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Characterizing mixing, weak mixing and transitivity of induced hyperspace dynamical systems. (English) Zbl 1131.54024

Authors’ summary: Hyperspace dynamical systems (2 E ,2 f ) induced by a given dynamical system (E,f) have been recently investigated regarding topological mixing, weak mixing and transitivity that characterize orbit structure. However, the Vietoris topology on 2 E employed in these studies is non-metrizable when E is not compact metrizable, e.g., E= n . Consequently, metric related dynamical concepts of (2 E ,2 f ) such as sensitivity on initial conditions and metric-based entropy could not even be defined. Moreover, a condition on (2 E ,2 f ) equivalent to the transitivity of (E,f) has not been established in the literature. On the other hand, Hausdorff locally compact second countable spaces (HLCSC) appear naturally in dynamics. When E is HLCSC, the hit-or-miss topology on 2 E 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 (E,f), these three conditions on (E,f) are, respectively, equivalent to mixing, weak mixing and transitivity on (2 E ,2 f ) (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.

MSC:
54H20Topological dynamics
54B20Hyperspaces (general topology)
37D45Strange attractors, chaotic dynamics