In this paper, it is shown that for a minimal system (X, G), if H is a normal subgroup of G with finite index n, then X can be decomposed into n components of closed sets such that each component is minimal under H-action. Meanwhile, we prove that for a residual set of points in a minimal system with finitely many commuting homeomorphisms, the set of return times to any non-empty open set contains arbitrarily long geometric progressions in multidimension, extending a previous result by Glasscock, Koutsogiannis and Richter.

中文翻译：

---

动态综合集合的多维乘法组合性质

在本文中，证明了对于最小系统（X，  G），如果H是具有有限索引n的G的一个正规子群，则X可以分解为n个封闭集合的分量，使得每个分量在H下都是最小的-行动。同时，我们证明了对于具有有限多个通态同胚的最小系统中的一组剩余点，返回到任何非空开放集的返回时间集包含任意长的多维几何级数，从而扩展了Glasscock，Koutsogiannis和里希特