Induced dynamics of non-autonomous dynamical 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-$\mathcal{F}$-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.