Given a dynamical system $(X,G)$,
A point $x\in X$ is called recurrent, if for any neibourhood $U$ of $x$, there exist a $g\in G$, $g\neq e$ such that $gx\in U$.
If $G$ is a topological group and $X$ is a topological sapce, then we can easily conclude that if $x$ is a recurrent point, then any point $y$ in the orbit $Gx$ is a recurrent point, too.
But if $G$ is a topological semi-group and $X$ is a topological sapce, is this always right? That is to say, if $x$ is a recurrent point, then any point $y$ in the orbit $Gx$ is a recurrent point, too?
$G$ is not always an Abelian group.
If If $G$ is a topological (semi-)group and $X$ is a topological sapce, if $x$ is a recurrent point, then any point $y$ in the closure of the orbit $Gx$ is a recurrent point, too?
Thanks a lot.