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Can every ergodic map be approximated by ergodic maps close to the identity?

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

Nate River
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