TITLE: Ergodicity of composition with a rotation
QUESTION [8 upvotes]: Let $T$ be an arbitrary Lebesgue measure-preserving automorphism of the unit interval $I$.  Let $R_{\alpha}$ denote rotation by $\alpha$, i.e. $R_{\alpha}(x)=x+\alpha \pmod{1}$ for $x \in I$ and $\alpha \in \mathbb{R}$.  Is it true that the composition $R_{\alpha} \circ T$ is ergodic for (Lebesgue) almost every $\alpha \in \mathbb{R}$?

REPLY [9 votes]: No, for trivial reasons. Take as your $T$ the map $x\mapsto-x\mod 1$.
Than the map $x\mapsto\alpha-x\mod 1$ is a reflection around $\alpha/2$, and is never ergodic.