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I'm currently studying the topic of endomorphisms and translations of topological groups within dynamical systems, and came across the following exercise:

Show that for any $k \in \mathbb{Z}, k \neq 0$, there is a continuous semi-conjugacy from $R_{\alpha}$ to $R_{k \alpha}$.

Here, $R_{\alpha}$ is the rotation of $S^1$ by angle $2 \pi \alpha$ for $\alpha \in \mathbb{R}$, so:

$ R_\alpha x = x + \alpha \mod 1$.

I have been struggling with dynamical systems in general, so any help or suggestion regarding this question would be much appreciated.

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You need to construct a continuous surjective (not necesserily injective) map $h$ from $S^1$ to itself such that $h \circ R_{\alpha} = R_{k\alpha} \circ h$, that is, transforms an $\alpha$ rotation into a $k$ times $\alpha$ rotation. An intuitive guess for that would be $h(x) = kx$ since it scales things up by a factor of $k$, and it indeed works:

$h(R_{\alpha}(x)) = h(x+\alpha) = kx + k\alpha = R_{k\alpha}(kx) = R_{k\alpha}(h(x))$

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