Suppose $F:X\to X$ is ergodic with probability measure $\mu$ and $G$ is a nonabelian compact connected Lie group with Haar measure $\nu$. Let $h\in G$, and define $f_h:X\times G\to X\times G$ by $f_h(x,g)=(Fx,gh)$.
I want to study the ergodicity of $f_h$. I believe that it should not be ergodic, but I am struggling to show why. Here is my methodology so far.
To show that $f_h$ is not ergodic, I want to a measurable set such that $H$ such that $f_h^{-1}(X\times H)=X\times H$ and $(\mu\times\nu)(X\times H)=\nu(H)\in (0,1)$.
Let $H=\bar{<h>}$, and note that this is an abelian (hence proper) Lie subgroup of $G$. Therefore $\nu(H)=0$. The idea is that if I can somehow "engulf" this group into some larger set of positive measure, then that should give me the required set. But I am unsure on how to do this (or whether it is even possible).
Thanks in advance!
Remark: The idea was from a comment in this question Showing measure is invariant/ergodic for a skew product.. The writer of the answer mentions "if there is a positive measure subgroup or $h$-invariant set, then ergodicity fails".
