交换作用的联合点远端性与结构理论
Joint Point-Distality and structure theory for commuting actions
- Tel Aviv University（特拉维夫大学）
- Dalian University of Technology（大连理工大学）
- Institute of Mathematics, Polish Academy of Sciences（波兰科学院数学研究所）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文证明交换极小作用的点远端性在联合作用下保持，构造公共高近端扩张，比较子作用与联合作用的典范结构，并给出联合Furstenberg塔的递归描述及严格性示例。
AI中文摘要：
设$G$和$H$在紧致可度量化空间$X$上交换地且极小地作用。我们证明，如果两个子作用是点远端的（等价地，HPI），则它们生成的联合作用也是点远端的。我们还构造了一个公共的高近端扩张，在该扩张上两个子作用都是严格HPI的。作为推论，混合积和线性迭代的传递性在点远端范畴中提升为极小性。然后我们比较两个子作用的典范结构与联合作用的结构。极大高近端运算一致，而公共因子上的极大联合等距因子是两个子作用等距因子的交。这给出了联合Furstenberg塔的递归描述。最后，我们构造$\mathbb T^3$上交换的极小远端同胚，其典范Furstenberg塔不同，表明交公式可以是严格的。
英文摘要：
Let $G$ and $H$ act commutatively and minimally on a compact metrizable space $X$. We prove that if the two subactions are point-distal, equivalently HPI, then the action generated by them is again point-distal. We also construct a common highly proximal extension on which both subactions are strictly HPI. As consequences, transitivity of mixed products and linear iterates upgrades to minimality in the point-distal category. We then compare the canonical structure of the two subactions with that of the joint action. The maximal highly proximal operations coincide, whereas the maximal joint isometric factor over a common factor is the meet of the two subaction-isometric factors. This gives a recursive description of the joint Furstenberg tower. Finally, we construct commuting minimal distal homeomorphisms of $\mathbb T^3$ whose canonical Furstenberg towers are different, showing that the meet formula can be strict.
