This question arises from my previous one.
If we have a measure-preserving system $(X, \mathcal B, \mu, T)$ it induces unitary operator $(U_Tf)(x)=f(Tx)$, known as Koopman operator. We can study it in many ways: for example, correspondence between $U_T$ properties and ergodicity.
But the only example I know that can be described explicitly is Koopman operator induced by the circle rotation. So I'm looking for other examples where I can write down Koopman operator explicitly (or find unitary equivalent). What about Markov or Bernoulli shifts? Or may be some other classic measure-preserving systems?