The rotation of the circle by a real number $\lambda$ is the function $f_\lambda:S^1 \to S^1$ defined by \begin{align} f_\lambda(x)=x+\lambda,\quad \text{mod}\,\mathbb{Z} \end{align} We know that if $\lambda$ is a rational number, then every point of $S^1$ is periodic and if $\lambda$ is irrational then $\text{Per(f)}=\emptyset$ and the orbit of every point in $S^1$ is dense in the whole space ($S^1$).
My question is about invertibility of this function. I think in the case the points rotated by the rational parameter the answer is positive, but I'm not sure about the "Irrational Rotation" ?