On $sncc$-Inheritance of Pointwise Almost Periodicity in Flows - Dai, Xiongping (Topology Proceedings, 2024)
Let $H$ be a subnormal co-compact closed subgroup of a Hausdorff topological group $T$ and $X$ a compact Hausdorff space. We prove the inheritance theorem: A point of $X$ is almost periodic (a.p.) for $T \curvearrowright X$ if and only if it is a.p. for $H \curvearrowright X$. Moreover, if $T \curvearrowright X$ is minimal with $H \triangleleft T$, then $\mathscr{O}_H: X → 2^X, x \mapsto \overline{H x}$ is a continuous mapping, and $T \curvearrowright X / H$ is an a.p. nontrivial factor of $T \curvearrowright X$ if and only if $T \curvearrowright X × T / H$ is not minimal.