Belinskaya-type reconstruction of measure-preserving flows
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Which measurable subsets of R have finite average symmetric-difference measure under translations in a unit interval? The classification target is agreement up to finite measure with the empty set, the whole line or one of the two half-lines; its flow analogue asks what an abstract L¹ full group determines.
Why it matters and what counts as progress
Why it matters: A commensuration classification can strengthen what an abstract L¹ full group reveals about the underlying flow.
What counts as progress: Prove the set classification and derive conjugacy of free ergodic measure-preserving flows up to nonzero scalar time change.
How the problem developed
- 1959 · Theorem [1]
Henry A. Dye Established foundational orbit-equivalence results for measure-preserving transformations, a precursor to reconstruction from orbit data.
- 1968 · Theorem [2]
A. A. Belinskaya Proved that integrably orbit-equivalent ergodic automorphisms are conjugate after possibly reversing time.
- 16 Jul 2026 · Reported claim [3]
Konstantin Slutsky, with an autonomous two-agent AI system Proved the flow analogue: isomorphic abstract L¹ full groups of free ergodic measure-preserving flows determine conjugacy up to a nonzero scalar time change.
AI contribution
The analogue of Belinskaya's theorem for measure-preserving flows
A measurable real-line set with finite average symmetric-difference measure under unit-interval translations is commensurate with exactly one of the empty set, the whole line, a right half-line or a left half-line. For free ergodic measure-preserving flows, this implies that abstractly isomorphic L¹ full groups determine conjugacy up to a nonzero scalar time change.
Sources
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Publication dates and editorial changes.
AI-found criterion strengthens Belinskaya’s theorem for flows