I'm taking a class on basic Ergodic theory and I was asked the following question:
"What is the spectral measure of operator, induced by rotation of acircle? Baker's map?"
More explicitly: let us consider rotation $T$ of the unit circle $\mathbb T$ with normed Lebesgue measure. Let's denote induced Koopman operator by $U_T$. It is unitary operator.
$$U_T:L^2(\mathbb T)\longrightarrow L^2(\mathbb T)\ \ \ \ \ \ \ \ \ (U_T f)(x) =f(Tx) $$
Now recall
Theorem For any unitary operator $V$ in Hilbert space $H$ with cyclic vector $f$ (which means $\textrm{cl span}\{V^if\}=H $) there exists measure $\sigma$ on unit circle such that operator $Uf(z) =zf(z)$ in $L^2(\sigma)$ is unitary equivalent to $V$.
So I tried using Fourier series to understand how rotation acts on functions - in $l^2$ it is basically diagonal operator: $$u=(u_k)\longrightarrow (u_ke^{ik\alpha}) $$ where $\alpha$ is a rotation angle. But I have no idea how to move further. Any hints are appreciated.
P. S. And with Baker's map I don't even know where shoud I start.