A framework of induced hyperspace dynamical systems equipped with the hit-or-miss topology

作者：Wang Yangeng; Wei Guo^*; Campbell William H; Bourquin Steven

来源：Chaos Solitons & Fractals, 2009, 41(4): 1708-1717.

DOI：10.1016/j.chaos.2008.07.014

For any dynamical system (E, d, f), where E is Hausdorff locally compact second countable (HLCSC), let F (resp., 2(E)) denote the space of all closed subsets (resp., non-empty closed subsets) of E equipped with the hit-or-miss topology tau(f). Both. F and 2(E) are again HLCSC (. F actually compact), thus metrizable. Let rho be such a metric (three metrics; available). The main purpose is to determine the conditions on f that ensure the continuity of the induced hyperspace maps 2(f) : F -> F and 2(f) : 2(E) -> 2(E) defined by 2(f)(F) = f(F). With this setting, the induced hyperspace systems (F, rho, 2(f)) and (2(E), rho, 2(f)) are compact and locally compact dynamical systems, respectively. Consequently, dynamical properties, particularly metric related dynamical properties, of the given system (E, d,f) can be explored through these hyperspace systems. In contrast, when the Vietoris topology tau(v) is equipped on 2(E), the space of the induced hyperspace topological dynamical system (2(E), tau(v) 2(f)) is not metrizable if E is not compact metrizable, e.g., E = R(n), implying that metric related dynamical concepts cannot be defined for (2(E), tau(v) ,2(f)). Moreover, two examples are provided to illustrate the advantages of the hit-or-miss topology as compared to the Vietoris topology.

• 出版日期2009-8-30
• 单位西北大学