If $R_{\alpha}:[0,1] \to [0,1]$ is defined by $$R_{\alpha}(x)=x+\alpha $$ then $R_{\alpha} $is called a circle rotation, and it is known that $R_{\alpha}$ is ergodic iff $\alpha$ is irrational. I interpret this to mean that the dynamical system determined by the orbit $R_\alpha^t(x)$ is ergodic, and I currently do not have a physical interpretation of this (either on $[0,1]$ or the circle) . Wikipedia (http://en.wikipedia.org/wiki/Ergodicity) says the dynamical system is ergodic if it "has the same behavior averaged over time as averaged over the space of the system's states." What does this mean on the circle? (My guess would be that it spends approximately the same time at each point)
Further, how would this interpretation differ in the case that our dynamical system is over a skew product of rotations of a circle: I.e., if we rotate by an irrtaional angle $\alpha$ and switch to another (identical) circle whenever our orbit lands in a predetermined subinterval of irrational length?