On the norm convergence of non-conventional ergodic averages
On the norm convergence of non-conventional ergodic averages
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DOI：
10.1017/s014338570900011x
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发表时间：
2008-05
影响因子：
0.9
通讯作者：
Tim Austin
中科院分区：
文献类型：
--
作者：
Tim Austin
摘要：我们对下述非传统的遍历定理给出一个证明：如果对于\(i = 1,2,\cdots,d\)，\(T_i:\mathbb{Z}^r\curvearrowright (X,\Sigma,\mu)\)是可交换的保概率\(\mathbb{Z}^r\)作用，\((I_N)_{N\geq1}\)是\(\mathbb{Z}^r\)的子集的一个Følner序列，\((a_N)_{N\geq1}\)是\(\mathbb{Z}^r\)中的一个基点序列，并且\(f_1,f_2,\cdots,f_d\in L^{\infty}(\mu)\)，那么非传统的遍历平均
\[
\frac{1}{|I_N|}\sum_{n\in I_N + a_N}\prod_{i = 1}^d f_i\circ T_i^n
\]
在\(L^2(\mu)\)中收敛到某个极限，该极限不依赖于\((a_N)_{N\geq1}\)或\((I_N)_{N\geq1}\)的选择。这个结果的主要情形，即\(r = 1\)且是平均集的标准序列，首先由陶哲轩证明，此前有孔泽（Conze）和莱西涅（Lesigne）、弗斯滕伯格（Furstenberg）和韦斯（Weiss）、张益唐、霍斯特（Host）和克拉（Kra）、弗兰齐基纳基斯（Frantzikinakis）和克拉以及齐格勒（Ziegler）对各种更特殊的情形及相关结果进行了分析。虽然陶哲轩的证明基于转化为一个有限性问题，但我们仅使用经典遍历理论的技术，从而对他的结果给出了一个新的证明。
Abstract We offer a proof of the following non-conventional ergodic theorem: If Ti:ℤr↷(X,Σ,μ) for i=1,2,…,d are commuting probability-preserving ℤr-actions, (IN)N≥1 is a Følner sequence of subsets of ℤr, (aN)N≥1 is a base-point sequence in ℤr and f1,f2,…,fd∈L∞(μ) then the non-conventional ergodic averages \[ \frac {1}{|I_N|}\sum _{n \in I_N + a_N} \prod _{i=1}^df_i\circ T_i^n \] converge to some limit in L2(μ) that does not depend on the choice of (aN)N≥1 or (IN)N≥1. The leading case of this result, with r=1 and the standard sequence of averaging sets, was first proved by Tao, following earlier analyses of various more special cases and related results by Conze and Lesigne, Furstenberg and Weiss, Zhang, Host and Kra, Frantzikinakis and Kra and Ziegler. While Tao’s proof rests on a conversion to a finitary problem, we invoke only techniques from classical ergodic theory, so giving a new proof of his result.