Invariant measures whose supports possess the strong open set property
Let \(X\) be a complete metric space, and \(S\) the union of a finite number of strict contractions on it. If \(P\) is a probability distribution on the maps, and \(K\) is the fractal determined by \(S\), there is a unique Borel probability measure \(\mu _P\) on \(X\) which is invariant under the a...
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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 |
| Subjects: | |
| Online Access: | http://www.opuscula.agh.edu.pl/vol28/4/art/opuscula_math_2835.pdf |
| _version_ | 1818503044087676928 |
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| author | Gerald S. Goodman |
| author_facet | Gerald S. Goodman |
| author_sort | Gerald S. Goodman |
| collection | DOAJ |
| description | Let \(X\) be a complete metric space, and \(S\) the union of a finite number of strict contractions on it. If \(P\) is a probability distribution on the maps, and \(K\) is the fractal determined by \(S\), there is a unique Borel probability measure \(\mu _P\) on \(X\) which is invariant under the associated Markov operator, and its support is \(K\). The Open Set Condition (OSC) requires that a non-empty, subinvariant, bounded open set \(V \subset X\) exists whose images under the maps are disjoint; it is strong if \(K \cap V \neq \emptyset\). In that case, the core of \(V\), \(\check{V}=\bigcap_{n=0}^{\infty} S^n (V)\), is non-empty and dense in \(K\). Moreover, when \(X\) is separable, \(\check{V}\) has full \(\mu _P\)-measure for every \(P\). We show that the strong condition holds for \(V\) satisfying the OSC iff \(\mu_P (\partial V) =0\), and we prove a zero-one law for it. We characterize the complement of \(\check{V}\) relative to \(K\), and we establish that the values taken by invariant measures on cylinder sets defined by \(K\),
or by the closure of \(V\), form multiplicative cascades. |
| first_indexed | 2024-12-10T21:18:35Z |
| format | Article |
| id | doaj.art-bd2abdd3fdf14e45bfd2b7324a0450e2 |
| institution | Directory Open Access Journal |
| issn | 1232-9274 |
| language | English |
| last_indexed | 2024-12-10T21:18:35Z |
| publishDate | 2008-01-01 |
| publisher | AGH Univeristy of Science and Technology Press |
| record_format | Article |
| series | Opuscula Mathematica |
| spelling | doaj.art-bd2abdd3fdf14e45bfd2b7324a0450e22022-12-22T01:33:12ZengAGH Univeristy of Science and Technology PressOpuscula Mathematica1232-92742008-01-012844714802835Invariant measures whose supports possess the strong open set propertyGerald S. Goodman0via Dazzi, 11, 50141 Firenze, ItalyLet \(X\) be a complete metric space, and \(S\) the union of a finite number of strict contractions on it. If \(P\) is a probability distribution on the maps, and \(K\) is the fractal determined by \(S\), there is a unique Borel probability measure \(\mu _P\) on \(X\) which is invariant under the associated Markov operator, and its support is \(K\). The Open Set Condition (OSC) requires that a non-empty, subinvariant, bounded open set \(V \subset X\) exists whose images under the maps are disjoint; it is strong if \(K \cap V \neq \emptyset\). In that case, the core of \(V\), \(\check{V}=\bigcap_{n=0}^{\infty} S^n (V)\), is non-empty and dense in \(K\). Moreover, when \(X\) is separable, \(\check{V}\) has full \(\mu _P\)-measure for every \(P\). We show that the strong condition holds for \(V\) satisfying the OSC iff \(\mu_P (\partial V) =0\), and we prove a zero-one law for it. We characterize the complement of \(\check{V}\) relative to \(K\), and we establish that the values taken by invariant measures on cylinder sets defined by \(K\), or by the closure of \(V\), form multiplicative cascades.http://www.opuscula.agh.edu.pl/vol28/4/art/opuscula_math_2835.pdfcorefractalfractal measureinvariant measurescaling functionscaling operatorstrong open set conditionzero-one law |
| spellingShingle | Gerald S. Goodman Invariant measures whose supports possess the strong open set property Opuscula Mathematica core fractal fractal measure invariant measure scaling function scaling operator strong open set condition zero-one law |
| title | Invariant measures whose supports possess the strong open set property |
| title_full | Invariant measures whose supports possess the strong open set property |
| title_fullStr | Invariant measures whose supports possess the strong open set property |
| title_full_unstemmed | Invariant measures whose supports possess the strong open set property |
| title_short | Invariant measures whose supports possess the strong open set property |
| title_sort | invariant measures whose supports possess the strong open set property |
| topic | core fractal fractal measure invariant measure scaling function scaling operator strong open set condition zero-one law |
| url | http://www.opuscula.agh.edu.pl/vol28/4/art/opuscula_math_2835.pdf |
| work_keys_str_mv | AT geraldsgoodman invariantmeasureswhosesupportspossessthestrongopensetproperty |