“保留代数结构的令人愉快的扩展”的另一种结尾
An alternative ending to "Pleasant extensions retaining algebraic structure''
摘要：
论文的高潮（arXiv：0905.0518，arXiv：0910.0909）是证明了与任意概率保持\bbZ^2 -系统（X，μ，T_1，T_2）相关联的二次非常规遍历平均\frac{1}{N} \sum_{n=1}^N（f_1 \circ T_1 ^{n^2}）（f_2 \circ T_1^{n^2}T_2^n）\quad \quad f_1，f_2 \in L^ \infty（μ）在$L^2(μ)$中的范数收敛。这是Bergelson-Leibman猜想关于多项式非常规遍历平均范数收敛的一个特例。这一证明依赖于一些新的机器，用于扩展概率保持$\bbZ^d$系统，以获得各种非常规平均值的简化渐近行为。这个机器的引擎是由一些详细的结构定理的“特征因子”，可用于一些这样的平均上升到一个适当的扩展系统后。然而，这些新的结构定理经历了两个不同的发展阶段，分离的发现，一些新的技术成果摩尔的上同调理论的局部紧群（arXiv：1004.4937）。这一发现使主结构定理（（arXiv：0905.0518）中的定理1.1）得到了显著的改进，这反过来又提供了一个大大缩短的收敛性证明。然而，由于使用原始结构定理证明收敛性需要一些完全不同的想法，这些想法现在在其他论文中没有，我在这里记录了它，以防它有一些独立的兴趣。
英文摘要：
The culmination of the papers (arXiv:0905.0518, arXiv:0910.0909) was a proof of the norm convergence in $L^2(μ)$ of the quadratic 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)\quad\quad f_1,f_2\in L^\infty(μ) associated to an arbitrary probability-preserving \bbZ^2-system (X,μ,T_1,T_2). This is a special case of the Bergelson-Leibman conjecture on the norm convergence of polynomial nonconventional ergodic averages. That proof relied on some new machinery for extending probability-preserving $\bbZ^d$-systems to obtain simplified asymptotic behaviour for various nonconventional averages such as the above. The engine of this machinery is formed by some detailed structure theorems for the `characteristic factors' that are available for some such averages after ascending to a suitably-extended system. However, these new structure theorems underwent two distinct phases of development, separated by the discovery of some new technical results in Moore's cohomology theory for locally compact groups (arXiv:1004.4937). That discovery enabled a significant improvement to the main structure theorem (Theorem 1.1 in (arXiv:0905.0518)), which in turn afforded a much shortened proof of convergence. However, since the proof of convergence using the original structure theorem required some quite different ideas that are now absent from these other papers, I have recorded it here in case it has some independent interest.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
