Universal characteristic factors and Furstenberg averages

Abstract

Let X = ( X 0 , B , μ , T ) X=(X^0,\mathcal {B},\mu ,T) be an ergodic probability measure-preserving system. For a natural number k k we consider the averages (*) 1 N ∑ n = 1 N ∏ j = 1 k f j ( T a j n x ) \begin{equation*} \tag {*} \frac {1}{N}\sum _{n=1}^N \prod _{j=1}^k f_j(T^{a_jn}x) \end{equation*} where f j ∈ L ∞ ( μ ) f_j \in L^{\infty }(\mu ) , and a j a_j are integers. A factor of X X is characteristic for averaging schemes of length k k (or k k -characteristic) if for any nonzero distinct integers a 1 , … , a k a_1,\ldots ,a_k , the limiting L 2 ( μ ) L^2(\mu ) behavior of the averages in (*) is unaltered if we first project the functions f j f_j onto the factor. A factor of X X is a k k -universal characteristic factor ( k k -u.c.f.) if it is a k k -characteristic factor, and a factor of any k k -characteristic factor. We show that there exists a unique k k -u.c.f., and it has the structure of a ( k − 1 ) (k-1) -step nilsystem, more specifically an inverse limit of ( k − 1 ) (k-1) -step nilflows. Using this we show that the averages in (*) converge in L 2 ( μ ) L^2(\mu ) . This provides an alternative proof to the one given by Host and Kra.