I am just starting to study dynamical systems so this might not be a particularly complicated question, but I've been scratching my head for days without much success.
I am looking at the map of rotations of the circle $R_\alpha:S^1\rightarrow S^1$, $R_\alpha(x)=x+\alpha$ mod 1, with irrational $\alpha$. My aim is to show that $R_{2\alpha}$ is a factor of $R_\alpha$ (as in, there is a topological semi-conjugacy $\psi$ with $\psi\circ R_\alpha=R_{2\alpha}\circ\psi$), but that the converse is not true (as in, there is no topological semi-conjugacy $\psi$ with $\psi\circ R_{2\alpha}=R_{\alpha}\circ\psi$). As a hint, I'm told to use the fact that all orbits under an irrational rotation are dense.
The first part was easy, just taking $\psi$ to be the doubling map $\psi(x)=2x$ mod 1 works fine. I'm struggling with the second part, especially because the hint confuses me. Every point in the circle has a dense orbit under an irrational rotation, but both $R_\alpha$ and $R_{2 \alpha}$ are irrational rotations, so I can't see how that would help here. I've tried starting from the definition of dense orbit and somehow using the semi-conjugacy property to reach some sort of contradiciton, but so far I've found nothing meaningful. I'd really appreciate any hints to get me started on this.
Thanks!
Rotations of the circle are uniquely ergodic, minimal and preserve the usual metric.
Suppose there is a conjugacy $\Phi$ between two rotations $R_\alpha$ and $R_\beta$. By potentially composing $\Phi$ with a rotation we can w.l.o.g. assume that $\Phi(1)=1$. Let us count how often we cross the point $1$. For that we consider the hitting times of the intervals $(\alpha,1]$ and $(\beta,1]$ respectively. By the Birkhoff ergodic theorem the density of those hitting times is $\alpha$ and $\beta$ respectively. (Look up the Keyword "Sturmian Sequences" for more information)
Proof that a conjugacy must preserve those times of crossing $1$.
Conclude that $\alpha = \beta$.