On the Continuity of the Hutchinson Operator
We investigate when the Hutchinson operator associated with an iterated function system is continuous. The continuity with respect to both the Hausdorff metric and Vietoris topology is carefully considered. An example showing that the Hutchinson operator on the hyperspace of nonempty closed bounded...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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MDPI AG
2015-10-01
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| Series: | Symmetry |
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| Online Access: | http://www.mdpi.com/2073-8994/7/4/1831 |
| _version_ | 1817994957455097856 |
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| author | Michael F. Barnsley Krzysztof Leśniak |
| author_facet | Michael F. Barnsley Krzysztof Leśniak |
| author_sort | Michael F. Barnsley |
| collection | DOAJ |
| description | We investigate when the Hutchinson operator associated with an iterated function system is continuous. The continuity with respect to both the Hausdorff metric and Vietoris topology is carefully considered. An example showing that the Hutchinson operator on the hyperspace of nonempty closed bounded sets need not be Hausdorff continuous is given. Infinite systems are also discussed. The work clarifies and generalizes several partial results scattered across the literature. |
| first_indexed | 2024-04-14T01:58:52Z |
| format | Article |
| id | doaj.art-6676655dca2c47ae86ba02d7588cb068 |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-04-14T01:58:52Z |
| publishDate | 2015-10-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-6676655dca2c47ae86ba02d7588cb0682022-12-22T02:18:53ZengMDPI AGSymmetry2073-89942015-10-01741831184010.3390/sym7041831sym7041831On the Continuity of the Hutchinson OperatorMichael F. Barnsley0Krzysztof Leśniak1Mathematical Sciences Institute, The Australian National University, Union Lane, Canberra, ACT 2601, AustraliaFaculty of Mathematics and Computer Science, Nicolaus Copernicus University in Toru´n, ul. Chopina 12/18, Toru´n 87-100, PolandWe investigate when the Hutchinson operator associated with an iterated function system is continuous. The continuity with respect to both the Hausdorff metric and Vietoris topology is carefully considered. An example showing that the Hutchinson operator on the hyperspace of nonempty closed bounded sets need not be Hausdorff continuous is given. Infinite systems are also discussed. The work clarifies and generalizes several partial results scattered across the literature.http://www.mdpi.com/2073-8994/7/4/1831iterated function systemHutchinson operatormultifunctionboundedly uniformly continuous mapupper semicontinuitystrict attractorVietoris continuitycompact-open topology |
| spellingShingle | Michael F. Barnsley Krzysztof Leśniak On the Continuity of the Hutchinson Operator Symmetry iterated function system Hutchinson operator multifunction boundedly uniformly continuous map upper semicontinuity strict attractor Vietoris continuity compact-open topology |
| title | On the Continuity of the Hutchinson Operator |
| title_full | On the Continuity of the Hutchinson Operator |
| title_fullStr | On the Continuity of the Hutchinson Operator |
| title_full_unstemmed | On the Continuity of the Hutchinson Operator |
| title_short | On the Continuity of the Hutchinson Operator |
| title_sort | on the continuity of the hutchinson operator |
| topic | iterated function system Hutchinson operator multifunction boundedly uniformly continuous map upper semicontinuity strict attractor Vietoris continuity compact-open topology |
| url | http://www.mdpi.com/2073-8994/7/4/1831 |
| work_keys_str_mv | AT michaelfbarnsley onthecontinuityofthehutchinsonoperator AT krzysztoflesniak onthecontinuityofthehutchinsonoperator |