Medidas SRB para Atratores Hiperbólicos

• Ano de defesa: 2017
• Autor(a) principal: CARVALHO, Péricles Rafael Pavão
• Orientador(a): RAMOS, Vanessa Ribeiro
• Tipo de documento: Dissertação
• Tipo de acesso: Acesso aberto
• Idioma: por
• Instituição de defesa: Universidade Federal do Maranhão
• Programa de Pós-Graduação: PROGRAMA DE PÓS-GRADUAÇÃO EM MATEMÁTICA/CCET
• Departamento: DEPARTAMENTO DE MATEMÁTICA/CCET
• País: Brasil
• Palavras-chave em Português: Medidas SRB; Aplicações expansoras; Atratores hiperbólicos
• Palavras-chave em Inglês: SRB measures; expanding maps; Hyperbolic attractors
• Área do conhecimento CNPq: Matemática
• Link de acesso: http://tedebc.ufma.br:8080/jspui/handle/tede/1735

This paper begins with the definition of SRB measures, and the introduction of several concepts in ergodic theory necessary for the development of the presented results. We prove the existence and uniqueness of SRB measures for uniformly expanding transformations in compact connected manifolds whose Jacobian is H¨older continuous. Then, we present the definition of hyperbolic sets, hyperbolic attractors and their respective fundamental properties. As a main result, we prove the existence of SRB measures for hyperbolic attractors contained in compact manifolds, and its uniqueness if the hyperbolic attractor is transitive. First, it is shown the existence of invariant measures absolutely continuous along the unstable foliation. Then, we note that the restriction of this measure over certain subsets have the SRB property. Using the transitivity of the hyperbolic attractor, it is shown that there exists a unique subset such that this restriction is an SRB measure. We conclude that the system supports a unique SRB measure.