Let $(X,d)$ be a compact metric space. In $(X,f)$, $x\in X$ is called minimal point if $N(x,U)=\{n|f^{n}(x)\in U\}$ is syndetic for every open set $U$ of $x$ i.e. there is $k\in N$ such that $\forall n\in N$, $N(x, U)\cap [n, n+k]\neq \emptyset$. It is known that $x\in X$ is minimal point if and only if $\omega(x,f)$ is minimal set (if $A\subseteq \omega(x,f)$ is close and $f(A)\subseteq A$ then $A=\omega(x,f)$.)
Cosider non-autonomous discrete system $F=\{f_n\}_{n=0}^{\infty}$, Similarly $x\in X$ is called minimal point if $F_n=f_n\circ f_{n-1}\circ \ldots f_1 \circ f_0$ then $N(x,U)=\{n|F_n(x)\in U\}$ is syndetic for every open set $U$ of $x$
Question. In nonautonomous discrete system $F=\{f_n\}_{n=0}^{\infty}$, $x\in X$ is minimal point if and only if $\omega(x,F)$ is minimal set?
$y\in \omega(x, F)$, whenever $\forall \epsilon>0$, there is $n$, with $d(F_n(x), y)<\epsilon$. $A\subseteq X$ is $F$- invariant if $F_n(A)\subseteq A$, $\forall n$