Shift limits of a non-autonomous system

TL;DR

This paper investigates the limits of non-autonomous systems generated by sequences over a finite alphabet, focusing on properties like shadowing and specification, and how these properties extend to generalized iterated function systems.

Contribution

It introduces a framework for analyzing the shift limits of non-autonomous systems and explores which dynamical properties are preserved in the generalized IFS setting.

Findings

Shadowing and specification properties extend to the limit systems.

Transitivity, mixing, and exactness do not necessarily extend.

Periodic points in the limit systems correspond to periodic points in some shift sequence.

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