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I have been reading the book "dynamical systems and semisimple groups an introduction". In this book, a point of a topological $G$-space $X$ is a periodic point if $G/G_x$ is compact, where $G$ is a topological group and $G_x$ is the isotropy group of $x$. And a point $x$ is recurrent if for each neightborhood $U$ of $x$ and each compact $K⊂G$ there is $g$ in the complement of $K$ such that $gx∈U$. According to the book, periodic points are recurrent. I do not see why this holds obviously.

I also find some other definitons of periodic points and recurrent points by using the term "syndetic". And I also do not kown how to prove that the definitions are equivalent?

Thank you!

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  • $\begingroup$ In addition, I think when $G$ is compact, then the periodic points may be not recurrent points. For example, let $G=S^1=\{z:|z|=1\}$ (with multiplication), then there is an action $S^1\times S^1\rightarrow S^1$ induced by $(z_1,z_2)\rightarrow z_1z_2$. Considering the point $1\in S^1$, we see that$G_1=\{id\}$. Thus $S^1/G_1$ is compact. So $1$ is a periodic point. Take $U=\{\exp(i\theta):|\theta|<\pi/8\}$ and $K=\{\exp(i\theta):|\theta|\le3\pi/4\}$, Then for each $z\in G-K$, we have $z\times 1=z\notin U$, which is a contradiction. Is there any thing wrong ? $\endgroup$
    – user57568
    Aug 29, 2014 at 2:48
  • $\begingroup$ In general, if $G$ is compact, with such definitions, every point is periodic and no point is recurrent! But in general, in dynamics one considers "large groups" acting on "small spaces". So, in most of the (dynamically interesting) cases, you can assume $G$ is non-compact. $\endgroup$
    – Alejandro
    Sep 1, 2014 at 22:01

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