通过幂零系统的局部刚性研究幂零系统的因子闭包

中文摘要 AI 辅助

幂零系统是在遍历理论中通过研究多重遍历平均自然产生的对象。Host-Kra结构定理指出，每个遍历s步幂零系统的因子仍是s步幂零系统，此前尚无独立于该结构定理的证明。本论文仅使用幂零系统的内在论证给出了这样的证明，主要新内容是关于幂零系统自联结的局部刚性结果：每个充分接近对角联结的遍历自联结必然是自同构的图联结。我们通过证明幂零流形不能包含任意小的非平凡子幂零流形，从几何上证明了这一结果。我们还为可迁拓扑幂零系统的类似因子闭包结果提供了直接证明，该结果此前是Host-Kra-Maass拓扑结构定理的推论。最后，我们详细阐述了Tao从Gowers范数的Green-Tao-Ziegler逆定理直接推导弱结构定理的过程。结合幂零系统的因子闭包，这给出了Host-Kra结构定理的新证明。本论文的主要新结果也出现在与Asgar Jamneshan的合作论文（arXiv:2604.02089）中，本论文提供了更自包含的处理，包括动力系统和幂零系统的必要背景，以及Tao论证的全部细节。

英文摘要

Pro-nilsystems arise naturally in ergodic theory through the study of multiple ergodic averages. The Host-Kra structure theorem implies that every factor of an ergodic $s$-step pro-nilsystem is again an $s$-step pro-nilsystem. No proof of this fact independent of the structure theorem was previously known. In this thesis, we give such a proof, using only arguments intrinsic to nilsystems. The main new ingredient is a local rigidity result for self-joinings of nilsystems: every ergodic self-joining sufficiently close to the diagonal joining is necessarily the graph joining of an automorphism. We prove this geometrically by showing that nilmanifolds cannot contain arbitrarily small nontrivial subnilmanifolds. We also obtain a direct proof of the analogous factor-closure result for transitive topological pro-nilsystems, previously known as a consequence of the Host-Kra-Maass topological structure theorem. Finally, we give a detailed account of Tao's deduction of a weak structure theorem directly from the Green-Tao-Ziegler inverse theorem for the Gowers norms. Combined with the factor-closure of pro-nilsystems, this yields a new proof of the Host-Kra structure theorem. The principal new results of this thesis also appear in a joint paper with Asgar Jamneshan (arXiv:2604.02089). This thesis provides a more self-contained treatment, including the necessary background on dynamical systems and nilsystems, as well as full details of Tao's argument.

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Master's thesis, University of Bonn, 2026, 98 pages