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.

F-sensitivity and (F_1, F_2)-sensitivity between dynamical systems and their induced hyperspace dynamical systems

“…The hit-or-miss topology τ f (see [28,29]) (also known as H-topology [9], Fell topology [4,5,21], Choquet-Matheron topology [26], or weak Vietoris topology [32]) on F is generated by the subbase…”

“…For any two topological spaces X and Y, a continuous map f : X → Y is perfect if f is a closed map and all fibers f −1 (y)(y ∈ Y) are compact (see [7,28,29]). …”

“…If E\U is compact, then U is called co-compact. If one of the U and V is (or both U and V are) co-compact, then U and V are said to be a pair of co-compact subsets of E, denoted by (U, V) (see [28,29]). …”

“…It is well-known that (E, f) is topologically conjugated to the subsystem of (2 E , 2 f ) that consists of the singleton sets of E when E satisfies certain conditions and an appropriate hyperspace topology is selected, e.g., hit-or-miss topology (see [28][29][30]) or Vietoris topology (see [3,16,36]). Clearly, an invariant subset of the original system (E, f) becomes a fixed point of the hyperspace system (2 E , 2 f ).…”

“…Recently, Wang et al [28] first used the hit-or-miss topology to study some dynamical properties of the hyperspace dynamical system (2 E , 2 f ) induced by a HLCSC dynamical system (E, f). This hyperspace topology is metrizable when E is HLCSC and a concrete metric is available.…”