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

Suppose first that the system is ergodic and $K_{F}g=g$. For each real $t$, the level setis invariant modulo a null set, so $\omega (E_t)\in \{0,1\}$. The distribution function of $\mathrm{Re}g$ can therefore jump only once, which makes $\mathrm{Re}g$ constant almost everywhere. The same argument applies to $\mathrm{Im}g$.

Conversely, if $E$ is invariant, then $K_F{1}_E=1_E$. If every invariant $L^2$ function is constant, the indicator function $1_E$ is almost everywhere zero or one, and hence $\omega (E)=0$ or $1$. This proves the invariant-function characterization of ergodicity.

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