Fractal sets satisfying the strong open set condition in complete metric spaces
Let \(K\) be a Hutchinson fractal in a complete metric space \(X\), invariant under the action \(S\) of the union of a finite number of Lipschitz contractions. The Open Set Condition states that \(X\) has a non-empty subinvariant bounded open subset \(V\), whose images under the maps are disjoint. I...
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| Format: | Article |
| Language: | English |
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AGH Univeristy of Science and Technology Press
2008-01-01
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| Series: | Opuscula Mathematica |
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| Online Access: | http://www.opuscula.agh.edu.pl/vol28/4/art/opuscula_math_2834.pdf |
| _version_ | 1819263622569787392 |
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| author | Gerald S. Goodman |
| author_facet | Gerald S. Goodman |
| author_sort | Gerald S. Goodman |
| collection | DOAJ |
| description | Let \(K\) be a Hutchinson fractal in a complete metric space \(X\), invariant under the action \(S\) of the union of a finite number of Lipschitz contractions. The Open Set Condition states that \(X\) has a non-empty subinvariant bounded open subset \(V\), whose images under the maps are disjoint. It is said to be strong if \(V\) meets \(K\). We show by a category argument that when \(K \not\subset V\) and the restrictions of the contractions to \(V\) are open, the strong condition implies that \(\check{V}=\bigcap_{n=0}^{\infty} S^n(V)\), termed the core of \(V\) , is non-empty. In this case, it is an invariant, proper, dense, subset of \(K\), made up of points whose addresses are unique. Conversely, \(\check{V}\neq \emptyset\) implies the SOSC, without any openness assumption. |
| first_indexed | 2024-12-23T20:16:31Z |
| format | Article |
| id | doaj.art-7777127ff09e4222b6cc70289e922729 |
| institution | Directory Open Access Journal |
| issn | 1232-9274 |
| language | English |
| last_indexed | 2024-12-23T20:16:31Z |
| publishDate | 2008-01-01 |
| publisher | AGH Univeristy of Science and Technology Press |
| record_format | Article |
| series | Opuscula Mathematica |
| spelling | doaj.art-7777127ff09e4222b6cc70289e9227292022-12-21T17:32:40ZengAGH Univeristy of Science and Technology PressOpuscula Mathematica1232-92742008-01-012844634702834Fractal sets satisfying the strong open set condition in complete metric spacesGerald S. Goodman0via Dazzi, 11, 50141 Firenze, ItalyLet \(K\) be a Hutchinson fractal in a complete metric space \(X\), invariant under the action \(S\) of the union of a finite number of Lipschitz contractions. The Open Set Condition states that \(X\) has a non-empty subinvariant bounded open subset \(V\), whose images under the maps are disjoint. It is said to be strong if \(V\) meets \(K\). We show by a category argument that when \(K \not\subset V\) and the restrictions of the contractions to \(V\) are open, the strong condition implies that \(\check{V}=\bigcap_{n=0}^{\infty} S^n(V)\), termed the core of \(V\) , is non-empty. In this case, it is an invariant, proper, dense, subset of \(K\), made up of points whose addresses are unique. Conversely, \(\check{V}\neq \emptyset\) implies the SOSC, without any openness assumption.http://www.opuscula.agh.edu.pl/vol28/4/art/opuscula_math_2834.pdfaddressBaire categoryfractalscaling functionscaling operatorstrong open set condition |
| spellingShingle | Gerald S. Goodman Fractal sets satisfying the strong open set condition in complete metric spaces Opuscula Mathematica address Baire category fractal scaling function scaling operator strong open set condition |
| title | Fractal sets satisfying the strong open set condition in complete metric spaces |
| title_full | Fractal sets satisfying the strong open set condition in complete metric spaces |
| title_fullStr | Fractal sets satisfying the strong open set condition in complete metric spaces |
| title_full_unstemmed | Fractal sets satisfying the strong open set condition in complete metric spaces |
| title_short | Fractal sets satisfying the strong open set condition in complete metric spaces |
| title_sort | fractal sets satisfying the strong open set condition in complete metric spaces |
| topic | address Baire category fractal scaling function scaling operator strong open set condition |
| url | http://www.opuscula.agh.edu.pl/vol28/4/art/opuscula_math_2834.pdf |
| work_keys_str_mv | AT geraldsgoodman fractalsetssatisfyingthestrongopensetconditionincompletemetricspaces |