Countable contraction mappings in metric spaces: Invariant sets and measure

We consider a complete metric space (X, d) and a countable number of contraction mappings on X, F = {F i: i ∈ ℕ}. We show the existence of a smallest invariant set (with respect to inclusion) for F. If the maps F i are of the form F i(x) = r ix + bi on X = ℝd, we prove a converse of the classic resu...

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Guardado en:
Detalles Bibliográficos
Publicado: 2014
Materias:
Contraction maps
Countable iterated function system
Invariant measure
Invariant set
Acceso en línea:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_18951074_v12_n4_p593_Barrozo
http://hdl.handle.net/20.500.12110/paper_18951074_v12_n4_p593_Barrozo
Aporte de:
Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires
id paper:paper_18951074_v12_n4_p593_Barrozo
record_format dspace
spelling paper:paper_18951074_v12_n4_p593_Barrozo2023-06-08T16:30:21Z Countable contraction mappings in metric spaces: Invariant sets and measure Contraction maps Countable iterated function system Invariant measure Invariant set We consider a complete metric space (X, d) and a countable number of contraction mappings on X, F = {F i: i ∈ ℕ}. We show the existence of a smallest invariant set (with respect to inclusion) for F. If the maps F i are of the form F i(x) = r ix + bi on X = ℝd, 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 r i 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. © 2014 Versita Warsaw and Springer-Verlag Wien. 2014 https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_18951074_v12_n4_p593_Barrozo http://hdl.handle.net/20.500.12110/paper_18951074_v12_n4_p593_Barrozo
institution Universidad de Buenos Aires
institution_str I-28
repository_str R-134
collection Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA)
topic Contraction maps
Countable iterated function system
Invariant measure
Invariant set
spellingShingle Contraction maps
Countable iterated function system
Invariant measure
Invariant set
Countable contraction mappings in metric spaces: Invariant sets and measure
topic_facet Contraction maps
Countable iterated function system
Invariant measure
Invariant set
description We consider a complete metric space (X, d) and a countable number of contraction mappings on X, F = {F i: i ∈ ℕ}. We show the existence of a smallest invariant set (with respect to inclusion) for F. If the maps F i are of the form F i(x) = r ix + bi on X = ℝd, 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 r i 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. © 2014 Versita Warsaw and Springer-Verlag Wien.
title Countable contraction mappings in metric spaces: Invariant sets and measure
title_short Countable contraction mappings in metric spaces: Invariant sets and measure
title_full Countable contraction mappings in metric spaces: Invariant sets and measure
title_fullStr Countable contraction mappings in metric spaces: Invariant sets and measure
title_full_unstemmed Countable contraction mappings in metric spaces: Invariant sets and measure
title_sort countable contraction mappings in metric spaces: invariant sets and measure
publishDate 2014
url https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_18951074_v12_n4_p593_Barrozo
http://hdl.handle.net/20.500.12110/paper_18951074_v12_n4_p593_Barrozo
_version_ 1768545392536846336

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