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doi: 10.3934/dcdsb.2022053
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## Inheritance of ${\mathscr F}-$chaos and ${\mathscr F}-$sensitivities under an iteration for non-autonomous discrete systems

 Faculty of Mathematics, Kim Il Sung University, Pyongyang, Democratic People's Republic of Korea

*Corresponding author: JinHyon Kim

Received  July 2021 Revised  December 2021 Early access March 2022

This paper is concerned with chaos and sensitivity via Furstenberg families in a non-autonomous discrete system defined by a sequence of continuous self-maps on a compact metric space $(X, \; d)$. First we consider the properties $P(k)$ and $Q(k)$ introduced in the literature. We show that if ${\mathscr F}$ is a Furstenberg family with the property $P(k)$ then its dual family $k{\mathscr F}$ has the property $Q(k)$ and that if ${\mathscr F}$ is a filter with the property $Q(k)$ then its dual family $k{\mathscr F}$ has the property $P(k)$. Next, for a given positive integer $k$, it is shown that $({\mathscr F}_{1} , \; {\mathscr F}_{2} )-$chaos, generically ${\mathscr F}-$chaos, dense ${\mathscr F}-$chaos and ${\mathscr F}-$sensitivities are inherited under the $k$th iteration when $\{ f_{n} \} _{n = 1}^{\infty }$ is equicontinuous on $X$ and, ${\mathscr F}_{1} , \; {\mathscr F}_{2}$ and ${\mathscr F}$ are translation invariant Furstenberg families with the properties $P(k)$ and $Q(k)$. It is to weaken the condition in the literature that $\{ f_{n} \} _{n = 1}^{\infty }$ uniformly converges on a compact metric space $X$.

Citation: JinHyon Kim, HyonHui Ju, WiJong An. Inheritance of ${\mathscr F}-$chaos and ${\mathscr F}-$sensitivities under an iteration for non-autonomous discrete systems. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022053
##### References:
 [1] E. Akin, Recurrence in Topological Dynamics, Furstenberg Families and Ellis Actions, Plenum Press, New York, 1997. [2] W. Huang, D. Khilko, S. Kolyada and G. Zhang, Dynamical compactness and sensitivity, J. Differ. Equ., 260 (2016), 6800-6827.  doi: 10.1016/j.jde.2016.01.011. [3] J. Kim and H. Ju, About two definitions of ${\mathscr F}-$sensitivity, Topol. Appl., 269 (2020), 106927, 10pp. doi: 10.1016/j.topol.2019.106927. [4] S. Kolyada and L. Snoha, Topological entropy of nonautonomous dynamical systems, Random. Comput. Dyn., 4 (1996), 205-233. [5] R. Li, Y. Zhao, H. Wang, R. Jiang and H. Liang, ${\mathscr F}-$sensitivity and $({\mathscr F}_{1}, \; {\mathscr F}_{2})-$sensitivity between dynamical systems and their induced hyperspace dynamical systems, J. Nonlinear Sci. Appl., 10 (2017), 1640-1651.  doi: 10.22436/jnsa.010.04.28. [6] Z. Li, H. Wang and J. Xiong, Some remarks on $({\mathscr F}_{1}, \; {\mathscr F}_{2})-$scrambled sets, ACTA Math. Sin. (Chin. Ser.), 53 (2010), 727-732. [7] T. Lu, P. Zhu and X. Wu, Distributional chaos in non-autonomous discrete systems, Acta Math. Sci. (Chin. Ser.), 35 (2015), 558-566. [8] M. Salman and R. Das, Multi-sensitivity and other stronger forms of sensitivity in non-autonomous discrete systems, Chaos Solitons Fractals, 115 (2018), 341-348.  doi: 10.1016/j.chaos.2018.07.031. [9] H. Shao, G. Chen and Y. Shi, Topological conjugacy between induced non-autonomous set-valued systems and subshifts of finite type, Qual. Theory Dyn. Syst., 19 (2020), Paper No. 34, 26 pp. [10] H. Shao and Y. Shi, Some weak versions of distributional chaos in non-autonomous discrete systems, Commun. Nonlinear Sci. Numer. Simulat., 70 (2019), 318-325.  doi: 10.1016/j.cnsns.2018.11.005. [11] H. Shao, Y. Shi and H. Zhu, On distributional chaos in non-autonomous discrete systems, Chaos Solitons Fractals, 107 (2018), 234-243.  doi: 10.1016/j.chaos.2018.01.005. [12] F. Tan and J. Xiong, Chaos via Furstenberg family couple, Topol. Appl., 156 (2009), 525-532.  doi: 10.1016/j.topol.2008.08.006. [13] F. Tan and R. Zhang, On ${\mathscr F}-$sensitive pairs, Acta Math. Sci., 31 (2011), 1425-1435.  doi: 10.1016/S0252-9602(11)60328-7. [14] X. Tang, G. Chen and T. Lu, Some iterative properties of $({\mathscr F}_{1}, \; {\mathscr F}_{2})-$chaos in non-autonomous discrete systems, Entropy, 20 (2018), Paper No. 188, 9 pp. doi: 10.3390/e20030188. [15] C. Tian and G. Chen, Chaos of a sequence of maps in a metric space, Chaos Solitons Fractals, 28 (2006), 1067-1075.  doi: 10.1016/j.chaos.2005.08.127. [16] R. Vasisht and R. Das, Exploring ${\mathscr F}-$Sensitivity for Non-Autonomous Systems, preprint, arXiv: 1806.00693v1. [17] X. Wu, J. Wang and G. Chen, ${\mathscr F}-$sensitivity and multi-sensitivity of hyperspatial dynamical systems, J. Math. Anal. Appl., 429 (2015), 16-26.  doi: 10.1016/j.jmaa.2015.04.009. [18] X. Wu and P. Zhu, Chaos in a class of non-autonomous discrete systems, Appl. Math. Lett., 26 (2013), 431-436.  doi: 10.1016/j.aml.2012.11.003.

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##### References:
 [1] E. Akin, Recurrence in Topological Dynamics, Furstenberg Families and Ellis Actions, Plenum Press, New York, 1997. [2] W. Huang, D. Khilko, S. Kolyada and G. Zhang, Dynamical compactness and sensitivity, J. Differ. Equ., 260 (2016), 6800-6827.  doi: 10.1016/j.jde.2016.01.011. [3] J. Kim and H. Ju, About two definitions of ${\mathscr F}-$sensitivity, Topol. Appl., 269 (2020), 106927, 10pp. doi: 10.1016/j.topol.2019.106927. [4] S. Kolyada and L. Snoha, Topological entropy of nonautonomous dynamical systems, Random. Comput. Dyn., 4 (1996), 205-233. [5] R. Li, Y. Zhao, H. Wang, R. Jiang and H. Liang, ${\mathscr F}-$sensitivity and $({\mathscr F}_{1}, \; {\mathscr F}_{2})-$sensitivity between dynamical systems and their induced hyperspace dynamical systems, J. Nonlinear Sci. Appl., 10 (2017), 1640-1651.  doi: 10.22436/jnsa.010.04.28. [6] Z. Li, H. Wang and J. Xiong, Some remarks on $({\mathscr F}_{1}, \; {\mathscr F}_{2})-$scrambled sets, ACTA Math. Sin. (Chin. Ser.), 53 (2010), 727-732. [7] T. Lu, P. Zhu and X. Wu, Distributional chaos in non-autonomous discrete systems, Acta Math. Sci. (Chin. Ser.), 35 (2015), 558-566. [8] M. Salman and R. Das, Multi-sensitivity and other stronger forms of sensitivity in non-autonomous discrete systems, Chaos Solitons Fractals, 115 (2018), 341-348.  doi: 10.1016/j.chaos.2018.07.031. [9] H. Shao, G. Chen and Y. Shi, Topological conjugacy between induced non-autonomous set-valued systems and subshifts of finite type, Qual. Theory Dyn. Syst., 19 (2020), Paper No. 34, 26 pp. [10] H. Shao and Y. Shi, Some weak versions of distributional chaos in non-autonomous discrete systems, Commun. Nonlinear Sci. Numer. Simulat., 70 (2019), 318-325.  doi: 10.1016/j.cnsns.2018.11.005. [11] H. Shao, Y. Shi and H. Zhu, On distributional chaos in non-autonomous discrete systems, Chaos Solitons Fractals, 107 (2018), 234-243.  doi: 10.1016/j.chaos.2018.01.005. [12] F. Tan and J. Xiong, Chaos via Furstenberg family couple, Topol. Appl., 156 (2009), 525-532.  doi: 10.1016/j.topol.2008.08.006. [13] F. Tan and R. Zhang, On ${\mathscr F}-$sensitive pairs, Acta Math. Sci., 31 (2011), 1425-1435.  doi: 10.1016/S0252-9602(11)60328-7. [14] X. Tang, G. Chen and T. Lu, Some iterative properties of $({\mathscr F}_{1}, \; {\mathscr F}_{2})-$chaos in non-autonomous discrete systems, Entropy, 20 (2018), Paper No. 188, 9 pp. doi: 10.3390/e20030188. [15] C. Tian and G. Chen, Chaos of a sequence of maps in a metric space, Chaos Solitons Fractals, 28 (2006), 1067-1075.  doi: 10.1016/j.chaos.2005.08.127. [16] R. Vasisht and R. Das, Exploring ${\mathscr F}-$Sensitivity for Non-Autonomous Systems, preprint, arXiv: 1806.00693v1. [17] X. Wu, J. Wang and G. Chen, ${\mathscr F}-$sensitivity and multi-sensitivity of hyperspatial dynamical systems, J. Math. Anal. Appl., 429 (2015), 16-26.  doi: 10.1016/j.jmaa.2015.04.009. [18] X. Wu and P. Zhu, Chaos in a class of non-autonomous discrete systems, Appl. Math. Lett., 26 (2013), 431-436.  doi: 10.1016/j.aml.2012.11.003.
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