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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 ($E,f\right)$ 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., $E={ℝ}^{n}$. 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 ($E,f\right)$ 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 ($E,f\right)$, these three conditions on ($E,f\right)$ 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:
 54H20 Topological dynamics 54B20 Hyperspaces (general topology) 37D45 Strange attractors, chaotic dynamics