IDEAS home Printed from https://ideas.repec.org/a/eee/chsofr/v41y2009i4p1708-1717.html
   My bibliography  Save this article

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

Author

Listed:
  • Wang, Yangeng
  • Wei, Guo
  • Campbell, William H.
  • Bourquin, Steven

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.

Suggested Citation

  • Wang, Yangeng & Wei, Guo & Campbell, William H. & Bourquin, Steven, 2009. "A framework of induced hyperspace dynamical systems equipped with the hit-or-miss topology," Chaos, Solitons & Fractals, Elsevier, vol. 41(4), pages 1708-1717.
  • Handle: RePEc:eee:chsofr:v:41:y:2009:i:4:p:1708-1717
    DOI: 10.1016/j.chaos.2008.07.014
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0960077908003184
    Download Restriction: Full text for ScienceDirect subscribers only

    File URL: https://libkey.io/10.1016/j.chaos.2008.07.014?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Peris, Alfredo, 2005. "Set-valued discrete chaos," Chaos, Solitons & Fractals, Elsevier, vol. 26(1), pages 19-23.
    2. Román-Flores, Heriberto & Chalco-Cano, Y., 2005. "Robinson’s chaos in set-valued discrete systems," Chaos, Solitons & Fractals, Elsevier, vol. 25(1), pages 33-42.
    3. Wang, Yangeng & Wei, Guo, 2008. "Conditions ensuring that hyperspace dynamical systems contain subsystems topologically (semi-)conjugate to symbolic dynamical systems," Chaos, Solitons & Fractals, Elsevier, vol. 36(2), pages 283-289.
    4. Dietrich Stoyan, 1998. "Random Sets: Models and Statistics," International Statistical Review, International Statistical Institute, vol. 66(1), pages 1-27, April.
    5. Liu, Lei & Wang, Yangeng & Wei, Guo, 2009. "Topological entropy of continuous functions on topological spaces," Chaos, Solitons & Fractals, Elsevier, vol. 39(1), pages 417-427.
    6. Banks, John, 2005. "Chaos for induced hyperspace maps," Chaos, Solitons & Fractals, Elsevier, vol. 25(3), pages 681-685.
    7. Kwietniak, Dominik & Oprocha, Piotr, 2007. "Topological entropy and chaos for maps induced on hyperspaces," Chaos, Solitons & Fractals, Elsevier, vol. 33(1), pages 76-86.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Ju, Hyonhui & Kim, Cholsan & Choe, Yunmi & Chen, Minghao, 2017. "Conditions for topologically semi-conjugacy of the induced systems to the subshift of finite type," Chaos, Solitons & Fractals, Elsevier, vol. 98(C), pages 1-6.
    2. Andres, Jan & Ludvík, Pavel, 2022. "Topological entropy of multivalued maps in topological spaces and hyperspaces," Chaos, Solitons & Fractals, Elsevier, vol. 160(C).

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Andres, Jan, 2020. "Chaos for multivalued maps and induced hyperspace maps," Chaos, Solitons & Fractals, Elsevier, vol. 138(C).
    2. Liu, Heng & Liao, Gongfu & Hou, Bingzhe, 2009. "The set-valued mapping induced by a non-minimal transitive system is Li–Yorke chaotic," Chaos, Solitons & Fractals, Elsevier, vol. 40(2), pages 826-830.
    3. Hou, Bingzhe & Liao, Gongfu & Liu, Heng, 2008. "Sensitivity for set-valued maps induced by M-systems," Chaos, Solitons & Fractals, Elsevier, vol. 38(4), pages 1075-1080.
    4. Fu, Heman & Xing, Zhitao, 2012. "Mixing properties of set-valued maps on hyperspaces via Furstenberg families," Chaos, Solitons & Fractals, Elsevier, vol. 45(4), pages 439-443.
    5. Cánovas Peña, Jose S. & López, Gabriel Soler, 2006. "Topological entropy for induced hyperspace maps," Chaos, Solitons & Fractals, Elsevier, vol. 28(4), pages 979-982.
    6. Kwietniak, Dominik & Oprocha, Piotr, 2007. "Topological entropy and chaos for maps induced on hyperspaces," Chaos, Solitons & Fractals, Elsevier, vol. 33(1), pages 76-86.
    7. Li, Risong, 2012. "A note on stronger forms of sensitivity for dynamical systems," Chaos, Solitons & Fractals, Elsevier, vol. 45(6), pages 753-758.
    8. Román-Flores, H. & Chalco-Cano, Y., 2008. "Some chaotic properties of Zadeh’s extensions," Chaos, Solitons & Fractals, Elsevier, vol. 35(3), pages 452-459.
    9. Román-Flores, H. & Chalco-Cano, Y. & Silva, G.N. & Kupka, Jiří, 2011. "On turbulent, erratic and other dynamical properties of Zadeh’s extensions," Chaos, Solitons & Fractals, Elsevier, vol. 44(11), pages 990-994.
    10. Andres, Jan & Ludvík, Pavel, 2022. "Topological entropy of multivalued maps in topological spaces and hyperspaces," Chaos, Solitons & Fractals, Elsevier, vol. 160(C).
    11. Daniel Jardón & Iván Sánchez & Manuel Sanchis, 2020. "Transitivity in Fuzzy Hyperspaces," Mathematics, MDPI, vol. 8(11), pages 1-9, October.
    12. Félix Martínez-Giménez & Alfred Peris & Francisco Rodenas, 2021. "Chaos on Fuzzy Dynamical Systems," Mathematics, MDPI, vol. 9(20), pages 1-11, October.
    13. Sánchez, Iván & Sanchis, Manuel & Villanueva, Hugo, 2017. "Chaos in hyperspaces of nonautonomous discrete systems," Chaos, Solitons & Fractals, Elsevier, vol. 94(C), pages 68-74.
    14. Daghar, Aymen & Naghmouchi, Issam, 2022. "Entropy of induced maps of regular curves homeomorphisms," Chaos, Solitons & Fractals, Elsevier, vol. 157(C).
    15. Lopez-Diaz, Miguel & Ralescu, Dan A., 2006. "Tools for fuzzy random variables: Embeddings and measurabilities," Computational Statistics & Data Analysis, Elsevier, vol. 51(1), pages 109-114, November.
    16. Chalco-Cano, Y. & Román-Flores, H., 2008. "On new solutions of fuzzy differential equations," Chaos, Solitons & Fractals, Elsevier, vol. 38(1), pages 112-119.
    17. López-Díaz, Miguel, 2006. "An indexed multivariate dispersion ordering based on the Hausdorff distance," Journal of Multivariate Analysis, Elsevier, vol. 97(7), pages 1623-1637, August.
    18. Molaei, M.R., 2009. "Observational modeling of topological spaces," Chaos, Solitons & Fractals, Elsevier, vol. 42(1), pages 615-619.
    19. Camargo, Javier & Rincón, Michael & Uzcátegui, Carlos, 2019. "Equicontinuity of maps on dendrites," Chaos, Solitons & Fractals, Elsevier, vol. 126(C), pages 1-6.
    20. Liu, Lei & Wang, Yangeng & Wei, Guo, 2009. "Topological entropy of continuous functions on topological spaces," Chaos, Solitons & Fractals, Elsevier, vol. 39(1), pages 417-427.

    More about this item

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:chsofr:v:41:y:2009:i:4:p:1708-1717. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Thayer, Thomas R. (email available below). General contact details of provider: https://www.journals.elsevier.com/chaos-solitons-and-fractals .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.