Tamar Ziegler. Universal Characteristic Factors and Furstenberg Averages
Let X=(X0,μ,T) be an ergodic measure preserving system. For a natural number k we consider the averages (*) 1/N ∑n=1N ∏j=1k fj (T n aj x) where the functions fj are bounded, and aj are integers. A factor of X is characteristic for averaging schemes of length k (or k-characteristic) if for any non zero distinct integers a1,...,ak, the limiting L2(μ) behavior of the averages in (*) is unaltered if we first project the functions fj onto the factor. A factor of X is a k-universal characteristic factor (k-u.c.f) if it is a k-characteristic factor, and a factor of any k-characteristic factor. We show that there exists a unique k-u.c.f, and it has a structure of a (k-1)-step nilsystem, more specifically an inverse limit of (k-1)-step nilflows. Using this we show that the averages in (*) converge in L2(μ). This provides an alternative proof to the one given by Host and Kra in 2002.