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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.

David Chan
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  • Are these definiutions correct? I mean, with topologies on $G$ and $X$ you certainly want the action to be continuous, which makes every point recurrent (just try $g\approx e$) – Hagen von Eitzen Jun 08 '15 at 13:07
  • @HagenvonEitzen I believe that it must be not only $g \neq e$, but $g$ should be out of some small neighbourhood of $e$. As far as I remember it's done this way for flows. – Evgeny Jun 08 '15 at 20:54

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