$(X,G)$ is a topological semi-group action, $G$ is a topological (abelian) semigroup, and $X$ is a Hausdorff space.
$x\in X$ is called almost periodic of $(X,G)$, if for any neibourhood $U$ of $x$, $\{g\in G: gx\in U\}$ is syndetic.
$x\in X$ is called uniformly recurrent of $(X,G)$, if for any neibourhood $U$ of $x$, there exist a compact subset $K$ of $G$, such that for any $g\in G$, we can find a $k\in K$ satisfying $gkx\in U$.
$S$ is called a syndetic subset of a topologcal semigroup $G$, if we can find a compact subset of $G$, such that for any $g\in G$, there exist an $s\in S$, and a $k\in K$ satisfying $g=sk$.
My question is: if $x$ is almost periodic, then $x$ must be unifomly recurrent?
