  Let $t=t_1t_2\cdots$ be an element of the full shift with shift map $\tau$ on
a finite set of characters $\mathcal{A}$ and let $ \Sigma=\text{ closure}
\{\tau^i(t):\;i\in\N\cup\{0\}\}$. Let $f_t=f_{t_1,\,\infty}=\cdots\circ
f_{t_2}\circ f_{t_1} $ be a non-autonomous system over a compact metric space $
X $ where $t_i\in \mathcal A $. The set $\F_t^+=\{f_{\tau^i(t)}:\; i\in\N\}$ is
called the shifted family of $f_t$. If $t$ is a transitive point of the full
shift on $\mathcal A$, then by introducing a natural topology,
$\overline{\F_t^+}$ is a classical IFS; otherwise,
$\overline{\F_t^+}=\{f_\sigma=f_{\sigma_1,\,\infty}:\; \sigma\in\Sigma\}$ is a
generalized IFS. We will show that if $ f_t$ has some various shadowing and
specification properties, then this is true for
$f_{\sigma}\in\overline{\F^+_t}$; however, this claim is not true for other
properties such as transitivity, mixing and exactness. Also, if $ \Sigma $ is
sofic and $x\in X$ is periodic point for some $f_\sigma\in\overline{\F^+_t}$,
then there is a periodic $\sigma'\in\Sigma$ such that $x$ is periodic for
$f_{\sigma'}\in\overline{\F^+_t}$.
