Let $T^n$ be the standard $n$-torus and $f:T^n\to K$ a surjective continuous mapping of $T^n$ onto a compact space. Let $g^t$ be a quasiperiodic action of the real line $R$ on $T^n$, i.e., $g^t(\varphi)=\varphi+\omega t$ where $\omega$ is a fixed vector with rationally independent components. Let also $G^t$ be a continuous action of $R$ on $K$. Suppose that $f\circ g^t=G^t$. Then $K$ is a torus of dimension no greater than $n$, and $G^t$ is quasiperiodic.
This is an easy theorem (well, easy modulo the Pontryagin duality needed to describe the compact subgroups of $T^n$), and I am sure that it is known in the literature. My question is: where is it present? I need a reference.
