Showing the ergodicity of a rotation on the unit sphere

Question

Consider the rotation $R_\alpha(z) = \alpha z, R_\alpha : S^1 \to S^1$. Show that

$R_\alpha$is ergodic with respect to Haar measure on $S^1$ $\iff$ $\alpha$ is not a root of unity.

I don't know how to show the $\Leftarrow$ implication. Can someone please help me? I know I have to consider some Fourier series, but I fail to solve this exercise.

Thank you!

Answer 1

Hint:

Show that the Fourier coefficients $c_k(\psi)$ of any $L^2$ function $\psi$ satisfy $c_k(\psi\circ R_\alpha)=\alpha^kc_k(\psi)$ for $k\in\mathbb Z$. After that use the fact that $\alpha^k\ne1$ (notice that if $\psi$ is invariant, the Fourier coefficients of $\psi$ and $\psi\circ R_\alpha$ are the same for all $k$).