Induced dynamics of non-autonomous dynamical systems

TL;DR

This paper investigates the entropy and dynamical properties of non-autonomous systems and their induced set-valued and fuzzified systems, establishing conditions under which entropy is preserved or becomes infinite, and exploring various mixing and shadowing properties.

Contribution

It provides new insights into how entropy and mixing properties transfer between non-autonomous systems and their induced set-valued and fuzzified counterparts, including specific conditions and counterexamples.

Findings

Positive entropy in the original system implies infinite entropy in induced systems.

Zero entropy in the original system implies zero entropy in certain induced subsystems.

Transitive interval maps induce infinite entropy in their set-valued and fuzzified systems.

Abstract

Let $f_{0,\infty}=\{f_n\}_{n=0}^{\infty}$ be a sequence of continuous self-maps on a compact metric space $X$. The non-autonomous dynamical system $(X,f_{0,\infty})$ induces the set-valued system $(\mathcal{K}(X), \bar{f}_{0,\infty})$ and the fuzzified system $(\mathcal{F}(X),\tilde{f}_{0,\infty})$. We prove that under some natural conditions, positive topological entropy of $(X,f_{0,\infty})$ implies infinite entropy of $(\mathcal{K}(X),\bar{f}_{0,\infty})$ and $(\mathcal{F}(X),\tilde{f}_{0,\infty})$, respectively; and zero entropy of $(S^1,f_{0,\infty})$ implies zero entropy of some invariant subsystems of $(\mathcal{K}(S^1),\bar{f}_{0,\infty})$ and $(\mathcal{F}(S^1),\tilde{f}_{0,\infty})$, respectively. We confirm that $(\mathcal{K}(I), \bar{f})$ and $(\mathcal{F}(I), \tilde{f})$ have infinite entropy for any transitive interval map $f$. In contrast, we construct a transitive non-autonomous system $(I, f_{0,\infty})$ such that both $(\mathcal{K}(I), \bar{f}_{0,\infty})$ and $(\mathcal{F}(I), \tilde{f}_{0,\infty})$ have zero entropy. We obtain that $(\mathcal{K}(X),\bar{f}_{0,\infty})$ is chain weakly mixing of all orders if and only if $(\mathcal{F}^1(X),\tilde{f}_{0,\infty})$ is so, and chain mixing (resp. $h$-shadowing and multi-$\mathscr{F}$-sensitivity) among $(X,f_{0,\infty})$, $(\mathcal{K}(X),\bar{f}_{0,\infty})$ and $(\mathcal{F}^1(X),\tilde{f}_{0,\infty})$ are equivalent, where $(\mathcal{F}^1(X),\tilde{f}_{0,\infty})$ is the induced normal fuzzification.