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 $T$