On equality of Hausdorff and affinity dimensions, via self-affine measures on positive subsystems

Autores
Morris, Ian D.; Shmerkin, Pablo Sebastian
Año de publicación
2018
Idioma
inglés
Tipo de recurso
artículo
Estado
versión publicada
Descripción
Under mild conditions we show that the affinity dimension of a planar self-affine set is equal to the supremum of the Lyapunov dimensions of self-affine measures supported on self-affine proper subsets of the original set. These self-affine subsets may be chosen so as to have stronger separation properties and in such a way that the linear parts of their affinities are positive matrices. Combining this result with some recent breakthroughs in the study of self-affine measures and their associated Furstenberg measures, we obtain new criteria under which the Hausdorff dimension of a self-affine set equals its affinity dimension. For example, applying recent results of Bárány, Hochman–Solomyak, and Rapaport, we provide many new explicit examples of self-affine sets whose Hausdorff dimension equals its affinity dimension, and for which the linear parts do not satisfy any positivity or domination assumptions.
Fil: Morris, Ian D.. University of Surrey; Reino Unido
Fil: Shmerkin, Pablo Sebastian. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Torcuato Di Tella. Departamento de Matemáticas y Estadística; Argentina
Materia
Self-affine sets
Affinity dimension
Hausdorff dimension
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:11336/177835
Acceder
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spelling On equality of Hausdorff and affinity dimensions, via self-affine measures on positive subsystemsMorris, Ian D.Shmerkin, Pablo SebastianSelf-affine setsAffinity dimensionHausdorff dimensionhttps://purl.org/becyt/ford/1.1https://purl.org/becyt/ford/1Under mild conditions we show that the affinity dimension of a planar self-affine set is equal to the supremum of the Lyapunov dimensions of self-affine measures supported on self-affine proper subsets of the original set. These self-affine subsets may be chosen so as to have stronger separation properties and in such a way that the linear parts of their affinities are positive matrices. Combining this result with some recent breakthroughs in the study of self-affine measures and their associated Furstenberg measures, we obtain new criteria under which the Hausdorff dimension of a self-affine set equals its affinity dimension. For example, applying recent results of Bárány, Hochman–Solomyak, and Rapaport, we provide many new explicit examples of self-affine sets whose Hausdorff dimension equals its affinity dimension, and for which the linear parts do not satisfy any positivity or domination assumptions.Fil: Morris, Ian D.. University of Surrey; Reino UnidoFil: Shmerkin, Pablo Sebastian. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Torcuato Di Tella. Departamento de Matemáticas y Estadística; ArgentinaAmerican Mathematical Society2018-10info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdfapplication/zipapplication/pdfhttp://hdl.handle.net/11336/177835Morris, Ian D.; Shmerkin, Pablo Sebastian; On equality of Hausdorff and affinity dimensions, via self-affine measures on positive subsystems; American Mathematical Society; Transactions of the American Mathematical Society; 371; 3; 10-2018; 1547-15820002-99471088-6850CONICET DigitalCONICETenginfo:eu-repo/semantics/altIdentifier/url/https://www.ams.org/journals/tran/2019-371-03/S0002-9947-2018-07334-2/home.htmlinfo:eu-repo/semantics/altIdentifier/doi/10.1090/tran/7334info:eu-repo/semantics/altIdentifier/arxiv/https://arxiv.org/abs/1602.08789info:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by-nc-sa/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2025-11-05T10:10:06Zoai:ri.conicet.gov.ar:11336/177835instacron: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:34982025-11-05 10:10:06.57CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse
dc.title.none.fl_str_mv On equality of Hausdorff and affinity dimensions, via self-affine measures on positive subsystems
title On equality of Hausdorff and affinity dimensions, via self-affine measures on positive subsystems
spellingShingle On equality of Hausdorff and affinity dimensions, via self-affine measures on positive subsystems
Morris, Ian D.
Self-affine sets
Affinity dimension
Hausdorff dimension
title_short On equality of Hausdorff and affinity dimensions, via self-affine measures on positive subsystems
title_full On equality of Hausdorff and affinity dimensions, via self-affine measures on positive subsystems
title_fullStr On equality of Hausdorff and affinity dimensions, via self-affine measures on positive subsystems
title_full_unstemmed On equality of Hausdorff and affinity dimensions, via self-affine measures on positive subsystems
title_sort On equality of Hausdorff and affinity dimensions, via self-affine measures on positive subsystems
dc.creator.none.fl_str_mv Morris, Ian D.
Shmerkin, Pablo Sebastian
author Morris, Ian D.
author_facet Morris, Ian D.
Shmerkin, Pablo Sebastian
author_role author
author2 Shmerkin, Pablo Sebastian
author2_role author
dc.subject.none.fl_str_mv Self-affine sets
Affinity dimension
Hausdorff dimension
topic Self-affine sets
Affinity dimension
Hausdorff dimension
purl_subject.fl_str_mv https://purl.org/becyt/ford/1.1
https://purl.org/becyt/ford/1
dc.description.none.fl_txt_mv Under mild conditions we show that the affinity dimension of a planar self-affine set is equal to the supremum of the Lyapunov dimensions of self-affine measures supported on self-affine proper subsets of the original set. These self-affine subsets may be chosen so as to have stronger separation properties and in such a way that the linear parts of their affinities are positive matrices. Combining this result with some recent breakthroughs in the study of self-affine measures and their associated Furstenberg measures, we obtain new criteria under which the Hausdorff dimension of a self-affine set equals its affinity dimension. For example, applying recent results of Bárány, Hochman–Solomyak, and Rapaport, we provide many new explicit examples of self-affine sets whose Hausdorff dimension equals its affinity dimension, and for which the linear parts do not satisfy any positivity or domination assumptions.
Fil: Morris, Ian D.. University of Surrey; Reino Unido
Fil: Shmerkin, Pablo Sebastian. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Torcuato Di Tella. Departamento de Matemáticas y Estadística; Argentina
description Under mild conditions we show that the affinity dimension of a planar self-affine set is equal to the supremum of the Lyapunov dimensions of self-affine measures supported on self-affine proper subsets of the original set. These self-affine subsets may be chosen so as to have stronger separation properties and in such a way that the linear parts of their affinities are positive matrices. Combining this result with some recent breakthroughs in the study of self-affine measures and their associated Furstenberg measures, we obtain new criteria under which the Hausdorff dimension of a self-affine set equals its affinity dimension. For example, applying recent results of Bárány, Hochman–Solomyak, and Rapaport, we provide many new explicit examples of self-affine sets whose Hausdorff dimension equals its affinity dimension, and for which the linear parts do not satisfy any positivity or domination assumptions.
publishDate 2018
dc.date.none.fl_str_mv 2018-10
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/11336/177835
Morris, Ian D.; Shmerkin, Pablo Sebastian; On equality of Hausdorff and affinity dimensions, via self-affine measures on positive subsystems; American Mathematical Society; Transactions of the American Mathematical Society; 371; 3; 10-2018; 1547-1582
0002-9947
1088-6850
CONICET Digital
CONICET
url http://hdl.handle.net/11336/177835
identifier_str_mv Morris, Ian D.; Shmerkin, Pablo Sebastian; On equality of Hausdorff and affinity dimensions, via self-affine measures on positive subsystems; American Mathematical Society; Transactions of the American Mathematical Society; 371; 3; 10-2018; 1547-1582
0002-9947
1088-6850
CONICET Digital
CONICET
dc.language.none.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv info:eu-repo/semantics/altIdentifier/url/https://www.ams.org/journals/tran/2019-371-03/S0002-9947-2018-07334-2/home.html
info:eu-repo/semantics/altIdentifier/doi/10.1090/tran/7334
info:eu-repo/semantics/altIdentifier/arxiv/https://arxiv.org/abs/1602.08789
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/zip
application/pdf
dc.publisher.none.fl_str_mv American Mathematical Society
publisher.none.fl_str_mv American Mathematical Society
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 13.087074

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