保留代数结构的令人愉快的扩展，II
Pleasant extensions retaining algebraic structure, II
摘要：
在本文中，我们结合（arXiv：0905.0518）中开发的通用工具和Furstenberg和Weiss，Host和Kra以及Ziegler早期关于一维非常规遍历平均的工作中的几个想法，来研究$f_1,f_2,f_3 \in L^\infty(μ)$的平均$\frac{1}{N}\sum_{n=1}^N(f_1\circ T^{n\bf{p}_1})(f_2\circ T^{n\bf{p}_2})(f_3\circ T^{n\bf{p}_3})$，该$\frac{1}{N}\sum_{n=1}^N(f_1\circ T^{n\bf{p}_1})(f_2\circ T^{n\bf{p}_2})(f_3\circ T^{n\bf{p}_3})$与位于一般位置的三个方向$\bf{p}_1,\bf{p}_2,\bf{p}_3 \in \mathbb{Z}^2$以及$0 \in \mathbb{Z}^2$相关联。我们将展示如何构建一个“令人愉快的”扩展的初始给定的$\mathbb{Z}^2$ -系统，这些平均承认特征因子与一个非常具体的描述，涉及一维各向同性因子和两步pro-nilsystems。我们还使用这种分析来构造令人愉快的扩展，然后证明规范收敛的多项式非常规遍历平均$\frac{1}{N}\sum_{n=1}^N(f_1\circ T_1^{n^2})(f_2\circ T_1^{n^2}T_2^n)$相关的两个交换变换$T_1$，$T_2$。
英文摘要：
In this paper we combine the general tools developed in (arXiv:0905.0518) with several ideas taken from earlier work on one-dimensional nonconventional ergodic averages by Furstenberg and Weiss, Host and Kra and Ziegler to study the averages $\frac{1}{N}\sum_{n=1}^N(f_1\circ T^{n\bf{p}_1})(f_2\circ T^{n\bf{p}_2})(f_3\circ T^{n\bf{p}_3})$ for $f_1,f_2,f_3 \in L^\infty(μ)$ associated to a triple of directions $\bf{p}_1,\bf{p}_2,\bf{p}_3 \in \mathbb{Z}^2$ that lie in general position along with $0 \in \mathbb{Z}^2$. We will show how to construct a `pleasant' extension of an initially-given $\mathbb{Z}^2$-system for which these averages admit characteristic factors with a very concrete description, involving one-dimensional isotropy factors and two-step pro-nilsystems. We also use this analysis to construct pleasant extensions and then prove norm convergence for the polynomial nonconventional ergodic averages $\frac{1}{N}\sum_{n=1}^N(f_1\circ T_1^{n^2})(f_2\circ T_1^{n^2}T_2^n)$ associated to two commuting transformations $T_1$, $T_2$.
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