Absolute continuity of self-similar measures, their projections and convolutions

Autores
Shmerkin, Pablo Sebastian; Solomyak, Boris
Año de publicación
2016
Idioma
inglés
Tipo de recurso
artículo
Estado
versión publicada
Descripción
We show that in many parametrized families of self-similar measures, their projections, and their convolutions, the set of parameters for which the measure fails to be absolutely continuous is very small—of co-dimension at least 1 in parameter space. This complements an active line of research concerning similar questions for dimension. Moreover, we establish some regularity of the density outside this small exceptional set, which applies in particular to Bernoulli convolutions; along the way, we prove some new results about the dimensions of self-similar measures and the absolute continuity of the convolution of two measures. As a concrete application, we obtain a very strong version of Marstrand’s projection theorem for planar self-similar sets.
Fil: Shmerkin, Pablo Sebastian. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Torcuato Di Tella. Departamento de Matemáticas y Estadística; Argentina
Fil: Solomyak, Boris. University of Washington; Estados Unidos
Materia
ABSOLUTE CONTINUITY
SELF-SIMILAR MEASURES
HAUSDORFF DIMENSION
CONVOLUTIONS
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:11336/90812
Acceder
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oai_identifier_str oai:ri.conicet.gov.ar:11336/90812
network_acronym_str CONICETDig
repository_id_str 3498
network_name_str CONICET Digital (CONICET)
spelling Absolute continuity of self-similar measures, their projections and convolutionsShmerkin, Pablo SebastianSolomyak, BorisABSOLUTE CONTINUITYSELF-SIMILAR MEASURESHAUSDORFF DIMENSIONCONVOLUTIONShttps://purl.org/becyt/ford/1.1https://purl.org/becyt/ford/1We show that in many parametrized families of self-similar measures, their projections, and their convolutions, the set of parameters for which the measure fails to be absolutely continuous is very small—of co-dimension at least 1 in parameter space. This complements an active line of research concerning similar questions for dimension. Moreover, we establish some regularity of the density outside this small exceptional set, which applies in particular to Bernoulli convolutions; along the way, we prove some new results about the dimensions of self-similar measures and the absolute continuity of the convolution of two measures. As a concrete application, we obtain a very strong version of Marstrand’s projection theorem for planar self-similar sets.Fil: Shmerkin, Pablo Sebastian. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Torcuato Di Tella. Departamento de Matemáticas y Estadística; ArgentinaFil: Solomyak, Boris. University of Washington; Estados UnidosAmerican Mathematical Society2016-07info: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/11336/90812Shmerkin, Pablo Sebastian; Solomyak, Boris; Absolute continuity of self-similar measures, their projections and convolutions; American Mathematical Society; Transactions Of The American Mathematical Society; 368; 7; 7-2016; 5125-51510002-9947CONICET DigitalCONICETenginfo:eu-repo/semantics/altIdentifier/url/https://www.ams.org/journals/tran/2016-368-07/S0002-9947-2015-06696-3/info:eu-repo/semantics/altIdentifier/doi/10.1090/tran6696info:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by-nc-sa/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2025-09-29T09:42:33Zoai:ri.conicet.gov.ar:11336/90812instacron: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:34982025-09-29 09:42:34.204CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse
dc.title.none.fl_str_mv Absolute continuity of self-similar measures, their projections and convolutions
title Absolute continuity of self-similar measures, their projections and convolutions
spellingShingle Absolute continuity of self-similar measures, their projections and convolutions
Shmerkin, Pablo Sebastian
ABSOLUTE CONTINUITY
SELF-SIMILAR MEASURES
HAUSDORFF DIMENSION
CONVOLUTIONS
title_short Absolute continuity of self-similar measures, their projections and convolutions
title_full Absolute continuity of self-similar measures, their projections and convolutions
title_fullStr Absolute continuity of self-similar measures, their projections and convolutions
title_full_unstemmed Absolute continuity of self-similar measures, their projections and convolutions
title_sort Absolute continuity of self-similar measures, their projections and convolutions
dc.creator.none.fl_str_mv Shmerkin, Pablo Sebastian
Solomyak, Boris
author Shmerkin, Pablo Sebastian
author_facet Shmerkin, Pablo Sebastian
Solomyak, Boris
author_role author
author2 Solomyak, Boris
author2_role author
dc.subject.none.fl_str_mv ABSOLUTE CONTINUITY
SELF-SIMILAR MEASURES
HAUSDORFF DIMENSION
CONVOLUTIONS
topic ABSOLUTE CONTINUITY
SELF-SIMILAR MEASURES
HAUSDORFF DIMENSION
CONVOLUTIONS
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 show that in many parametrized families of self-similar measures, their projections, and their convolutions, the set of parameters for which the measure fails to be absolutely continuous is very small—of co-dimension at least 1 in parameter space. This complements an active line of research concerning similar questions for dimension. Moreover, we establish some regularity of the density outside this small exceptional set, which applies in particular to Bernoulli convolutions; along the way, we prove some new results about the dimensions of self-similar measures and the absolute continuity of the convolution of two measures. As a concrete application, we obtain a very strong version of Marstrand’s projection theorem for planar self-similar sets.
Fil: Shmerkin, Pablo Sebastian. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Torcuato Di Tella. Departamento de Matemáticas y Estadística; Argentina
Fil: Solomyak, Boris. University of Washington; Estados Unidos
description We show that in many parametrized families of self-similar measures, their projections, and their convolutions, the set of parameters for which the measure fails to be absolutely continuous is very small—of co-dimension at least 1 in parameter space. This complements an active line of research concerning similar questions for dimension. Moreover, we establish some regularity of the density outside this small exceptional set, which applies in particular to Bernoulli convolutions; along the way, we prove some new results about the dimensions of self-similar measures and the absolute continuity of the convolution of two measures. As a concrete application, we obtain a very strong version of Marstrand’s projection theorem for planar self-similar sets.
publishDate 2016
dc.date.none.fl_str_mv 2016-07
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/11336/90812
Shmerkin, Pablo Sebastian; Solomyak, Boris; Absolute continuity of self-similar measures, their projections and convolutions; American Mathematical Society; Transactions Of The American Mathematical Society; 368; 7; 7-2016; 5125-5151
0002-9947
CONICET Digital
CONICET
url http://hdl.handle.net/11336/90812
identifier_str_mv Shmerkin, Pablo Sebastian; Solomyak, Boris; Absolute continuity of self-similar measures, their projections and convolutions; American Mathematical Society; Transactions Of The American Mathematical Society; 368; 7; 7-2016; 5125-5151
0002-9947
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/2016-368-07/S0002-9947-2015-06696-3/
info:eu-repo/semantics/altIdentifier/doi/10.1090/tran6696
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 13.070432

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