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 scalar multiples 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? More precisely: let $X$ be compact metrizable and $T\colon X\to X$ an homeomorphism such that $(X,T)$ is minimal and $H^1(X,T)$ trivial (in the sense of above). Then, is $(X,T)$ conjugated, in some sense, to $(S^1,+\alpha)$, for some $\alpha$? or at least to a more general group rotation?