<p style='text-indent:20px;'>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 <inline-formula><tex-math id="M3">\begin{document}$ (X, \; d) $\end{document}</tex-math></inline-formula>. First we consider the properties <inline-formula><tex-math id="M4">\begin{document}$ P(k) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M5">\begin{document}$ Q(k) $\end{document}</tex-math></inline-formula> introduced in the literature. We show that if <inline-formula><tex-math id="M6">\begin{document}$ {\mathscr F} $\end{document}</tex-math></inline-formula> is a Furstenberg family with the property <inline-formula><tex-math id="M7">\begin{document}$ P(k) $\end{document}</tex-math></inline-formula> then its dual family <inline-formula><tex-math id="M8">\begin{document}$ k{\mathscr F} $\end{document}</tex-math></inline-formula> has the property <inline-formula><tex-math id="M9">\begin{document}$ Q(k) $\end{document}</tex-math></inline-formula> and that if <inline-formula><tex-math id="M10">\begin{document}$ {\mathscr F} $\end{document}</tex-math></inline-formula> is a filter with the property <inline-formula><tex-math id="M11">\begin{document}$ Q(k) $\end{document}</tex-math></inline-formula> then its dual family <inline-formula><tex-math id="M12">\begin{document}$ k{\mathscr F} $\end{document}</tex-math></inline-formula> has the property <inline-formula><tex-math id="M13">\begin{document}$ P(k) $\end{document}</tex-math></inline-formula>. Next, for a given positive integer <inline-formula><tex-math id="M14">\begin{document}$ k $\end{document}</tex-math></inline-formula>, it is shown that <inline-formula><tex-math id="M15">\begin{document}$ ({\mathscr F}_{1} , \; {\mathscr F}_{2} )- $\end{document}</tex-math></inline-formula>chaos, generically <inline-formula><tex-math id="M16">\begin{document}$ {\mathscr F}- $\end{document}</tex-math></inline-formula>chaos, dense <inline-formula><tex-math id="M17">\begin{document}$ {\mathscr F}- $\end{document}</tex-math></inline-formula>chaos and <inline-formula><tex-math id="M18">\begin{document}$ {\mathscr F}- $\end{document}</tex-math></inline-formula>sensitivities are inherited under the <inline-formula><tex-math id="M19">\begin{document}$ k $\end{document}</tex-math></inline-formula>th iteration when <inline-formula><tex-math id="M20">\begin{document}$ \{ f_{n} \} _{n = 1}^{\infty } $\end{document}</tex-math></inline-formula> is equicontinuous on <inline-formula><tex-math id="M21">\begin{document}$ X $\end{document}</tex-math></inline-formula> and, <inline-formula><tex-math id="M22">\begin{document}$ {\mathscr F}_{1} , \; {\mathscr F}_{2} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M23">\begin{document}$ {\mathscr F} $\end{document}</tex-math></inline-formula> are translation invariant Furstenberg families with the properties <inline-formula><tex-math id="M24">\begin{document}$ P(k) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M25">\begin{document}$ Q(k) $\end{document}</tex-math></inline-formula>. It is to weaken the condition in the literature that <inline-formula><tex-math id="M26">\begin{document}$ \{ f_{n} \} _{n = 1}^{\infty } $\end{document}</tex-math></inline-formula> uniformly converges on a compact metric space <inline-formula><tex-math id="M27">\begin{document}$ X $\end{document}</tex-math></inline-formula>.</p>
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.