Colloquium: Tuesday, March 22, 2022. Speaker: Speaker: Anush Tserunyan (McGill). Title: “A story about pointwise ergodic theorems”.

Will be on Zoom Time: 14:00

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 theorems 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, T x, …, T^nx}. 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. This backward theorem for a single transformation surprisingly has applications to actions of free groups, which we will also discuss. This is joint work with Jenna Zomback.