$k$-type Chaos for Induced Group Actions on Hyperspaces

This paper investigates the correlation between $k$-type dynamical properties of $\mathbb{Z}^d$-actions on compact metric spaces and their induced actions on the corresponding hyperspaces. We extend the classical results from discrete dynamical systems and general group actions to the specific setting of $k$-type dynamics. Specifically, we define and study $k$-type transitivity, $k$-type mixing, $k$-type weak mixing, and $k$-type Li-Yorke chaos for induced hyperspace actions, establishing that these properties transfer from the base system to the hyperspace under appropriate conditions.