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 $\alpha \in \mathbb{R}$?



Ergodicity of composition with a rotation
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. 


