Shift limits of a non-autonomous system

Abstract

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 $f_{\sigma'}\in\overline{\F^+_t}$.