Let (X, μ) be a probability space and T: X to X be an ergodic measure preserving transformation. Birkhoff ergodic theorem says that given an L1 function f on the space X, almost every x in X with respect to μ is generic, i.e. the average of f evaluated at the trajectory { Tnx: 0 ≤ n < N} converges to the integral of f as N tends to infinity. Given a μ measure zero set one can ask whether most points in this set are generic.
In this talk after reviewing some background material on ergodic theory we report some recent progress on this question in the setting of homogeneous space with respect to the action of diagonal elements.