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 |
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Wiley
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 |
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| _version_ | 1832550255168585728 |
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| author | T. H. Steele |
| author_facet | T. H. Steele |
| author_sort | T. H. Steele |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-138d23e3bbd544b5be66ce50a894fe76 |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 2006-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-138d23e3bbd544b5be66ce50a894fe762025-02-03T06:07:12ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04252006-01-01200610.1155/IJMMS/2006/8262382623Continuity of the maps f↦∪x∈Iω(x,f) and f↦{ω(x,f):x∈I}T. H. Steele0Department of Mathematics, Weber State University, Ogden 84408-1702, UT, USAWe 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.http://dx.doi.org/10.1155/IJMMS/2006/82623 |
| spellingShingle | T. H. Steele Continuity of the maps f↦∪x∈Iω(x,f) and f↦{ω(x,f):x∈I} International Journal of Mathematics and Mathematical Sciences |
| title | Continuity of the maps f↦∪x∈Iω(x,f) and f↦{ω(x,f):x∈I} |
| title_full | Continuity of the maps f↦∪x∈Iω(x,f) and f↦{ω(x,f):x∈I} |
| title_fullStr | Continuity of the maps f↦∪x∈Iω(x,f) and f↦{ω(x,f):x∈I} |
| title_full_unstemmed | Continuity of the maps f↦∪x∈Iω(x,f) and f↦{ω(x,f):x∈I} |
| title_short | Continuity of the maps f↦∪x∈Iω(x,f) and f↦{ω(x,f):x∈I} |
| title_sort | continuity of the maps f↦∪x∈iω x f and f↦ ω x f x∈i |
| url | http://dx.doi.org/10.1155/IJMMS/2006/82623 |
| work_keys_str_mv | AT thsteele continuityofthemapsfxiōxfandfōxfxi |