极小零旋转的多项式对角轨道闭包的连通性及其应用
Connectedness of polynomial diagonal orbit closures for minimal nilrotations and applications
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中文总结 AI 辅助
该研究证明了极小零旋转的多项式对角轨道闭包的连通性，解决了相关猜想，结合等价定理得到拓扑动力学中多项式多重回归的结果，还建立了测度论对应结论并构造了反例。
中文摘要 AI 辅助
针对紧连通幂零流形上的极小零旋转，我们证明了：对任意一组在原点处消失的整系数多项式，其对应的多项式对角轨道闭包是连通的。这解决了Glasscock、Koutsogiannis、Le、Moreira、Richter和Robertson提出的一个猜想。结合他们的等价定理，若变换的对应幂是极小的，我们的结果可得到拓扑动力学中每个指定剩余类的多项式多重回归。此外，我们独立建立了该回归现象的测度论对应结果。最后，我们构造了一个完全极小幂零系统，其中Leibman提出的下中心序列恒等式不成立。
英文摘要
For a minimal nilrotation on a compact connected nilmanifold, we prove that the polynomial diagonal orbit closure associated with any finite family of polynomials with integer coefficients vanishing at the origin is connected. This resolves a conjecture of Glasscock, Koutsogiannis, Le, Moreira, Richter, and Robertson. Combined with their equivalence theorem, our result yields polynomial multiple recurrence in every prescribed residue class in topological dynamics, provided that the corresponding power of the transformation is minimal. Furthermore, we independently establish the measure-theoretic counterpart of this recurrence phenomenon. Finally, we construct a totally minimal nilsystem for which the lower central series identity proposed by Leibman fails.
