Characterizing mixing, weak mixing and transitivity of induced hyperspace dynamical systems

Abstract

Hyperspace dynamical system ( 2 E , 2 f ) induced by a given dynamical system ( E , f ) has 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 ) 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.