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!
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.