It is well known that a rotation $f(z)=e^{i\theta}$ of the unit circle, is ergodic if and only if $e^{i\theta}$ is not a root of unity.

Now, what happens if we let $\theta$ depend on $\theta$ (say, continuously)?

To be more explicit: let $\theta:S^1\subset{\mathbb C}\rightarrow [0,2\pi)$ be continuous, and let $$f:S^1\rightarrow S^1,\qquad f(z)=e^{i\theta(z)}.$$ Does there exist criteria (or some sort of classification) on whether $f$ is ergodic or not?