If $\alpha \in \mathbb{R}\setminus \mathbb{Q}$ is an irrational number, then the rotation $X = (S^1, +\alpha)$ has "trivial" cohomology i.e. $$H^1(X) := C(X,\mathbb{C})/\beta C(X,\mathbb{C})$$ consists only of the class $[1]$ of the constant function (where $C(X,\mathbb{C})$ is the additive group of complex-valued continuous functions on $X$ and $\beta g(x) = g(x+\alpha)-g(x)$ is the coboundary map). My question is, does this property characterizes irrational rotations? I don't know if other minimal group rotations satisfy this.