Shift limits of a non-autonomous system

Let $t=t_1{t}_2\cdots$ be an element of the full shift with shift map $\tau$ on a finite set of characters $\mathcal{A}$ and let $\mathrm{\Sigma }=\text{ closure}\{{\tau }^i(t):i\in \mathbb{N}\cup \{0\}\}$. Let $f_t=f_{t_1,\mathrm{\infty }}=\cdots \circ f_{t_2}\circ f_{t_1}$ be a non-autonomous system over a compact metric space $X$ where $t_i\in \mathcal{A}$. The set ${\text{F}}_t^{+}=\{f_{{\tau }^i(t)}:i\in \mathbb{N}\}$ 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, $\overline{{\text{F}}_t^{+}}$ is a classical IFS; otherwise, $\overline{{\text{F}}_t^{+}}=\{f_{\sigma }=f_{{\sigma }_1,\mathrm{\infty }}:\sigma \in \mathrm{\Sigma }\}$ is a generalized IFS. We will show that if $f_t$ has some various shadowing and specification properties, then this is true for $f_{\sigma }\in \overline{{\text{F}}_t^{+}}$; however, this claim is not true for other properties such as transitivity, mixing and exactness. Also, if $\mathrm{\Sigma }$ is sofic and $x\in X$ is periodic point for some $f_{\sigma }\in \overline{{\text{F}}_t^{+}}$, then there is a periodic ${\sigma }^{\mathrm{\prime }}\in \mathrm{\Sigma }$ such that $x$ is periodic for $f_{{\sigma }^{\mathrm{\prime }}}\in \overline{{\text{F}}_t^{+}}$.