arXiv:2607.14444v1

The analogue of Belinskaya's theorem for measure-preserving flows

Konstantin Slutsky

math.DS37A1037A0537A1537A20

Paper metadata

Original submission: July 16, 2026
Claimed MSC: 37A10, 37A05, 37A15, 37A20
arXiv categories: math.DS
Paper license: arXiv non-exclusive distribution license ↗

Abstract

We prove the analogue of Belinskaya's theorem for measure-preserving flows: two free ergodic measure-preserving flows whose $\mathrm{L}^1$ full groups are isomorphic as abstract groups are conjugate up to a scalar time change. This answers a question posed by François Le Maître and the author. We show that whenever two free ergodic flows generate the same orbit equivalence relation and one is contained in the other's $\mathrm{L}^1$ full group, their positive half-orbits are commensurate after possibly reversing time. Katznelson's criterion then yields conjugacy after a scalar time change. The key new ingredient is a commensuration criterion asserting that a measurable subset of the real line whose symmetric differences with its translates have finite average measure over the unit interval is commensurate with exactly one of the empty set, the whole line, and the two half-lines. This criterion and its application were discovered autonomously by a two-agent AI system. The author independently verified the proofs and prepared the final text.

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Detailed mathematical audit

Generated August 18, 2026

01Statements3 reported findingsCorrect

Both rigidity statements are supported. Theorem 3.1 turns a one-sided inclusion of $L^1$ full groups into conjugacy of the underlying free ergodic flows up to a nonzero scalar time change. Theorem 3.3 starts from an abstract topological group isomorphism, spatially reconstructs the orbit relation, obtains the required full-group inclusion, and then invokes Theorem 3.1.

Theorem 3.1Correct

One-sided $L^1$ -full-group inclusion forces flow conjugacy

Pages 2–3 and 6–8 · Theorems 3.1 and 3.3 · arXiv:2607.14444v1

If the $L^1$ full group of one free ergodic measure-preserving flow is included in that of another in the one-sided sense stated, Theorem 3.1 proves that the flows are conjugate after multiplication of time by a nonzero scalar. The orbit cocycle is integrable in the required direction, and the conclusion allows the orientation reversal represented by a negative scalar. Ergodicity and freeness are used to make the scalar constant almost everywhere and to exclude orbitwise periodic exceptions.

Theorem 3.3Correct

Abstract $L^1$ -full-group isomorphism reconstructs the flow

Pages 7–8 · Theorem 3.3 · arXiv:2607.14444v1

The automatic-continuity/spatial-reconstruction theorem represents the abstract isomorphism by conjugation with a measure-space isomorphism. This map carries the first flow's orbit relation to the second and preserves the $L^1$ topology, giving the one-sided inclusion required by Theorem 3.1 in both directions. Applying that theorem yields a nonzero scalar time change. The inverse isomorphism excludes a degenerate zero slope.

Orientation alternativesCorrect

Positive and negative scalar time changes

Pages 6–8 · proofs of Theorems 3.1 and 3.3 · arXiv:2607.14444v1

The orbitwise commensuration lemma has two tail alternatives, corresponding to preservation or reversal of the linear order on almost every orbit. Ergodicity makes the choice constant. The former gives a positive time scalar and the latter a negative one; both conjugacies induce the same abstract $L^1$ -full-group structure. Thus the statement's nonzero scalar, rather than an unjustified positive scalar, is correct.

02Proofs3 reported findingsCorrect

Lemma 2.1, Corollary 2.2, and the orbitwise application. The $L^1$ inclusion gives an integrable orbit cocycle between the two flows. Testing it on interval exchanges along each orbit and applying Tonelli turns the average translation defect into a summable sequence of cell oscillations. Lemma 2.1 shows that a measurable subset of the line with finite total boundary variation has two tail limits and is, modulo finite measure, a half-line or its complement. This orbitwise classification makes the cocycle monotone with a constant slope. Ergodicity makes that slope global, and the cocycle identity yields the scalar-time conjugacy.

Commensuration criterionCorrect and complete

Lemma 2.1, Corollary 2.2, and the orbitwise application

Pages 3–8 · Sections 2–3 · arXiv:2607.14444v1

The $L^1$ inclusion gives an integrable orbit cocycle between the two flows. Testing it on interval exchanges along each orbit and applying Tonelli turns the average translation defect into a summable sequence of cell oscillations. Lemma 2.1 shows that a measurable subset of the line with finite total boundary variation has two tail limits and is, modulo finite measure, a half-line or its complement. This orbitwise classification makes the cocycle monotone with a constant slope. Ergodicity makes that slope global, and the cocycle identity yields the scalar-time conjugacy.

Section 2Correct and complete

Finite-variation commensuration criterion

Pages 3–6 · Lemma 2.1 and Corollary 2.2 · arXiv:2607.14444v1

Partitioning the line into unit cells and integrating translation symmetric differences bounds the sum of adjacent cell oscillations. Hence the cell densities have limits at both tails. Finite total symmetric difference under translations forces those limits to be zero and one in one of the two orders, so the set differs by finite measure from a half-line. The converse is immediate. Null boundary cases are absorbed by the modulo-measure formulation.

Section 3Correct and complete

Orbitwise application and spatial reconstruction

Pages 6–8 · Section 3 · arXiv:2607.14444v1

The full-group norm identity expresses cocycle integrability as the averaged symmetric-difference quantity required by Corollary 2.2. Applying it on almost every orbit gives an order orientation and additive slope; the cocycle law and ergodicity make the slope constant. For an abstract isomorphism, the cited reconstruction theorem supplies the conjugating map before this argument is used. All null sets are chosen invariantly, so the final conjugacy holds globally modulo measure.

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No non-novelty findings.

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Paper: arXiv:2607.14444v1
Authors listed: Konstantin Slutsky
Audit date: August 18, 2026

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The analogue of Belinskaya's theorem for measure-preserving flows

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

One-sided L1L^1L1-full-group inclusion forces flow conjugacy

Abstract L1L^1L1-full-group isomorphism reconstructs the flow

Positive and negative scalar time changes

Lemma 2.1, Corollary 2.2, and the orbitwise application

Finite-variation commensuration criterion

Orbitwise application and spatial reconstruction

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One-sided $L^1$ -full-group inclusion forces flow conjugacy

Abstract $L^1$ -full-group isomorphism reconstructs the flow