度量空间上一些上同调\(C^0\)稳定连续群作用的分类
Classification of some cohomologically $C^0$-stable continuous group actions on metric spaces
AI总结：
研究度量空间上连续群作用的上同调\(C^0\)稳定性，将Katok关于同胚映射的分类扩展到有限生成群作用，证明作用是上同调\(C^0\)稳定当且仅当其像为有限群，无需顺从性假设，还讨论了光滑范畴及相关猜想。
AI中文摘要：
在Katok之后，紧致度量空间的同胚映射\(f\colon M\to M\)若其实\(C^0\)上边缘空间在\(C^0(M)\)中是闭的，则称其为上同调\(C^0\)稳定的。Kocsard证明当且仅当\(f\)是周期的时才是这种情况。我们将该分类扩展到任意有限生成群的作用：作用\(\alpha: G \to Homeo(M)\)是上同调\(C^0\)稳定的当且仅当像\(\alpha(G)\)是有限群。特别解决了由有限多个可交换同胚生成的\(\mathbb Z^k\)作用的情况。不需要顺从性假设，解释了非顺从作用在Hölder、Sobolev和\(L^2\)范畴中产生上同调稳定性的谱隙现象为何在一致范数下不可见。还讨论了真正不同的光滑范畴，并陈述了关于无周期轨道的光滑圆微分同胚的\(\mathbb Z^k\)作用的上同调\(C^\infty\)稳定性的猜想，将该问题与Moser、Fayad - Khanin、Avila - Kocsard和Petković的工作联系起来。
英文摘要：
After Katok, a homeomorphism $f\colon M\to M$ of a compact metric space is said to be cohomologically $C^0$-stable if its space of real $C^0$-coboundaries is closed in $C^0(M)$. Kocsard proved that this is the case if and only if $f$ is periodic. We extend the classification to actions of arbitrary finitely generated groups: an action $α: G \to Homeo(M)$ is cohomologically $C^0$ - stable if and only if the image $α(G)$ is a finite group. In particular this settles the case of $\mathbb Z^k$-actions generated by finitely many commuting homeomorphisms. Notably, no amenability assumption is needed: we explain why spectral-gap phenomena for non-amenable actions, which do produce cohomological stability in Hölder, Sobolev and $L^2$ categories, are invisible to the uniform norm. We also discuss the genuinely different smooth category and state a conjecture regarding cohomological $C^\infty$-stability of $\mathbb Z^k$-action by smooth circle diffeomorphisms without periodic orbits, connecting the problem with works of Moser, Fayad-Khanin, Avila-Kocsard and Petković.
