelian semigroup actions of several stronger versions of sensitivity, such as syndetically sensitive, thickly sensitive and thickly syndetically sensitive. We derive some sufficient conditions for a dynamical system to have these sensitivities. Also, we prove that the minimal and sensitive system is syndetically sensitive and non-minimal M-system is thickly syndetically sensitive. Some other significant properties of this new class are obtained.

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References

  1. Auslander J, Yorke JA. Interval map, factors of maps and chaos. Tô,hoku Math J 1980;32:177–88.

    MathSciNet  MATH  Google Scholar 

  2. Banks J, Brooks J, Cairns G, Davis G, Stacey P. On Devaney’s definition of chaos. Amer Math Mon 1992;99:332–4.

    Article  MathSciNet  Google Scholar 

  3. Ceccherini-Silberstein T, Coornaert M. Sensitivity and Devaney’s chaos in uniform spaces. J Dyn Control Syst 2013;19:349–57.

    Article  MathSciNet  Google Scholar 

  4. Devaney RL. An introduction to chaotic dynamical systems, 2nd ed. Redwood City: Addison-Wesley; 1989.

    MATH  Google Scholar 

  5. Fu H, Long X, Wang H. Sensitivity and chaos of semigroup actions. Semigroup Forum 2012;84:81–90.

    Article  MathSciNet  Google Scholar 

  6. Glasner E, Weiss B. Sensitive dependence on initial conditions. Nonlinearity. 1993;6:1067–75.

    Article  MathSciNet  Google Scholar 

  7. Good C, Macías S. What is topological about topological dynamics?. Discrete Contin Dyn Syst 2018;38:1007–31.

    Article  MathSciNet  Google Scholar 

  8. Haju T. An introduction to semigroup theory. Cambridge: Academic Press; 1976.

    Google Scholar 

  9. John B, Pan SJ. An introduction to topological semigroups. Gaz Mat. 1983;19-25.

  10. Kontorovich E, Megrelishvili M. A note on sensitivity of semigroup actions. Semigroup Forum 2008;76:133–41.

    Article  MathSciNet  Google Scholar 

  11. Li TY, Yorke JA. Period three implies chaos. Amer Math Mon 1975;82:985–92.

    Article  MathSciNet  Google Scholar 

  12. Moothathu TS. Diagonal points having dense orbit. Colloq Math 2010;120:127–38.

    Article  MathSciNet  Google Scholar 

  13. Niven I, Zuckerman HS, Montgomery HI. A introduction to the theory of numbers, 5th ed. New York: Wiley; 1991.

    MATH  Google Scholar 

  14. Schneider FM, Kerkhoff S, Behrisch M, Siegmund S. Chaotic actions of topologcal semigroups. Semigroup Forum 2013;87:590–98.

    Article  MathSciNet  Google Scholar 

  15. Willard S. General topology. New York: Addison-Wesley; 1970.

    MATH  Google Scholar 

  16. Wang H. Thickly syndetic sensitivity of semigroup actions. Bull Korean Math Soc 2018;55(4):1125–35.

    MathSciNet  MATH  Google Scholar 

  17. Wang H, Zhong Y. A Note on sensitivity in uniform spaces. J Dyn Control Syst 2018;24:625–33.

    Article  MathSciNet  Google Scholar 

  18. Wang T, Yin JD, Yan Q. Sufficient conditions for dynamical systems of semigroup actions to have some stronger forms of sensitivities. J Nonlinear Sci Appl 2016;9:989–97.

    Article  MathSciNet  Google Scholar 

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Acknowledgments

The authors would like to thank the referees for the valuable comments.

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Authors and Affiliations

  1. Department of Mathematics, Central University of Tamil Nadu, Thiruvarur, Tamil Nadu, 610 005, India

    V. Renukadevi

  2. Ayya Nadar Janaki Ammal College (Autonomous), Sivakasi, Tamil Nadu, 626 124, India

    S. Tamilselvi

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  1. V. Renukadevi
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  2. S. Tamilselvi
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Correspondence to V. Renukadevi.

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Renukadevi, V., Tamilselvi, S. Stronger Forms of Sensitivity in the Dynamical System of Abelian Semigroup Actions. J Dyn Control Syst 28, 151–162 (2022). https://doi.org/10.1007/s10883-020-09527-w

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  • DOI: https://doi.org/10.1007/s10883-020-09527-w

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