Apparently the following is not ergodic, and a very sketchy argument is given, so I was hoping someone here would be able to explain/give a better argument.
Suppose $G$ is a compact connected non-abelian Lie group with Haar measure $m$ and fix $h\in G$. Let $X$ be a bounded metric space and $F:X\to X$ ergodic. Let $F_h:X\times G\to X\times G$ be given by $F_h(x,g)=(Fx,gh)$.
Apparently this is not ergodic. The argument I am referring to says: $H=\langle h\rangle$ is abelian, hence is a proper subspace of $G$. It follows that $m(H)=0$. (This part is clear).
$H$ is at most a maximal torus. In particular, the ergodic components have measure zero and ergodicity fails.
The part which confuses me is the previous sentence. Could anyone shed any light on this?