This is probably trivial (in which case I apologize), but it's late and I would really like a quick proof/counterexample for this (for a different problem that I'm doing): if $(X,\mathcal{M},\mu,T)$ is a measure preserving ergodic dynamical system with $T$ invertible and $\mu(X)=1$ (i.e. $\mu(T^{-1}(A))=\mu(A)$ for every measurable set $A$ in $\mathcal{M}$ and ergodic), is it true that $A'=T^{-k}(A)$ implies $(A')^{c}=T^{-k}(A^{c})$?
Here $A^{c}$ denotes the complement of $A$ in $X$. Any help appreciated!
This is not strictly related to dynamical systems. Indeed $$T^{-1}(A\setminus B)=T^{-1}A\setminus T^{-1}B$$ for any sets $A$ and $B$. Since $T^{-1}X=X$ you get what you want.