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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 $K\subset G$ there is $g$ in the complement of $K$ such that $gx\in U$. According to the book, periodic points are recurrent. I do not see why this holds obviously.

I have 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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The term "syndetic" seems completely new to me and hence unable to say anything about the equivalence of definitions

But in the first case, I think I have found some counter examples. Let G be the multiplicative group {1,-1} (undoubtedly it is a compact hausdorff second countable topological group) and X be S^1, i.e. the unit circle in R^2. Clearly X is a complete, second countable and hence separable metric space. We define the action by (1,x) goes to x and (-1,x) maps to -x for each x in S^1.

Since G is compact, any x in S^1 is a periodic point. We claim that it fails to be recurrent. Why? Take a very small arc around x which avoids -x as U and consider K={1}. Since -x does not belong to U, no point in the complement of K can bring x back inside U.

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