Countable contraction mappings in metric spaces: Invariant Sets and Measures
We consider a complete metric space (X, d) and a countable number of contraction mappings on X, F = {Fi : i ∈ N}. We show the existence of a smallest invariant set (with respect to inclusion) for F. If the maps Fi are of the form Fi(x) = rix + bi on X = R d , we prove a converse of the classic resul...
| Autores: | ; |
|---|---|
| Tipo de documento: | Artículo |
| Estado: | Published version |
| Fecha de publicación: | 2014 |
| País: | Argentina |
| Institución: | Consejo Nacional de Investigaciones Científicas y Técnicas |
| Repositorio: | CONICET Digital (CONICET) |
| Lenguaje: | Inglés |
| OAI Identifier: | oai:ri.conicet.gov.ar:11336/18774 |
| Acceso en línea: | http://hdl.handle.net/11336/18774 |
| Nivel de acceso: | Acceso abierto |
| Palabra clave: | Countable Iterated Function Systems Invariant Measure Atractor https://purl.org/becyt/ford/1.1 https://purl.org/becyt/ford/1 |
| _version_ | 1878403694067712000 |
|---|---|
| author | Barrozo, María Fernanda |
| author2 | Molter, Ursula Maria |
| author2_role | author |
| author_browse | Barrozo, María Fernanda Molter, Ursula Maria |
| author_facet | Barrozo, María Fernanda Molter, Ursula Maria |
| author_role | author |
| collection | CONICET Digital (CONICET) |
| description | We consider a complete metric space (X, d) and a countable number of contraction mappings on X, F = {Fi : i ∈ N}. We show the existence of a smallest invariant set (with respect to inclusion) for F. If the maps Fi are of the form Fi(x) = rix + bi on X = R d , we prove a converse of the classic result on contraction mappings, more precisely, there exists a unique bounded invariant set if and only if r = supi ri is strictly smaller than 1. Further, if ρ = {ρk}k∈N is a probability sequence, we show that if there exists an invariant measure for the system (F, ρ), then its support must be precisely this smallest invariant set. If in addition there exists any bounded invariant set, this invariant measure is unique, even though there may be more than one invariant set. |
| eu_rights_str_mv | openAccess |
| format | article |
| id | AR_ae823dfd12a4e2a81318fce32c186aff |
| identifier_str_mv | Barrozo, María Fernanda; Molter, Ursula Maria; Countable contraction mappings in metric spaces: Invariant Sets and Measures; Versita; CENTRAL EUROPEAN JOURNAL OF MATHEMATICS - (Print); 12; 4; 4-2014; 593-602 1895-1074 CONICET Digital CONICET |
| institution | Consejo Nacional de Investigaciones Científicas y Técnicas |
| instname_str | Consejo Nacional de Investigaciones Científicas y Técnicas |
| language | eng |
| network_acronym_str | AR |
| network_name_str | Argentina |
| oai_identifier_str | oai:ri.conicet.gov.ar:11336/18774 |
| publishDate | 2014 |
| publishDateSort | 2014 |
| publisher | Versita |
| publisherStr | Versita |
| reponame_str | CONICET Digital (CONICET) |
| spelling | Countable contraction mappings in metric spaces: Invariant Sets and Measures Barrozo, María Fernanda Molter, Ursula Maria Countable Iterated Function Systems Invariant Measure Atractor https://purl.org/becyt/ford/1.1 https://purl.org/becyt/ford/1 We consider a complete metric space (X, d) and a countable number of contraction mappings on X, F = {Fi : i ∈ N}. We show the existence of a smallest invariant set (with respect to inclusion) for F. If the maps Fi are of the form Fi(x) = rix + bi on X = R d , we prove a converse of the classic result on contraction mappings, more precisely, there exists a unique bounded invariant set if and only if r = supi ri is strictly smaller than 1. Further, if ρ = {ρk}k∈N is a probability sequence, we show that if there exists an invariant measure for the system (F, ρ), then its support must be precisely this smallest invariant set. If in addition there exists any bounded invariant set, this invariant measure is unique, even though there may be more than one invariant set. Fil: Barrozo, María Fernanda. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - San Luis. Instituto de Matemática Aplicada de San Luis "Prof. Ezio Marchi". Universidad Nacional de San Luis. Facultad de Ciencias Físico, Matemáticas y Naturales. Instituto de Matemática Aplicada de San Luis "Prof. Ezio Marchi"; Argentina Fil: Molter, Ursula Maria. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santalo". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santalo"; Argentina Versita http://hdl.handle.net/11336/18774 Barrozo, María Fernanda; Molter, Ursula Maria; Countable contraction mappings in metric spaces: Invariant Sets and Measures; Versita; CENTRAL EUROPEAN JOURNAL OF MATHEMATICS - (Print); 12; 4; 4-2014; 593-602 1895-1074 CONICET Digital CONICET |
| spellingShingle | Countable contraction mappings in metric spaces: Invariant Sets and Measures Barrozo, María Fernanda Countable Iterated Function Systems Invariant Measure Atractor https://purl.org/becyt/ford/1.1 https://purl.org/becyt/ford/1 |
| status_str | publishedVersion |
| title | Countable contraction mappings in metric spaces: Invariant Sets and Measures |
| title_full | Countable contraction mappings in metric spaces: Invariant Sets and Measures |
| title_fullStr | Countable contraction mappings in metric spaces: Invariant Sets and Measures |
| title_full_unstemmed | Countable contraction mappings in metric spaces: Invariant Sets and Measures |
| title_short | Countable contraction mappings in metric spaces: Invariant Sets and Measures |
| title_sort | Countable contraction mappings in metric spaces: Invariant Sets and Measures |
| topic | Countable Iterated Function Systems Invariant Measure Atractor https://purl.org/becyt/ford/1.1 https://purl.org/becyt/ford/1 |
| topic_facet | Countable Iterated Function Systems Invariant Measure Atractor https://purl.org/becyt/ford/1.1 https://purl.org/becyt/ford/1 |
| url | http://hdl.handle.net/11336/18774 |