Perfect measurement device for a dynamical system

For a map $F:X\to X$, a perfect measurement device may query any $x\in X$ and any precision $m$, receiving $y$ with $d(F(x),y)\le 2^{-m}$. It is perfect in the information-theoretic sense: there is no sampling restriction and measurement error can be made arbitrarily small, although every terminating algorithm makes only finitely many queries.