Ergodic components of a rational circle rotation
= Ergodic components of a rational circle rotation
{title2=$\mu_x=\frac1q\sum_{j=0}^{q-1}\delta_{x+jp/q}$}
For rotation by $p/q$ with coprime integers and Haar probability on the circle, components are the equal measures on the $q$-point orbits. The quotient map is $x\mapsto qx\pmod1$. Integration over the quotient recovers Haar measure, and the rotation permutes each orbit transitively, making its <probability measure> <ergodic>.