The expansivity and sensitivity of the set-valued discrete dynamical systems

Let $(X,d)$ be a metric space and $\mathcal{H}(X)$ represent all non-empty, compact subsets of $X$. The expansivity of the multivalued map sequence ${\overline{f}}_{1,\mathrm{\infty }}:\mathcal{H}(X)\to \mathcal{H}(X)$, including expansivity, positive ${\mathrm{\aleph }}_0$-expansivity, were investigated. Also, stronger forms of sensitivities, such as multi-sensitivity and syndetical sensitivity, were explored. This research demonstrated that some chaotic properties can be mutually derived between $(f_{1,\mathrm{\infty }},X)$ and $({\overline{f}}_{1,\mathrm{\infty }},\mathcal{H}(X))$, showing fundamental similarities between these systems. Conversely, the inability to derive other properties underlined essential differences between them. These insights are crucial for simplifying theoretical models and enhancing independent research. Lastly, the relationship between expansivity and sensitivity was discussed and the concept of topological conjugacy to the system $({\overline{f}}_{1,\mathrm{\infty }},\mathcal{H}(X))$ was extended.