Irrational rotation of the circle

For irrational $\alpha$, the circle rotation $T(x)=x+\alpha (\mathrm{mod}1)$ preserves Lebesgue measure and is an ergodic transformation.

The rotation $x\mapsto x+a$ of a circle of circumference $2\pi$ is ergodic exactly when $a/\pi$ is irrational. In the Fourier basis, an invariant function can have a nonzero $k$th coefficient only when $e^{ika}=1$.

Every orbit of an irrational rotation of the circle visits a half-open interval $I$ with limiting frequency equal to its length. One proof sandwiches the orbit of an arbitrary point between inner and outer intervals along a nearby orbit for which the Birkhoff ergodic theorem holds.