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 if $d_{TV} \ (T\mu, F\mu) < \delta$ where $d_{TV}$ denotes the total variation distance, and $T\mu$ (resp. $F\mu$) denotes the pushforward of mu under $T$ (resp. $F$).

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 and $G^n$ is $\varepsilon$-close to $T$?