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

For any dynamical system ðE; d; f Þ, 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 sf . Both F and 2E are again HLCSC (F actually compact), thus metrizable. Let q 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 ðFÞ ¼ f ðFÞ. With this setting, the induced hyperspace systems ðF; q; 2f Þ and ð2E; q; 2f Þ are compact and locally compact dynamical systems, respectively. Consequently, dynamical properties, particularly metric related dynamical properties, of the given system ðE; d; f Þ can be explored through these hyperspace systems. In contrast, when the Vietoris topology sv is equipped on 2E, the space of the induced hyperspace topological dynamical system ð2E; sv; 2f Þ is not metrizable if E is not compact metrizable, e.g., E ¼ Rn, implying that metric related dynamical concepts cannot be defined for ð2E; sv; 2f Þ. Moreover, two examples are provided to illustrate the advantages of the hit-or-miss topology as compared to the Vietoris topology.