Let $A$ be a finite set endowed with the discrete topology. Then, the pair $(A^{\mathbb{Z}}, \sigma)$ is said to be the full shift over the alphabet $A$ where $A^{\mathbb{Z}}$ is endowed with the product topology and $\sigma:A^{\mathbb{Z}} \to A^{\mathbb{Z}}$ is the shift map $\left((x_i)_{i \in \mathbb{Z}} \mapsto (x_{i+1})_{i \in \mathbb{Z}}\right)$. A shift space $X$ over the alphabet $A$ is a closed subset of $A^{\mathbb{Z}}$ invariant by the shift map. A point $x \in X$ is said to be periodic if $\sigma^n(x)=x$ for some posite integer $n$.
I am trying, with no success so far, to construct a shift space $X$ with no periodic points. But I couldn't also prove that there is not such a shift space. So that bring my question: is there a shift space over a finite alphabet with no periodic points? If yes, could you give me an example? If not, could you give me some hints on how to prove that?
Thank you!