Continuity of the maps f ↦ ∪ x ∈ I ω ( x, f ) and f ↦ { ω ( x, f ): x ∈ I }

T. H. Steele

International Journal of Mathematics and Mathematical Sciences, 2006, vol. 2006, 1-15

Abstract:

We study the behavior of two maps in an effort to betterunderstand the stability of ω -limit sets ω ( x , f ) as we perturb either x or f , or both. The first map is theset-valued function Λ taking f in C ( I , I ) to its collection of ω -limit points Λ ( f ) = ∪ x ∈ I ω ( x , f ) , and the second is the map Ω taking f in C ( I , I ) to its collection of ω -limit sets Ω ( f ) = { ω ( x , f ) : x ∈ I } . We characterize those functions f in C ( I , I ) at which each of our maps Λ and Ω is continuous, and then go on to show that both Λ and Ω are continuous on a residual subset of C ( I , I ) . We theninvestigate the relationship between the continuity of Λ and Ω at some function f in C ( I , I ) with the chaoticnature of that function.

Date: 2006
References: Add references at CitEc
Citations:

Downloads: (external link)
http://downloads.hindawi.com/journals/IJMMS/2006/082623.pdf (application/pdf)
http://downloads.hindawi.com/journals/IJMMS/2006/082623.xml (text/xml)

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:hin:jijmms:082623

DOI: 10.1155/IJMMS/2006/82623

Access Statistics for this article

More articles in International Journal of Mathematics and Mathematical Sciences from Hindawi
Bibliographic data for series maintained by Mohamed Abdelhakeem ().