Let be $T:X\to X$ a topological dynamical system, $X$ a compact space and $T$ is also a isometry. Let be $\mathcal{R}(T)$ the chain recurrent set of $T$.
Theorem: $\mathcal{R}(T)=X$
There is a simple demonstration of this fact? I know a proof of this result, which I read in the Terence Tao's blog:
http://terrytao.wordpress.com/2008/01/24/254a-lecture-6-isometric-systems-and-isometric-extensions/
but hoped that this demonstration was simpler.
I also would like a simple example, if possible, a compact dynamical system $ T $ satisfying:
$$ \varnothing \neq \overline{Per(T)}\subsetneq \mathcal{R}(T)=X $$
Obs: $Per(T)$= Periodic points of $T$