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Pointwise Dynamics Under Orbital Convergence

  • Abdul Gaffar Khan
  • Pramod Kumar Das
  • Tarun DasEmail author
Article
  • 19 Downloads

Abstract

We obtain sufficient conditions under which the limit of a sequence of functions exhibits a particular dynamical behaviour at a point like expansivity, shadowing, mixing, sensitivity and transitivity. We provide examples to show that the set of all expansive, positively expansive and sensitive points are neither open nor closed in general. We also observe that the set of all transitive and mixing points are closed but not open in general. We give examples to show that properties like expansivity, sensitivity, shadowing, transitivity and mixing at a point need not be preserved under uniform convergence and properties like topological stability and \(\alpha \)-persistence at a point need not be preserved under pointwise convergence.

Keywords

Expansivity Shadowing Transitivity Topological Stability Chaos 

Mathematics Subject Classification

Primary 54H20 Secondary 40A30 

Notes

Acknowledgements

The first author is supported by CSIR-Junior Research Fellowship (File No.-09/045(1558)/ 2018-EMR-I) of Government of India. The authors express sincere thanks to the reviewer for suggestions.

References

  1. Abu-Saris, R., Al-Hami, K.: Uniform convergence and chaotic behavior. Nonlinear Anal. Theory Methods Appl. 65(4), 933–937 (2006)MathSciNetCrossRefGoogle Scholar
  2. Abu-Saris, R.M., Martinez-Gimenez, F., Peris, A.: Erratum to “Uniform convergence and chaotic behavior” [Nonlinear Anal. TMA 65 (4)(2006) 933–937]. Nonlinear Anal. Theory Methods Appl. 68(5), 1406–1407 (2008)CrossRefGoogle Scholar
  3. Akin, E.: On chain continuity. Discret. Contin. Dyn. Syst. A. 2(1), 111–120 (1996)MathSciNetCrossRefGoogle Scholar
  4. Chen, L., Li, S.H.: Shadowing property for inverse limit spaces. Proc. Am. Math. Soc. 115(2), 573–580 (1992)MathSciNetCrossRefGoogle Scholar
  5. Das, P., Khan, A.G., Das, T.: Measure expansivity and specification for pointwise dynamics. Bull. Braz. Math. Soc., New Series. 50(4), 933–948 (2019)MathSciNetCrossRefGoogle Scholar
  6. Fedeli, A., Donne, A.L.: A note on the uniform limit of transitive dynamical systems. Bull. Belg. Math. Soc. Simon Stevin. 16(1), 59–66 (2009)MathSciNetzbMATHGoogle Scholar
  7. Koo, N., Lee, K., Morales, C.A.: Pointwise Topological Stability. Proc. Edinb. Math. Soc. 61(4), 1179–1191 (2018)MathSciNetCrossRefGoogle Scholar
  8. Kawaguchi, N.: Properties of shadowable points: Chaos and equicontinuity. Bull. Braz. Math. Soc. New Ser. 48(4), 599–622 (2017)MathSciNetCrossRefGoogle Scholar
  9. Kawaguchi, N.: Quantitative shadowable points. Dyn. Syst. 32(4), 504–518 (2017)MathSciNetCrossRefGoogle Scholar
  10. Li, R.: A note on uniform convergence and transitivity. Chaos Solitons Fractals. 45(6), 759–764 (2012)MathSciNetCrossRefGoogle Scholar
  11. Moothathu, T.K.S.: Implications of pseudo-orbit tracing property for continuous maps on compacta. Topol. Appl. 158(16), 2232–2239 (2011)MathSciNetCrossRefGoogle Scholar
  12. Mandelkern, M.: Metrization of the one-point compactification. Proc. Am. Math. Soc. 107(4), 1111–1115 (1989)MathSciNetCrossRefGoogle Scholar
  13. Morales, C.A.: Shadowable points. Dyn. Syst. 31(3), 347–56 (2016)MathSciNetCrossRefGoogle Scholar
  14. Reddy, W.L.: Pointwise expansion homeomorphisms. J. Lond. Math. Soc. 2(2), 232–236 (1970)MathSciNetCrossRefGoogle Scholar
  15. Sharma, P.: Uniform convergence and dynamical behavior of a discrete dynamical system. J. Appl. Math. Phys. 3(07), 766–770 (2015)CrossRefGoogle Scholar
  16. Utz, W.R.: Unstable homeomorphisms. Proc. Am. Math. Soc. 1(6), 769–774 (1950)MathSciNetCrossRefGoogle Scholar
  17. Walters, P.: On the pseudo orbit tracing property and its relationship to stability. In the structure of attractors in dynamical systems, pp. 231–244. Springer, Berlin, Heidelberg (1978)Google Scholar
  18. Yan, K., Zeng, F., Zhang, G.: Devaney’s chaos on uniform limit maps. Chaos Solitons Fractals. 44(7), 522–525 (2011)MathSciNetCrossRefGoogle Scholar
  19. Ye, X., Zhang, G.: Entropy points and applications. Trans. Am. Math. Soc. 359(12), 6167–6186 (2007)MathSciNetCrossRefGoogle Scholar

Copyright information

© Sociedade Brasileira de Matemática 2019

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of Mathematical SciencesUniversity of DelhiDelhiIndia
  2. 2.School of Mathematical SciencesNarsee Monjee Institute of Management StudiesMumbaiIndia

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