Nonstandard Smooth Realizations of Liouville Rotations
  We augment the method of $C^\infty$ conjugation approximation with explicit estimates on the conjugacy map. This allows us to construct ergodic volume preserving diffeomorphisms measure-theoretically isomorphic to any apriori given Liouville rotation on a variety of manifolds. In the special case of tori the maps can be made uniquely ergodic. 
Bassam Fayad, Maria Saprykina, Alistair Windsor