Title: Ergodic Theory on Foliations
Speaker: Professor Dan Rudolph
Speaker Info: Colorado State University
I will describe an approach to a very general ergodic theorem on foliations. This general picture builds on joint work with Elon Lindenstrauss. To begin we will define a Borel foliation of a Polish space X by R^n or Z^n. These are a generalization of the orbits of a free Borel action of these groups. We next show how to take any Borel probability measure and diffuse it onto the leaves of such a foliation, giving Borel leaf measures mu_x on each leaf. The goal of this plan then is to attempt to prove convergence of averages over large balls on leaves with respect to these leaf measures as the size of the ball grows. Both the Birkhoff and Hurewicz ergodic theorems are cases of this for invariant or nonsingular measures.Date: Tuesday, January 30, 2007
The main result I will describe is a fully general maximal lemma, using the theory of Besicovitch coverings. This reduces the question to whether or not a dense family of functions in L_1 exists for which one has convergence. I will describe the work of Feldman which is the best known on this issue. This is enough to prove the general result is true for one dimensional foliations.