Inheritance of $ {\mathscr F}- $chaos and $ {\mathscr F}- $sensitivities under an iteration for non-autonomous discrete systems

Abstract

<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>