A framework of induced hyperspace dynamical systems equipped with the hit-or-miss topology

For any dynamical system (E,d,f), where E is Hausdorff locally compact second countable (HLCSC), let F (resp., 2E) denote the space of all closed subsets (resp., non-empty closed subsets) of E equipped with the hit-or-miss topology τf. Both F and 2E are again HLCSC (F actually compact), thus metrizable. Let ρ be such a metric (three metrics available). The main purpose is to determine the conditions on f that ensure the continuity of the induced hyperspace maps 2f:F→F and 2f:2E→2E defined by 2f(F)=f(F). With this setting, the induced hyperspace systems (F,ρ,2f) and (2E,ρ,2f) are compact and locally compact dynamical systems, respectively. Consequently, dynamical properties, particularly metric related dynamical properties, of the given system (E,d,f) can be explored through these hyperspace systems. In contrast, when the Vietoris topology τv is equipped on 2E, the space of the induced hyperspace topological dynamical system (2E,τv,2f) is not metrizable if E is not compact metrizable, e.g., E=Rn, implying that metric related dynamical concepts cannot be defined for (2E,τv,2f). Moreover, two examples are provided to illustrate the advantages of the hit-or-miss topology as compared to the Vietoris topology.