On sncc-inheritance of pointwise almost periodicity in flows

Abstract

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 $T\curvearrowright X$ iff it is a.p. for $H\curvearrowright X$. Moreover, if $T\curvearrowright X$ is minimal with $H\lhd T$, then $\mathscr{O}_H\colon X\rightarrow2^X$, ${x\mapsto\overline{Hx}}$ is a continuous mapping, and, $T\curvearrowright X/H$ is an a.p. nontrivial factor of $T\curvearrowright X$ iff $T\curvearrowright X\times T/H$ is not minimal.