Given a probability space $(\Omega, \mathcal{F}, \mu)$ we call $T:\Omega \rightarrow \Omega$ with $\mu(T^{-1} (A)) = \mu(A)$ for all $A\in\mathcal{F}$ ergodic if for all $A\in\mathcal{F}$ with $T^{-1}(A)=A$ we get $\mu(A)\in \{0,1\}$.
So is it possible to construct $T$ ergodic with $T^2$ not ergodic? Any ideas?
Consider the two-point space $A=\{x,y\}$ with $\mu(\{x\})=\mu(\{y\})=1/2$. Take $T$ to be the map swapping $x$ and $y$. Clearly $T$ is ergodic, but $T^2$ fixes $\{x\}$ of measure $1/2$, so is not.