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Universal Characteristic Factors and Furstenberg Averages | | Tamar Ziegler
; | | Date: |
12 Mar 2004 | | Subject: | Dynamical Systems MSC-class: 37A30;37A25 | math.DS | | Abstract: | Let X=(X^0,mu,T) be an ergodic measure preserving system. For a natural number k we consider the averages (*) 1/N sum_{n=1}^N prod_{j=1}^k f_j(T^{n a_j}x) where the functions f_j are bounded, and a_j are integers. A factor of X is characteristic for averaging schemes of length k (or k-characteristic) if for any non zero distinct integers a_1,...,a_k, the limiting L^2(mu) behavior of the averages in (*) is unaltered if we first project the functions f_j 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 L^2(mu). This provides an alternative proof to the one given by Host and Kra in 2002. | | Source: | arXiv, math.DS/0403212 | | Services: | Forum | Review | PDF | Favorites | |
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