[Math] Product of ergodic transformations

Question

I'm asked to give an example, that the product of two ergodic systems is not ergodic in general.
I know that for $X_1=X_2=(S^1,B,m,R_a)$ (the irrational rotation on the unit circle with Lebesgue measure), the product is not ergodic, while the irrational rotation is.
But how to show?
I don't know, but my guess is that maybe the orbit of any point $x \in S^1\times S^1$ is not dense? (I start at a point $(x,y)$, while iterating the process i get somehow a "ellipse on the torus" or not?). This would imply non-ergodicity. Any ideas?

Answer 1

Let $f(x,y)=x-y$ on the torus $S^1\times S^1$ (I consider $S^1=\mathbb{R}/\mathbb{Z}$). This function is not constant, but it is invariant: it is easy to check that $f(x+\alpha, y+\alpha)=f(x,y)$.