Measure Preserving Diffeomorphisms of the Torus are Unclassifiable

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