Chaos for Induced Group Actions on Hyperspaces
Anshid Aboobacker
TL;DR
The paper investigates how a group action $(X,\Gamma)$ on a compact metric space relates to its induced action on the hyperspace $(Sub_X,\widehat{\Gamma})$, extending classical transitivity, mixing, and chaos results from single-map dynamics to group actions. It defines the induced action using the Hausdorff metric $H$ and Vietoris topology on $Sub_X$, and develops transfer principles showing that transitivity, weak mixing, and mixing properties are mirrored between the base space and the hyperspace (with abelian $\Gamma$ for some results). It also proves that density of periodic points transfers to the hyperspace, and that, when $(X,\Gamma)$ is weakly mixing with dense periodic points, $(Sub_X,\widehat{\Gamma})$ becomes Devaney chaotic. Overall, the work provides a unified framework to deduce hyperspace dynamics from base dynamics, enriching the theory of induced group actions in topological dynamics.
Abstract
This paper investigates the correlation between dynamical properties of group actions on compact metric spaces and their induced actions on the corresponding hyperspaces. We extend classical results from discrete dynamical systems, particularly those concerning transitivity, mixing, and chaos, to the setting of group actions.