Measure preserving diffeomorphisms of the torus are unclassifiable

The isomorphism problem in ergodic theory was formulated by von Neumann 1932 his pioneering paper Zur Operatorenmethode der klassischen Mechanik (Ann. of Math. (2), 33(3):587--642, 1932). has been solved for some classes transformations that have special properties, such as the collection with discrete spectrum or Bernoulli shifts. This shows a general classification is impossible (even concrete settings) showing $E$ pairs ergodic, Lebesgue measure preserving diffeomorphisms $(S,T)$ 2-torus are isomorphic complete analytic set $C^\infty$- topology (and hence not Borel).