平均维数与比较半径
Mean dimension and radius of comparison
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中文总结 AI 辅助
针对离散顺从群上满足一致Rokhlin性质和开集Cuntz比较的极小自由拓扑动力系统，证明了其平均维数与比较半径的相关结论，该结论也适用于带对角映射的单单位AH代数，研究基于OpenAI关于$\mathbb Z$作用的预印本及相关拓扑思想。
中文摘要 AI 辅助
考虑一个极小且自由的拓扑动力系统$(X, \Gamma)$，其中$\Gamma$是离散顺从群。假设$(X, \Gamma)$具有一致Rokhlin性质（URP）和开集的Cuntz比较（COS），则证明了$$\mathrm{rc}(\mathrm{C}(X) \rtimes \Gamma) = \frac{1}{2} \mathrm{mdim}(X, \Gamma).$$。该结论同样适用于带对角映射的单单位AH代数。两种情形的论证均基于OpenAI近期关于$\mathbb Z$作用的预印本及该论文中用到的拓扑思想。
英文摘要
Consider a minimal and free topological dynamical system $(X, Γ)$, where $Γ$ is a discrete amenable group. Assume $(X, Γ)$ has the uniform Rokhlin property (URP) and the Cuntz comparison of open sets (COS), then it is shown that $$\mathrm{rc}(\mathrm{C}(X) \rtimes Γ) = \frac{1}{2} \mathrm{mdim}(X, Γ).$$ The same statement also holds for unital simple AH algebras with diagonal maps. The arguments in both cases are based on the recent preprint of OpenAI dealing with $\mathbb Z$-actions and the topological ideas used in that paper.
