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

Yangeng Wang, Guo Wei, William H. Campbell and Steven Bourquin

Chaos, Solitons & Fractals, 2009, vol. 41, issue 4, 1708-1717

Abstract: For any dynamical system (E,d,f), where E is Hausdorff locally compact second countable (HLCSC), let F (resp., 2E) denote the space of all closed subsets (resp., non-empty closed subsets) of E equipped with the hit-or-miss topology τf. Both F and 2E are again HLCSC (F actually compact), thus metrizable. Let ρ 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 2f:F→F and 2f:2E→2E defined by 2f(F)=f(F). With this setting, the induced hyperspace systems (F,ρ,2f) and (2E,ρ,2f) 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 τv is equipped on 2E, the space of the induced hyperspace topological dynamical system (2E,τv,2f) is not metrizable if E is not compact metrizable, e.g., E=Rn, implying that metric related dynamical concepts cannot be defined for (2E,τv,2f). Moreover, two examples are provided to illustrate the advantages of the hit-or-miss topology as compared to the Vietoris topology.

Date: 2009
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (2)

Downloads: (external link)
http://www.sciencedirect.com/science/article/pii/S0960077908003184
Full text for ScienceDirect subscribers only

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:eee:chsofr:v:41:y:2009:i:4:p:1708-1717

DOI: 10.1016/j.chaos.2008.07.014

Access Statistics for this article

Chaos, Solitons & Fractals is currently edited by Stefano Boccaletti and Stelios Bekiros

More articles in Chaos, Solitons & Fractals from Elsevier
Bibliographic data for series maintained by Thayer, Thomas R. ().