Give Hit - Or - Miss Topology Hyperspace Chaotic Nature Of The Power System

Let E be a topological space, and F (resp.,2E) be the set of all closed subsets (resp., all non-empty closed subsets) of E equipped with the hit-or-miss topology. Assuming that (E, d) is a locally compact second countable metric space, d is a compact-type metric of E, f:E�'E is a perfect mapping, it is proved that (1) if (E,d,f) has Li-Yorke chaos, w-chaos, or distributive chaos (type1and type2), then so does the induced hyperspace dynamical system (2E,ρ,2f)(resp.,(F,ρ,2f), if existing), but the converse assertion does not hold (relevant counter examples are provided);(2) if (E,d,f) is exact Devaney-chaotic, so is (2E, ρ,2f), but the converse assertion does not hold;(3) if (E, d, f) is co-compact Devaney-chaotic, then (2E,ρ,2f)(resp.,(F, ρ,2f), if existing) is Devaney-chaotic; under certain conditions, if (E, d, f) is distributive chaos type3, so is (2E,ρ,2f)(resp.,(F, ρ,2f), if existing), but the converse assertion does not hold;(4)(2E,ρ,2f)(resp.,(F, ρ,2f), if existing) and (E,d,f) do not imply each other regarding Devaney chaos. Moreover, a condition ensuring that hyper-space topological dynamical systems contain subsystems topologically conjugate to the two-sided symbolic dynamical system (∑(2),σ) is investigated.