Authors’ summary: Hyperspace dynamical systems () induced by a given dynamical system ( have been recently investigated regarding topological mixing, weak mixing and transitivity that characterize orbit structure. However, the Vietoris topology on employed in these studies is non-metrizable when is not compact metrizable, e.g., . Consequently, metric related dynamical concepts of () such as sensitivity on initial conditions and metric-based entropy could not even be defined. Moreover, a condition on () equivalent to the transitivity of ( has not been established in the literature. On the other hand, Hausdorff locally compact second countable spaces (HLCSC) appear naturally in dynamics. When is HLCSC, the hit-or-miss topology on is again HLCSC, thus metrizable.
In this paper, the concepts of co-compact mixing, co-compact weak mixing and co-compact transitivity are introduced for dynamical systems. For any HLCSC system (, these three conditions on ( are, respectively, equivalent to mixing, weak mixing and transitivity on () (hit-or-miss topology equipped). Other noticeable properties of co-compact mixing, co-compact weak mixing and co-compact transitivity such as invariants for topological conjugacy, as well as their relations to mixing, weak mixing and transitivity, are also explored.