Measure Preserving Diffeomorphisms of the Torus are Unclassifiable

2017/05/12 by Foreman, Matthew, Weiss, Benjamin · 2 citations
#Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.1705.04414

Abstract

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^∞- topology (and hence not Borel).

Cited by

• Unbalancedness of the conjugacy relation of ergodic measure-preserving transformations
• Non-classifiability of mixing zero-entropy diffeomorphisms up to isomorphism

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