Fractal dimensions of hyperbolic graphs and horseshoes
Autor(a) principal: | |
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Data de Publicação: | 2022 |
Tipo de documento: | Dissertação |
Idioma: | eng |
Título da fonte: | Repositório Institucional da UFMG |
Texto Completo: | http://hdl.handle.net/1843/46723 |
Resumo: | Fractal Geometry is a field of Mathematics under development, which is open for expansion and new insights. Moreover, fractals are objects that arouses curiosity, since these objects have unorthodox properties, an irregular geometry and some self-similarities. On the other hand, since it is an area in development, there are some distinct fractal dimension calculations for hyperbolic graphs, as we are going to present here, based on two articles. The first paper calculates the Hausdorff dimension of a horseshoe using its intersection with the unstable manifold, with a strong usage of ergodic theory and hyperbolic dynamics. However, the second paper calculates the box dimension of an attractor by almost periodic functions and Fourier series, with a bit of hyperbolic dynamics. |
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Bernardo Melo de Carvalhohttp://lattes.cnpq.br/4074889152879220Rafael da Costa PereiraAlexandre Miranda AlvesPablo Daniel Carrasco Correahttp://lattes.cnpq.br/1380764077639340Deberton Moura de Oliveira2022-10-27T21:04:36Z2022-10-27T21:04:36Z2022-08-19http://hdl.handle.net/1843/46723Fractal Geometry is a field of Mathematics under development, which is open for expansion and new insights. Moreover, fractals are objects that arouses curiosity, since these objects have unorthodox properties, an irregular geometry and some self-similarities. On the other hand, since it is an area in development, there are some distinct fractal dimension calculations for hyperbolic graphs, as we are going to present here, based on two articles. The first paper calculates the Hausdorff dimension of a horseshoe using its intersection with the unstable manifold, with a strong usage of ergodic theory and hyperbolic dynamics. However, the second paper calculates the box dimension of an attractor by almost periodic functions and Fourier series, with a bit of hyperbolic dynamics.A Geometria Fractal é uma área da Matemática que está em desenvolvimento e, dessa forma, apresenta bastante espaço para crescimento e descobertas. Além disso, os fractais são objetos que despertam curiosidade, pois apresentam propriedades pouco ortodoxas, uma geometria irregular e algumas autossimilaridades. Por outro lado, por estar em desenvolvimento, a área apresenta distintas formas de cálculo das dimensões fractais de conjuntos hiperbólicos, como será apresentado nesse texto, que é baseado em dois artigos. O primeiro calcula a dimensão de Hausdorff de uma ferradura usando a sua interseção com a variedade instável, com uma forte base em teoria ergódica e dinâmica hiperbólica. O outro calcula a dimensão caixa de um atrator a partir de funções quase periódicas e séries de Fourier, com um toque de dinâmica hiperbólica.CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível SuperiorengUniversidade Federal de Minas GeraisPrograma de Pós-Graduação em MatemáticaUFMGBrasilICX - DEPARTAMENTO DE MATEMÁTICAMatemática – TesesGeometria – TesesFractais – TesesGrupos hiperbólicos – TesesFractal GeometryHausdorff DimensionBox DimensionHyperbolic GraphsHorseshoeGeometria FractalDimensão HausdorffDimensão CaixaConjuntos HiperbólicosFerraduraFractal dimensions of hyperbolic graphs and horseshoesDimensões fractais de gráficos hiperbólicos e ferradurasinfo:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/masterThesisinfo:eu-repo/semantics/openAccessreponame:Repositório Institucional da UFMGinstname:Universidade Federal de Minas Gerais (UFMG)instacron:UFMGORIGINALFractal Dimensions of Hyperbolic Graphs and Horseshoes.pdfFractal Dimensions of Hyperbolic Graphs and Horseshoes.pdfapplication/pdf2535808https://repositorio.ufmg.br/bitstream/1843/46723/1/Fractal%20Dimensions%20of%20Hyperbolic%20Graphs%20and%20Horseshoes.pdfaa42aafa5a7f057010e47fa66132c42eMD51LICENSElicense.txtlicense.txttext/plain; charset=utf-82118https://repositorio.ufmg.br/bitstream/1843/46723/2/license.txtcda590c95a0b51b4d15f60c9642ca272MD521843/467232022-10-27 18:04:36.773oai:repositorio.ufmg.br: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ório de PublicaçõesPUBhttps://repositorio.ufmg.br/oaiopendoar:2022-10-27T21:04:36Repositório Institucional da UFMG - Universidade Federal de Minas Gerais (UFMG)false |
dc.title.pt_BR.fl_str_mv |
Fractal dimensions of hyperbolic graphs and horseshoes |
dc.title.alternative.pt_BR.fl_str_mv |
Dimensões fractais de gráficos hiperbólicos e ferraduras |
title |
Fractal dimensions of hyperbolic graphs and horseshoes |
spellingShingle |
Fractal dimensions of hyperbolic graphs and horseshoes Deberton Moura de Oliveira Fractal Geometry Hausdorff Dimension Box Dimension Hyperbolic Graphs Horseshoe Geometria Fractal Dimensão Hausdorff Dimensão Caixa Conjuntos Hiperbólicos Ferradura Matemática – Teses Geometria – Teses Fractais – Teses Grupos hiperbólicos – Teses |
title_short |
Fractal dimensions of hyperbolic graphs and horseshoes |
title_full |
Fractal dimensions of hyperbolic graphs and horseshoes |
title_fullStr |
Fractal dimensions of hyperbolic graphs and horseshoes |
title_full_unstemmed |
Fractal dimensions of hyperbolic graphs and horseshoes |
title_sort |
Fractal dimensions of hyperbolic graphs and horseshoes |
author |
Deberton Moura de Oliveira |
author_facet |
Deberton Moura de Oliveira |
author_role |
author |
dc.contributor.advisor1.fl_str_mv |
Bernardo Melo de Carvalho |
dc.contributor.advisor1Lattes.fl_str_mv |
http://lattes.cnpq.br/4074889152879220 |
dc.contributor.advisor-co1.fl_str_mv |
Rafael da Costa Pereira |
dc.contributor.referee1.fl_str_mv |
Alexandre Miranda Alves |
dc.contributor.referee2.fl_str_mv |
Pablo Daniel Carrasco Correa |
dc.contributor.authorLattes.fl_str_mv |
http://lattes.cnpq.br/1380764077639340 |
dc.contributor.author.fl_str_mv |
Deberton Moura de Oliveira |
contributor_str_mv |
Bernardo Melo de Carvalho Rafael da Costa Pereira Alexandre Miranda Alves Pablo Daniel Carrasco Correa |
dc.subject.por.fl_str_mv |
Fractal Geometry Hausdorff Dimension Box Dimension Hyperbolic Graphs Horseshoe Geometria Fractal Dimensão Hausdorff Dimensão Caixa Conjuntos Hiperbólicos Ferradura |
topic |
Fractal Geometry Hausdorff Dimension Box Dimension Hyperbolic Graphs Horseshoe Geometria Fractal Dimensão Hausdorff Dimensão Caixa Conjuntos Hiperbólicos Ferradura Matemática – Teses Geometria – Teses Fractais – Teses Grupos hiperbólicos – Teses |
dc.subject.other.pt_BR.fl_str_mv |
Matemática – Teses Geometria – Teses Fractais – Teses Grupos hiperbólicos – Teses |
description |
Fractal Geometry is a field of Mathematics under development, which is open for expansion and new insights. Moreover, fractals are objects that arouses curiosity, since these objects have unorthodox properties, an irregular geometry and some self-similarities. On the other hand, since it is an area in development, there are some distinct fractal dimension calculations for hyperbolic graphs, as we are going to present here, based on two articles. The first paper calculates the Hausdorff dimension of a horseshoe using its intersection with the unstable manifold, with a strong usage of ergodic theory and hyperbolic dynamics. However, the second paper calculates the box dimension of an attractor by almost periodic functions and Fourier series, with a bit of hyperbolic dynamics. |
publishDate |
2022 |
dc.date.accessioned.fl_str_mv |
2022-10-27T21:04:36Z |
dc.date.available.fl_str_mv |
2022-10-27T21:04:36Z |
dc.date.issued.fl_str_mv |
2022-08-19 |
dc.type.status.fl_str_mv |
info:eu-repo/semantics/publishedVersion |
dc.type.driver.fl_str_mv |
info:eu-repo/semantics/masterThesis |
format |
masterThesis |
status_str |
publishedVersion |
dc.identifier.uri.fl_str_mv |
http://hdl.handle.net/1843/46723 |
url |
http://hdl.handle.net/1843/46723 |
dc.language.iso.fl_str_mv |
eng |
language |
eng |
dc.rights.driver.fl_str_mv |
info:eu-repo/semantics/openAccess |
eu_rights_str_mv |
openAccess |
dc.publisher.none.fl_str_mv |
Universidade Federal de Minas Gerais |
dc.publisher.program.fl_str_mv |
Programa de Pós-Graduação em Matemática |
dc.publisher.initials.fl_str_mv |
UFMG |
dc.publisher.country.fl_str_mv |
Brasil |
dc.publisher.department.fl_str_mv |
ICX - DEPARTAMENTO DE MATEMÁTICA |
publisher.none.fl_str_mv |
Universidade Federal de Minas Gerais |
dc.source.none.fl_str_mv |
reponame:Repositório Institucional da UFMG instname:Universidade Federal de Minas Gerais (UFMG) instacron:UFMG |
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Universidade Federal de Minas Gerais (UFMG) |
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UFMG |
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UFMG |
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Repositório Institucional da UFMG |
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Repositório Institucional da UFMG |
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