Definition: Let $(X,B,\mu)$ and $(Y,C,\nu)$ be probability spaces where $B$ and $C$ are the Borel $\sigma$- Algebras of their respective spaces. Let $g:X \rightarrow X$ and $h: Y \rightarrow Y$ be measure preserving transformations. $h$ is a measure preserving factor of $g$ if $\exists \text{ measurable } F:X \rightarrow Y$ ($F$ is called factor map) such that for $\mu$-a.e. $x \in X$ holds $F(g(x)) = h(F(x))$ and $\forall A \in C$ $\nu(A) = \mu(F^{-1}(A))$.
Now there is following problem to solve: Let $\alpha = \frac{p}{q}$ and $\beta = \frac{r}{s}$ in $\mathbb{Q}$ with $(p,q)=1$, $(r,s)=1$, $(q,s) = 1$ and $q \neq s$. Let $h: \mathbb{R}/\mathbb{Z} \rightarrow \mathbb{R}/\mathbb{Z}$, $x \mapsto x + \alpha$ and $g:\mathbb{R}/\mathbb{Z} \rightarrow \mathbb{R}/\mathbb{Z}$, $x \mapsto x + \beta$ be circle rotations. Show that there is no factor map between $g$ and $h$.
My attempt. By contradiction I assumed such a factor map $F$ exists. Then I could show that for $\mu$ - a.e. $x \in X$ holds $F(x+\frac{1}{s}) = F(x+\frac{1}{q}) = F(x)$ and I tried to find a contradiction to measure preserving from here. But I failed.
Could someone give a hint or tell me if I am on the wrong path?
By contradiction assume a factor map $F: \mathbb{R}/\mathbb{Z} \rightarrow \mathbb{R}/\mathbb{Z}$ between $g$ and $h$ exists.
Observe $g^q = Id_{\mathbb{R}/\mathbb{Z}}$. Thus by $q$ times applying $F(g(x)) = h(g(x))$ for $\mu$ - a.e $x \in \mathbb{R}/\mathbb{Z}$ we get $F(x) = F(g^q(x)) = h^q(F(x))$ for $\mu$ - a.e $x \in \mathbb{R}/\mathbb{Z}$.
Using that $F(\mathbb{R}/\mathbb{Z}) = \mathbb{R}/\mathbb{Z}$ $\mu$-a.e. we get that $x = h^q(x) = x+q \frac{r}{s}$ for $\mu$-a.e $x \in \mathbb{R}/\mathbb{Z}$. Contradiction, since $q \frac{r}{s} \notin \mathbb{Z}$!
We conclude there is no factor map between $h$ and $g$.