In 1932 von Neumann proposed classifying the statistical behavior of differentiable systems. In modern language this is interpreted as classifying diffeomorphisms of compact manifolds up to measure isomorphism. This paper proves that this is impossible in a rigorous sense.
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Matthew Foreman, Benjamin Weiss, Measure preserving diffeomorphisms of the torus are unclassifiable. J. Eur. Math. Soc. 24 (2022), no. 8, pp. 2605–2690DOI 10.4171/JEMS/1151