It is well known that a rotation $f(z)=e^{i\theta}z$ 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
$$f:S^1\rightarrow S^1,\qquad f(z)=e^{i\theta(z)}z.$$
Does there exist criteria (or some sort of classification) on whether $f$ is ergodic or not?