Shadowing and structural stability for operators
Autor(a) principal: | |
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Data de Publicação: | 2021 |
Outros Autores: | |
Tipo de documento: | Artigo |
Idioma: | eng |
Título da fonte: | Repositório Institucional da UNESP |
Texto Completo: | http://dx.doi.org/10.1017/etds.2019.107 http://hdl.handle.net/11449/232955 |
Resumo: | A well-known result in the area of dynamical systems asserts that any invertible hyperbolic operator on any Banach space is structurally stable. This result was originally obtained by Hartman in 1960 for operators on finite-dimensional spaces. The general case was independently obtained by Palis and Pugh around 1968. We will exhibit a class of examples of structurally stable operators that are not hyperbolic, thereby showing that the converse of the above-mentioned result is false in general. We will also prove that an invertible operator on a Banach space is hyperbolic if and only if it is expansive and has the shadowing property. Moreover, we will show that if a structurally stable operator is expansive, then it must be uniformly expansive. Finally, we will characterize the weighted shifts on the spaces and (<![CDATA[1\leq p) that satisfy the shadowing property. |
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Repositório Institucional da UNESP |
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Shadowing and structural stability for operators37B99 (Secondary)37C2037C5047A16 (Primary)expansivityhyperbolicitylinear operatorsshadowingstructural stability 2010 Mathematics Subject ClassificationA well-known result in the area of dynamical systems asserts that any invertible hyperbolic operator on any Banach space is structurally stable. This result was originally obtained by Hartman in 1960 for operators on finite-dimensional spaces. The general case was independently obtained by Palis and Pugh around 1968. We will exhibit a class of examples of structurally stable operators that are not hyperbolic, thereby showing that the converse of the above-mentioned result is false in general. We will also prove that an invertible operator on a Banach space is hyperbolic if and only if it is expansive and has the shadowing property. Moreover, we will show that if a structurally stable operator is expansive, then it must be uniformly expansive. Finally, we will characterize the weighted shifts on the spaces and (<![CDATA[1\leq p) that satisfy the shadowing property.1 Departamento de Matemática Aplicada Instituto de Matemática Universidade Federal Do Rio de Janeiro, Caixa Postal 68530Departamento de Matemática Universidade Estadual Paulista, Rua Cristóvão Colombo, 2265Departamento de Matemática Universidade Estadual Paulista, Rua Cristóvão Colombo, 2265Universidade Federal do Rio de Janeiro (UFRJ)Universidade Estadual Paulista (UNESP)Bernardes Jr, Nilson C.Messaoudi, Ali [UNESP]2022-04-30T22:28:33Z2022-04-30T22:28:33Z2021-04-01info:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/article961-980http://dx.doi.org/10.1017/etds.2019.107Ergodic Theory and Dynamical Systems, v. 41, n. 4, p. 961-980, 2021.1469-44170143-3857http://hdl.handle.net/11449/23295510.1017/etds.2019.1072-s2.0-85078036077Scopusreponame:Repositório Institucional da UNESPinstname:Universidade Estadual Paulista (UNESP)instacron:UNESPengErgodic Theory and Dynamical Systemsinfo:eu-repo/semantics/openAccess2022-04-30T22:28:33Zoai:repositorio.unesp.br:11449/232955Repositório InstitucionalPUBhttp://repositorio.unesp.br/oai/requestopendoar:29462024-08-05T18:05:18.183966Repositório Institucional da UNESP - Universidade Estadual Paulista (UNESP)false |
dc.title.none.fl_str_mv |
Shadowing and structural stability for operators |
title |
Shadowing and structural stability for operators |
spellingShingle |
Shadowing and structural stability for operators Bernardes Jr, Nilson C. 37B99 (Secondary) 37C20 37C50 47A16 (Primary) expansivity hyperbolicity linear operators shadowing structural stability 2010 Mathematics Subject Classification |
title_short |
Shadowing and structural stability for operators |
title_full |
Shadowing and structural stability for operators |
title_fullStr |
Shadowing and structural stability for operators |
title_full_unstemmed |
Shadowing and structural stability for operators |
title_sort |
Shadowing and structural stability for operators |
author |
Bernardes Jr, Nilson C. |
author_facet |
Bernardes Jr, Nilson C. Messaoudi, Ali [UNESP] |
author_role |
author |
author2 |
Messaoudi, Ali [UNESP] |
author2_role |
author |
dc.contributor.none.fl_str_mv |
Universidade Federal do Rio de Janeiro (UFRJ) Universidade Estadual Paulista (UNESP) |
dc.contributor.author.fl_str_mv |
Bernardes Jr, Nilson C. Messaoudi, Ali [UNESP] |
dc.subject.por.fl_str_mv |
37B99 (Secondary) 37C20 37C50 47A16 (Primary) expansivity hyperbolicity linear operators shadowing structural stability 2010 Mathematics Subject Classification |
topic |
37B99 (Secondary) 37C20 37C50 47A16 (Primary) expansivity hyperbolicity linear operators shadowing structural stability 2010 Mathematics Subject Classification |
description |
A well-known result in the area of dynamical systems asserts that any invertible hyperbolic operator on any Banach space is structurally stable. This result was originally obtained by Hartman in 1960 for operators on finite-dimensional spaces. The general case was independently obtained by Palis and Pugh around 1968. We will exhibit a class of examples of structurally stable operators that are not hyperbolic, thereby showing that the converse of the above-mentioned result is false in general. We will also prove that an invertible operator on a Banach space is hyperbolic if and only if it is expansive and has the shadowing property. Moreover, we will show that if a structurally stable operator is expansive, then it must be uniformly expansive. Finally, we will characterize the weighted shifts on the spaces and (<![CDATA[1\leq p) that satisfy the shadowing property. |
publishDate |
2021 |
dc.date.none.fl_str_mv |
2021-04-01 2022-04-30T22:28:33Z 2022-04-30T22:28:33Z |
dc.type.status.fl_str_mv |
info:eu-repo/semantics/publishedVersion |
dc.type.driver.fl_str_mv |
info:eu-repo/semantics/article |
format |
article |
status_str |
publishedVersion |
dc.identifier.uri.fl_str_mv |
http://dx.doi.org/10.1017/etds.2019.107 Ergodic Theory and Dynamical Systems, v. 41, n. 4, p. 961-980, 2021. 1469-4417 0143-3857 http://hdl.handle.net/11449/232955 10.1017/etds.2019.107 2-s2.0-85078036077 |
url |
http://dx.doi.org/10.1017/etds.2019.107 http://hdl.handle.net/11449/232955 |
identifier_str_mv |
Ergodic Theory and Dynamical Systems, v. 41, n. 4, p. 961-980, 2021. 1469-4417 0143-3857 10.1017/etds.2019.107 2-s2.0-85078036077 |
dc.language.iso.fl_str_mv |
eng |
language |
eng |
dc.relation.none.fl_str_mv |
Ergodic Theory and Dynamical Systems |
dc.rights.driver.fl_str_mv |
info:eu-repo/semantics/openAccess |
eu_rights_str_mv |
openAccess |
dc.format.none.fl_str_mv |
961-980 |
dc.source.none.fl_str_mv |
Scopus reponame:Repositório Institucional da UNESP instname:Universidade Estadual Paulista (UNESP) instacron:UNESP |
instname_str |
Universidade Estadual Paulista (UNESP) |
instacron_str |
UNESP |
institution |
UNESP |
reponame_str |
Repositório Institucional da UNESP |
collection |
Repositório Institucional da UNESP |
repository.name.fl_str_mv |
Repositório Institucional da UNESP - Universidade Estadual Paulista (UNESP) |
repository.mail.fl_str_mv |
|
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1808128892473442304 |