$\mathbb{Z}$ action on the unit circle $S^1$

Question

Let $\alpha$ be an irrational number and consider the $\mathbb{Z}$ action on the unit circle $S^1$ given by $e^{2\pi ix}.n=e^{2\pi i(x+n\alpha)}, n\in \mathbb{Z}$. Can we get an example of a subspace of $S^1$ that is invariant under the restricted action by $\mathbb{N}$?