紧度量空间同胚群中的Rokhlin性质
Rokhlin properties in homeomorphism groups of compact metric spaces
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中文总结 AI 辅助
本文利用组合合并方法，刻画紧度量空间同胚群的Rokhlin性质，重新证明Hjorth定理并确立两类扇的通有同胚存在性，还发现其通有同胚无Li-Yorke对、拓扑熵为零。
中文摘要 AI 辅助
我们利用受Fraïssé理论启发的组合合并方法，开发了一套研究紧度量空间的一般框架。该方法将Rosendal的通有共轭类准则与由Bartoš-Bice-Vignati对偶性产生的紧空间的组合编码相结合。我们引入了一种新的合并性质，称为弱极小合并，该性质在编码同胚局部动力学数据的合适的范畴中表述，由此得到了Rokhlin性质和强Rokhlin性质的刻画，即紧度量空间同胚群中稠密且通有共轭类的存在性。作为应用，我们重新证明了Hjorth的定理，即Homeo₊([0,1])存在一个通有同胚，并确立了Cantor扇和Lelek扇的通有同胚的存在性。此外，我们证明这些空间的通有同胚没有Li-Yorke对，因此通有地具有零拓扑熵。更广泛地说，本文在拓扑群论和拓扑动力学中的组合合并性质与通有现象之间建立了新的联系。
英文摘要
We develop a general framework for studying generic homeomorphisms of compact metric spaces using combinatorial amalgamation methods inspired by Fraïssé theory. Our approach combines Rosendal's criterion for comeager conjugacy classes with combinatorial codings of compact spaces arising from the Bartoš-Bice-Vignati duality. We introduce a new amalgamation property, called weak minimal amalgamation, formulated in suitable paracategories encoding local dynamical data of homeomorphisms. This yields characterizations of the Rokhlin and the strong Rokhlin properties, i.e., the existence of dense and comeager conjugacy classes in homeomorphism groups of compact metric spaces. As applications, we reprove Hjorth's theorem that $\mbox{Homeo}_+([0,1])$ admits a generic homeomorphism, and establish the existence of generic homeomorphisms for the Cantor fan and the Lelek fan. Moreover, we show that generic homeomorphisms of these spaces have no Li--Yorke pairs, and therefore generically have zero topological entropy. More broadly, the paper develops new connections between combinatorial amalgamation properties and generic phenomena in topological group theory and topological dynamics.
