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Master's Dissertation
DOI
https://doi.org/10.11606/D.45.2018.tde-08012021-001207
Document
Author
Full name
Lucas Amorim Vilas Boas
Institute/School/College
Knowledge Area
Date of Defense
Published
São Paulo, 2018
Supervisor
Committee
Tal, Fabio Armando (President)
Freire Junior, Ricardo dos Santos
Kocsard, Alejandro
Title in English
Ergodic theorems from probabilistic and topological viewpoints
Keywords in English
Baire category
Ergodic theorem
Law of large numbers
Topological analogue
Abstract in English
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.
Title in Portuguese
Teoremas ergódicos dos pontos de vista probabilístico e topológico
Keywords in Portuguese
Análogo topológico
Categoria de Baire
Lei dos grandes números
Teorema ergódico
Abstract in Portuguese
Teoremas ergódicos são resultados medida-teóricos clássicos em sistemas dinâmicos ou, mais precisamente, teoria ergódica. Eles afirmam que a convergência das médias de Birkhoff é típica, em um sentido de medida. Este trabalho objetiva explicar como esses resultados podem ser reinterpretados à luz da topologia e da teoria das probabilidades. A primeira relação é apresentada por meio de um análogo em categoria de Baire de uma versão habitual do teorema ergódico de Birkhoff (assumindo ergodicidade). Ao invés de convergência das médias de Birkhoff, o comportamento topologicamente típico será oposto: as médias não convergirão de modo dramático. A segunda relação é apresentada examinando como a lei dos grandes números interage com o teorema ergódico de Birkhoff (assumindo ergodicidade). A lei dos grandes números pode ser obtida como corolário do teorema ergódico de Birkhoff. Entretanto, ela permite um novo ponto de vista, pois mantém as conclusões do teorema ergódico de Birkhoff (assumindo ergodicidade) mesmo no caso não-ergódico, ao custo de que haja um certo tipo de independência.
 
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LAVB.pdf (1.08 Mbytes)
Publishing Date
2021-01-20
 
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