Continuity of the maps f↦∪x∈Iω(x,f) and f↦{ω(x,f):x∈I}
We study the behavior of two maps in an effort to better understand the stability of ω-limit sets ω(x,f) as we perturb either x or f, or both. The first map is the set-valued function Λ taking f in C(I,I) to its collection of ω-limit points Λ(f)=∪x∈Iω(x,f), and the second is the map Ω taking f in C(...
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| Format: | Article |
| Language: | English |
| Published: |
Hindawi Limited
2006-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/IJMMS/2006/82623 |
| Summary: | We study the behavior of two maps in an effort to better
understand the stability of ω-limit sets ω(x,f) as we perturb either x or f, or both. The first map is the
set-valued function Λ taking f in C(I,I) to its collection of ω-limit points Λ(f)=∪x∈Iω(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 then
investigate the relationship between the continuity of Λ and Ω at some function f in C(I,I) with the chaotic
nature of that function. |
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| ISSN: | 0161-1712 1687-0425 |