令人愉快的扩展保留代数结构，我
Pleasant extensions retaining algebraic structure, I
摘要：
在最近的两篇论文中，我们介绍了构造概率保持系统$T:\mathbb{Z}^d\curvearrowright (X,μ)$的扩展的一些新技术，该系统具有某些期望的性质，并与某些相关的非常规遍历平均的渐近行为相联系。本文件是两份文件中的第一份，将探讨这些想法的各种改进和扩展。第一部分致力于一些更通用的机器，用于构建可用于恢复各种早期结果的扩展。它还包含两个相对简单的新的应用，这一机制的研究某些家庭的非常规平均，一个在离散和一个在连续时间（收敛是一个新的结果，后者）。在即将到来的第二部分（arXiv：0910.0907）中，我们将介绍当方向$\bf{p}_1$、$\bf{p}_2$、\ldots、$\bf{p}_k \in \mathbb{Z}^d$不假定为线性无关时描述线性非常规平均值$\frac{1}{N}\sum_{n=1}^N \prod_{i=1}^kf_i\circ T^{n\bf{p}_i}$的特征因子和极限的问题，并在k = 3、d = 2和任何一对方向为线性无关的情况下提供相当详细的解。这将被用来证明在$L^2(μ)$的二次非常规平均$\frac{1}{N}\sum_{n=1}^N (f_1\circ T_1^{n^2})(f_2\circ T_1^{n^2}T_2^n)$的收敛。
英文摘要：
In two recent papers we introduced some new techniques for constructing an extension of a probability-preserving system $T:\mathbb{Z}^d\curvearrowright (X,μ)$ that enjoys certain desirable properties in connexion with the asymptotic behaviour of some related nonconventional ergodic averages. The present paper is the first of two that will explore various refinements and extensions of these ideas. This first part is dedicated to some much more general machinery for the construction of extensions that can be used to recover various earlier results. It also contains two relatively simple new applications of this machinery to the study of certain families of nonconventional averages, one in discrete and one in continuous time (convergence being a new result for the latter). In the forthcoming second part (arXiv:0910.0907) we will introduce the problem of describing the characteristic factors and the limit of the linear nonconventional averages $\frac{1}{N}\sum_{n=1}^N \prod_{i=1}^kf_i\circ T^{n\bf{p}_i}$ when the directions $\bf{p}_1$, $\bf{p}_2$, \ldots, $\bf{p}_k \in \mathbb{Z}^d$ are not assumed to be linearly independent, and provide a fairly detailed solution in the case when k = 3, d = 2 and any pair of directions is linearly independent. This will then be used to prove the convergence in $L^2(μ)$ of the quadratic nonconventional averages $\frac{1}{N}\sum_{n=1}^N (f_1\circ T_1^{n^2})(f_2\circ T_1^{n^2}T_2^n)$.
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