Let $\mathbf X := (X, \mathcal S, \mu)$ be a probability space without atoms. We say two measure preserving transformations $T$ and $F$ on $\mathbf X$ are $\delta$-close, for $\delta > 0$, if $ \int_X |Tg - Fg| d\mu < \delta ||g||_{L^\infty}$ for all $g \in L^\infty(X)$.

For any ergodic transformation $T$, and real numbers $\varepsilon, \delta > 0$ does there exist an integer $n > 0$ and an ergodic transformation $G$ such that $G$ is $\delta$-close to the identity transformation and $G^n$ is $\varepsilon$-close to $T$?

Note: $Tg$ (respectively $Fg$) denotes the function composition $g \circ T$ (respectively $f \circ T$).