Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 358 / 3 / b / ii / Solution

Use the Fourier basisof $L^2([-\pi ,\pi {]}_{\mathrm{per}})$. For the rotation $F(x)=x+a$,If $a/\pi$ is irrational, $e^{ika}=1$ implies $k=0$. Hence every fixed $L^2$ function has only its constant Fourier coefficient, and part (i) proves ergodicity.

If $a/\pi =p/q$ is rational, thenis a nonconstant fixed function because $e^{i2qa}=e^{i2\pi p}=1$. Part (i) now shows that the system is not ergodic. Therefore the ergodicity criterion for a circle rotation is