The smooth realization problem: does finite entropy suffice for a smooth volume-preserving model?

Claims Theorem 1.1: there is an ergodic invertible transformation of a standard nonatomic probability space with $h_\mu(T)=0$ that is not measurably conjugate on invariant conull sets to any $C^\infty$ diffeomorphism of a compact finite-dimensional manifold (boundary allowed, orientation not required) preserving a strictly positive smooth probability density. A single symbolic example excludes every dimension. So finite entropy, even zero entropy, does not suffice. It does not exclude models whose invariant measure is only equivalent to volume, and it does not characterize which systems are smoothly realizable.

Claims Theorem 1.1: there is an ergodic invertible transformation of a standard nonatomic probability space with $h_\mu(T)=0$ that is not measurably conjugate on invariant conull sets to any $C^\infty$ diffeomorphism of a compact finite-dimensional manifold (boundary allowed, orientation not required) preserving a strictly positive smooth probability density. A single symbolic example excludes every dimension. So finite entropy, even zero entropy, does not suffice. It does not exclude models whose invariant measure is only equivalent to volume, and it does not characterize which systems are smoothly realizable.