Logic Seminar

We discuss two equivalence relations for diffeomorphisms of finite dimensional smooth manifolds. The first is measure isomorphism between ergodic Lebesgue measure preserving diffeomorphisms of the 2-torus. The second is topological conjugacy for diffeomorphisms of smooth manifolds. In both cases we show that the equivalence relation is unclassifiable. For topological conjugacy, the known results differ for dimension at least two and dimension at least 5.
We finish by showing that the equivalence relations are Π01-hard: for measure preserving diffeomorphisms of the torus we exhibit a primitive recursive association of Π01-statements with diffeomorphisms of the torus so that for each ϕ,

So, as Erdős used to say, Independence rears its ugly head.