Analysis Seminar

In this talk I will consider actions of non-abelian groups on n-dimensional tori, explain the notions of stiffness and stationary measures, and show how under fairly general assumptions stationary measures can be classified. A key ingredient is a result of Bourgain related to the sum product phenomena on the reals. In particular, we prove the following: let A, B be two non commuting 2x2 integer matrices of determinant one. Consider a random product X_r....X_1.y where y is a point in the two torus. We show that as r--> infinity this random product is distributed in an increasingly uniform manner. Based on joint work with Bourgain, Furman and Mozes.