Shift limits of a non-autonomous system

Abstract

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 $\Sigma =\text{ closure}\{{\tau }^i(t):i\in \mathbb{N}\cup \{0\}\}$. 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 $\F_t^+=\{f_{τ^i(t)}:\; i\in\N\}$ 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, $\overline{\F_t^+}=\{f_σ=f_{σ_1,\,\infty}:\; σ\inΣ\}$ is a generalized IFS. We will show that if $f_t$ has some various shadowing and specification properties, then this is true for $f_σ\in\overline{\F^+_t}$; 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 $f_σ\in\overline{\F^+_t}$, then there is a periodic ${\sigma }^{\prime }\in \Sigma$ such that $x$ is periodic for $f_{σ'}\in\overline{\F^+_t}$.