Individual chaos implies collective chaos for weakly mixing discrete dynamical systems

Gongfu Liao, Xianfeng Ma and Lidong Wang

Chaos, Solitons & Fractals, 2007, vol. 32, issue 2, 604-608

Abstract: Let X be a metric space, (X,f) a discrete dynamical system, where f:X→X is a continuous function. Let f¯ denote the natural extension of f to the space of all non-empty compact subsets of X endowed with Hausdorff metric induced by d. In this paper we investigate some dynamical properties of f and f¯. It is proved that f is weakly mixing (mixing) if and only if f¯ is weakly mixing (mixing, respectively). From this, we deduce that weak-mixing of f implies transitivity of f¯, further, if f is mixing or weakly mixing, then chaoticity of f (individual chaos) implies chaoticity of f¯ (collective chaos) and if X is a closed interval then f¯ is chaotic (in the sense of Devaney) if and only if f is weakly mixing.

Date: 2007
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (4)

Downloads: (external link)
http://www.sciencedirect.com/science/article/pii/S0960077905010891
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:32:y:2007:i:2:p:604-608

DOI: 10.1016/j.chaos.2005.11.002

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. ().