OAI.SmoothObstruction.main

The theorem states that there exist a type X with a measurable space structure, a probability measure μ on X, and a measurable equivalence T : X ≃ᵐ X (a bimeasurable bijection) such that four things hold. First, μ is standard nonatomic, meaning that after removing measurable null sets it is isomorphic, via a bimeasurable measure-preserving bijection between measurable conull subsets, to Lebesgue measure on the interval [0,1]. Second, T is ergodic with respect to μ. Third, the system has finite Kolmogorov–Sinai entropy in the source's sense: there is one real bound C such that for every measurable finite-labelled partition P, the entropy rate of the join of the first n iterates of P, measured in bits as the block entropy divided by n, converges as n tends to infinity to some limit h with h ≤ C. Fourth, the system has no smooth positive-volume model: there is no compact, Hausdorff, second-countable smooth (C^∞) manifold M of any finite dimension d, either closed (including dimension zero) or with boundary (for d ≥ 1), carrying a Borel probability measure ν with smooth positive densities in every chart (with respect to Lebesgue measure), together with a C^∞ diffeomorphism S of M preserving ν, such that (μ,T) is measurably conjugate to (ν,S) on invariant conull measurable subsets by a measure-preserving bimeasurable bijection intertwining T and S. Orientability is not required of the manifolds. The statement is admitted without proof in the source.