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.
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Abu-Saris, R., Al-Hami, K.: Uniform convergence and chaotic behavior. Nonlinear Anal. Theory Methods Appl. 65(4), 933–937 (2006)
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)
Akin, E.: On chain continuity. Discret. Contin. Dyn. Syst. A. 2(1), 111–120 (1996)
Chen, L., Li, S.H.: Shadowing property for inverse limit spaces. Proc. Am. Math. Soc. 115(2), 573–580 (1992)
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)
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)
Koo, N., Lee, K., Morales, C.A.: Pointwise Topological Stability. Proc. Edinb. Math. Soc. 61(4), 1179–1191 (2018)
Kawaguchi, N.: Properties of shadowable points: Chaos and equicontinuity. Bull. Braz. Math. Soc. New Ser. 48(4), 599–622 (2017)
Kawaguchi, N.: Quantitative shadowable points. Dyn. Syst. 32(4), 504–518 (2017)
Li, R.: A note on uniform convergence and transitivity. Chaos Solitons Fractals. 45(6), 759–764 (2012)
Moothathu, T.K.S.: Implications of pseudo-orbit tracing property for continuous maps on compacta. Topol. Appl. 158(16), 2232–2239 (2011)
Mandelkern, M.: Metrization of the one-point compactification. Proc. Am. Math. Soc. 107(4), 1111–1115 (1989)
Morales, C.A.: Shadowable points. Dyn. Syst. 31(3), 347–56 (2016)
Reddy, W.L.: Pointwise expansion homeomorphisms. J. Lond. Math. Soc. 2(2), 232–236 (1970)
Sharma, P.: Uniform convergence and dynamical behavior of a discrete dynamical system. J. Appl. Math. Phys. 3(07), 766–770 (2015)
Utz, W.R.: Unstable homeomorphisms. Proc. Am. Math. Soc. 1(6), 769–774 (1950)
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)
Yan, K., Zeng, F., Zhang, G.: Devaney’s chaos on uniform limit maps. Chaos Solitons Fractals. 44(7), 522–525 (2011)
Ye, X., Zhang, G.: Entropy points and applications. Trans. Am. Math. Soc. 359(12), 6167–6186 (2007)
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.
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Khan, A.G., Das, P.K. & Das, T. Pointwise Dynamics Under Orbital Convergence. Bull Braz Math Soc, New Series 51, 1001–1016 (2020). https://doi.org/10.1007/s00574-019-00178-5
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DOI: https://doi.org/10.1007/s00574-019-00178-5