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 $z$ (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?

uniquelyergodic. (2) the maps you define are degree 1 maps of the circle. If you put a few more restrictions on $z$ so that they must be homeomorphisms, then there's a very rich theory of orientation preserving homeomorphisms of the circle. See en.wikipedia.org/wiki/Rotation_number, for instance. – Vaughn Climenhaga Feb 16 '12 at 1:07