Ergodic theorems from probabilistic and topological viewpoints

• Ano de defesa: 2018
• Autor(a) principal: Boas, Lucas Amorim Vilas
• Tipo de documento: Dissertação
• Tipo de acesso: Acesso aberto
• Idioma: eng
• Instituição de defesa: Biblioteca Digitais de Teses e Dissertações da USP
• Palavras-chave em Português: Análogo topológico; Baire category; Categoria de Baire; Ergodic theorem; Law of large numbers; Lei dos grandes números; Teorema ergódico; Topological analogue
• Link de acesso: https://www.teses.usp.br/teses/disponiveis/45/45132/tde-08012021-001207/

Ergodic theorems are classic measure theoretical results in dynamical systems or, more precisely, ergodic theory. They state that the convergence of Birkhoff averages is typical, in a measure theoretical sense. This work aims to explain how these results can be re-ìnterpreted in light of topology and probability theory. The first relationship is presented through a Baire category analogue of a standard version of Birkhoff\'s ergodic (assuming ergodicity). Instead of convergence of Birkhoff averages, the topological typical behavior will be the opposite: averages do not converge in a dramatic way. The second relationship is presented by examining how the law of large numbers interacts with Birkhoff\'s ergodic theorem (assuming ergodicity). The law of large numbers can be obtained as a corollary of Birkhoff\'s ergodic theorem. However, the law provides a new point of view, as it guarantees the conclusions of Birkhoff\'s ergodic theorem (assuming ergodicity) will hold even in the non-ergodic case, at the cost of requiring some sort of independence.