Can anyone help me to understand this example. I have already proved that irrational rotation is ergodic. This example is showing that $f \times f$ is not ergodic. But I am not understanding what this transformation does not satisfy so that it is not ergodic
The transformation $(x,y)\mapsto(\theta x,\theta y)$, where $\theta$ is a rotation, leaves the ratio $x^{-1} y$ invariant.
The set of $(\exp(ix),\exp(iy))$ for which (say) $\Re\exp(i(x-y))>0$ is invariant under the translations $(\exp(ix),\exp(iy))\mapsto(\exp(ix+i\theta),\exp(iy+i\theta))$. But that set does not have measure zero or full measure.
Said another way: $T$ is ergodic if the only $T$-invariant functions are a.e. constant. But $(a,b)\mapsto a/b$ is invariant and not constant.