Countable contraction mappings in metric spaces: Invariant Sets and Measures

Autores
Barrozo, María Fernanda; Molter, Ursula Maria
Año de publicación
2014
Idioma
inglés
Tipo de recurso
artículo
Estado
versión publicada
Descripción
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
Materia
Countable Iterated Function Systems
Invariant Measure
Atractor
Nivel de accesibilidad
acceso abierto
Condiciones de uso
https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
Repositorio
CONICET Digital (CONICET)
Institución
Consejo Nacional de Investigaciones Científicas y Técnicas
OAI Identificador
oai:ri.conicet.gov.ar:11336/18774
Acceder
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network_name_str CONICET Digital (CONICET)
spelling Countable contraction mappings in metric spaces: Invariant Sets and MeasuresBarrozo, María FernandaMolter, Ursula MariaCountable Iterated Function SystemsInvariant MeasureAtractorhttps://purl.org/becyt/ford/1.1https://purl.org/becyt/ford/1We 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"; ArgentinaFil: 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"; ArgentinaVersita2014-04info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdfapplication/pdfhttp://hdl.handle.net/11336/18774Barrozo, 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; 593-6021895-1074CONICET DigitalCONICETenginfo:eu-repo/semantics/altIdentifier/doi/10.2478/s11533-013-0371-0info:eu-repo/semantics/altIdentifier/url/https://www.degruyter.com/view/j/math.2014.12.issue-4/s11533-013-0371-0/s11533-013-0371-0.xmlinfo:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by-nc-sa/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2025-09-10T13:10:30Zoai:ri.conicet.gov.ar:11336/18774instacron:CONICETInstitucionalhttp://ri.conicet.gov.ar/Organismo científico-tecnológicoNo correspondehttp://ri.conicet.gov.ar/oai/requestdasensio@conicet.gov.ar; lcarlino@conicet.gov.arArgentinaNo correspondeNo correspondeNo correspondeopendoar:34982025-09-10 13:10:30.724CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse
dc.title.none.fl_str_mv Countable contraction mappings in metric spaces: Invariant Sets and Measures
title Countable contraction mappings in metric spaces: Invariant Sets and Measures
spellingShingle Countable contraction mappings in metric spaces: Invariant Sets and Measures
Barrozo, María Fernanda
Countable Iterated Function Systems
Invariant Measure
Atractor
title_short 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_sort Countable contraction mappings in metric spaces: Invariant Sets and Measures
dc.creator.none.fl_str_mv Barrozo, María Fernanda
Molter, Ursula Maria
author Barrozo, María Fernanda
author_facet Barrozo, María Fernanda
Molter, Ursula Maria
author_role author
author2 Molter, Ursula Maria
author2_role author
dc.subject.none.fl_str_mv Countable Iterated Function Systems
Invariant Measure
Atractor
topic Countable Iterated Function Systems
Invariant Measure
Atractor
purl_subject.fl_str_mv https://purl.org/becyt/ford/1.1
https://purl.org/becyt/ford/1
dc.description.none.fl_txt_mv 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
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.
publishDate 2014
dc.date.none.fl_str_mv 2014-04
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/publishedVersion
http://purl.org/coar/resource_type/c_6501
info:ar-repo/semantics/articulo
format article
status_str publishedVersion
dc.identifier.none.fl_str_mv 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
url http://hdl.handle.net/11336/18774
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
dc.language.none.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv info:eu-repo/semantics/altIdentifier/doi/10.2478/s11533-013-0371-0
info:eu-repo/semantics/altIdentifier/url/https://www.degruyter.com/view/j/math.2014.12.issue-4/s11533-013-0371-0/s11533-013-0371-0.xml
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
eu_rights_str_mv openAccess
rights_invalid_str_mv https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
dc.format.none.fl_str_mv application/pdf
application/pdf
dc.publisher.none.fl_str_mv Versita
publisher.none.fl_str_mv Versita
dc.source.none.fl_str_mv reponame:CONICET Digital (CONICET)
instname:Consejo Nacional de Investigaciones Científicas y Técnicas
reponame_str CONICET Digital (CONICET)
collection CONICET Digital (CONICET)
instname_str Consejo Nacional de Investigaciones Científicas y Técnicas
repository.name.fl_str_mv CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicas
repository.mail.fl_str_mv dasensio@conicet.gov.ar; lcarlino@conicet.gov.ar
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score 12.993085

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