I'm looking for a simple example in discrete dynamical systems whose periodic points set is not necessary closed.
I've seen some example in websites but they are not that simple and discrete.
Note that :
$(X,f)$ is a Dynamical System if $f:X \to X$ is a homeomorphism and $X$ is a compact metric space. \begin{align} Per(f):=\{x \in X ; f^n(x)=x ,\text{ for some } n \in \mathbb{Z}\} \end{align}
For example I found this example :
Let $X$ be the unit disk $\{z\in\Bbb C: |z|\le 1\}$ and $f:X\to X$, $x\mapsto xe^{|x|i}$. Then $Per(f)=\{x\in X: |x|/\pi\in\Bbb Q\}.$
But It's not discrete and also It's not that simple to me to see why the periodic points is that and why it is not closed.
Could you please help me find a simple example with this property in discrete dynamical systems ?