Topological sequence entropy of nonautonomous dynamical systems

Abstract

Let $f_{0,\infty}=\{f_n\}_{n=0}^{\infty}$ be a sequence of continuous self-maps on a compact metric space $X$. Firstly, we obtain the relations between topological sequence entropy of a nonautonomous dynamical system $(X,f_{0,\infty})$ and that of its finite-to-one extension. We then prove that the topological sequence entropy of $(X,f_{0,\infty})$ is no less than its corresponding measure sequence entropy if $X$ has finite covering dimension. Secondly, we study the supremum topological sequence entropy of $(X,f_{0,\infty})$, and confirm that it equals to that of its $n$-th compositions system if $f_{0,\infty}$ is equi-continuous; and we prove the supremum topological sequence entropy of $(X,f_{i,\infty})$ is no larger than that of $(X,f_{j,\infty})$ if $i\leq j$, and they are equal if $f_{0,\infty}$ is equi-continuous and surjective. Thirdly, we investigate the topological sequence entropy relations between $(X,f_{0,\infty})$ and $(\mathcal{M}(X),\hat{f}_{0,\infty})$ induced on the space $\mathcal{M}(X)$ of all Borel probability measures, and obtain that given any sequence, the topological sequence entropy of $(X,f_{0,\infty})$ is zero if and only if that of $(\mathcal{M}(X),\hat{f}_{0,\infty})$ is zero; the topological sequence entropy of $(X,f_{0,\infty})$ is positive if and only if that of $(\mathcal{M}(X),\hat{f}_{0,\infty})$ is infinite. By applying this result, we obtain some big differences between entropies of nonautonomous dynamical systems and that of autonomous dynamical systems. Finally, we study whether multi-sensitivity of $(X,f_{0,\infty})$ imply positive or infinite topological sequence entropy.