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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.

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