发表机构
Faculty of Mathematics and Physics, Charles University（查理大学数学物理学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文探讨紧致可度量化空间的同胚群被稠密非闭子群取代时轨道等价关系分类复杂性的变化，分析三类子群及两类空间的超空间作用，发现其复杂性与闭子群不同。
AI 中文摘要
本文研究当紧致可度量化空间的全同胚群被一个稠密非闭子群取代时，自然轨道等价关系的分类复杂性如何变化。对于紧致空间$X$和子群$G \leq \mathcal{H}(X)$，我们考虑三个典型作用：在$\mathcal{H}(X)$上的左移作用、在$\mathcal{F}(X)$上的诱导超空间作用，以及在$G$上的共轭作用。我们首先分析递增区间同胚群$\mathcal{H}^+([0,1])$的子群，重点关注双利普希茨同胚、微分同胚和双绝对连续同胚。结果表明，与闭子群的行为相反，过渡到这些子群会严格增加相关分类问题的复杂性，或使其与对应全群关系不可比。第二部分，我们研究双绝对连续同胚在Cantor空间和希尔伯特立方体上的超空间作用，针对某些Borel概率测度进行分析，发现这些空间上也会出现类似行为。
英文摘要
In this paper, we study how the classification complexity of natural orbit equivalence relations changes when the full homeomorphism group of a compact metrizable space is replaced by a dense non-closed subgroup. For a compact space $X$ and a subgroup $G \leq \mathcal{H}(X)$, we consider three canonical actions: the left shift action on $\mathcal{H}(X)$, the induced hyperspace action on $\mathcal{F}(X)$, and the conjugation action on $G$ We first analyze subgroups of the group $\mathcal{H}^+([0,1])$ of increasing interval homeomorphisms, focusing on bi-Lipschitz homeomorphisms, diffeomorphisms, and bi-absolutely continuous homeomorphisms. We show that, in contrast to the behavior of closed subgroups, passing to these subgroups strictly increases the complexities of the associated classification problems or makes them incomparable with the corresponding full-group relations. In the second part, we investigate hyperspace actions of bi-absolutely continuous homeomorphisms on the Cantor space and the Hilbert cube with respect to some Borel probability measure and show that a similar behavior occurs on these spaces as well.
Comments49 pages
