Zoltán Buczolich (Eötvös Loránd University, Budapest)
Kakutani-Rokhlin towers, rotations, ergodic averages
Abstract:
In this course first I plan to cover some classical results in Ergodic Theory and the Ornstein-Weiss Kakutani-Rokhlin tower theorem for amenable groups. After this we will consider ergodic averages of measurable functions along rotations of the circle. Using the Ornstein-Weiss theorem for the free Z^2 action generated by two independent irrational rotations, alpha and beta, we will see that there might be some measurable functions for which the ergodic averages converge but the alpha averages to zero and the beta averages to one. If there are many rotations (positive Lebesgue measure) for which the ergodic averages converge then the measurable function should be in L^1. We will discuss an example of a non-L^1 measurable function for which there is a set of Hausdorff dimension one of alphas for which the ergodic averages converge. If time permits some related results for rotations of compact Abelian groups might also be discussed.