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Title:听A story about pointwise ergodic theorems

Abstract:Pointwise ergodic theorems provide a bridge between the global behaviour of the dynamical system and the local combinatorial statistics of the system at a point. Such theorem have been proven in different contexts, but typically for actions of semigroups on a probability space. Dating back to Birkhoff (1931), the first known pointwise ergodic theorem states that for a measure-preserving ergodic transformation T on a probability space, the mean of a function (its global average) can be approximated by taking local averages of the function at a point x over finite sets in the forward-orbit of x, namely {x, Tx, ..., T^n x}. Almost a century later, we revisit Birkhoff's theorem and turn it backwards, showing that the averages along trees of possible pasts also approximate the global average.听This backward theorem for a single transformation surprisingly has applications to actions of free groups, which we will also discuss.听This is joint work with Jenna Zomback.

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Colloquium Colloque des sciences math茅matiques du Qu茅bec
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