Ergodic theorems from probabilistic and topological viewpoints - Boas, Lucas Amorim Vilas (2018)
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.