arXiv:2607.11549v1

Positive Rokhlin Entropy Implies Infinite $L^1$ -Orbit Multiplicity: A Negative Answer to Thouvenot's Question

Zongrui Hu, Leiye Xu, Shuhao Zhang

math.DSmath.FA37A3537A1537A3041A4646B20

Paper metadata

Original submission: July 13, 2026
Claimed MSC: 37A35, 37A15, 37A30, 41A46, 46B20
arXiv categories: math.DS, math.FA
Paper license: CC BY-NC-ND 4.0 ↗
Comments: 8 pages

Abstract

We prove that every free ergodic measure-preserving action of a countably infinite amenable group with positive Rokhlin entropy has infinite complex $L^1$ -orbit multiplicity, both on $L^1$ and on its mean-zero subspace $L^1_0$ . This gives a negative answer to a question of J.-P. Thouvenot recorded by Iwanik and establishes the corresponding endpoint statement at $p=1$ of Iwanik's theorem that positive entropy implies infinite $L^p$ -multiplicity for every $p>1$ . The proof combines Malykhin's rigidity theorem for independent random variables, Seward's Bernoulli factor theorem, and a Følner set argument.

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Audited against arXiv v1

Not a correctness certificate. A “Correct” result may include yellow typos or minor formal corrections that do not affect substantive soundness. It means this audit found no unresolved substantive error under the stated criteria; it does not replace expert scrutiny or formal verification.

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Detailed mathematical audit

Generated August 18, 2026

01Statements3 reported findingsCorrect

The positive-Rokhlin-entropy obstruction to finite $L^1$ and mean-zero $L^1$ orbit multiplicity, together with its Bernoulli-shift consequence, is correct under the stated freeness and countability hypotheses.

Theorem 1.1 and Corollary 4.2Correct

Positive entropy forces infinite orbit multiplicity

Pages 2 and 6–8 · Theorem 1.1, Proposition 4.1, and Corollary 4.2 · arXiv:2607.11549v1

For a hypothetical generating family of $m$ functions, the Seward factor theorem supplies a Bernoulli factor with $2m+1$ independent coordinates. Malykhin's independent-variable rigidity gives a uniform positive distance from every real space of dimension at most $2m|FK|$ . Choosing a Følner set with $|FK|\leq(1+\delta)|F|$ makes the orbit-span approximation contradict that lower bound. Passing between $L^1$ and its mean-zero subspace changes the generated module by only the constants, so both multiplicities are infinite.

Full paper, version 1 ↗

Proposition 3.1Correct

A Bernoulli shift with enough independent coordinates has multiplicity larger than $m$

Pages 3–4 · Proposition 3.1 · arXiv:2607.11549v1

For $r>2m$ , the $r|F|$ centered coordinate variables are independent and normalized. Approximating them by the orbit span of $m$ complex generators produces a real subspace of dimension at most $2m|FK|$ . The Følner inequality makes this strictly smaller than the dimension threshold in Malykhin's theorem, so the positive lower distance contradicts density of the proposed orbit span.

Lemma 4.1 and Corollary 4.2Correct

Multiplicity descends to factors and applies to every nontrivial Bernoulli shift

Pages 5–6 · Lemma 4.1 and Corollary 4.2 · arXiv:2607.11549v1

Conditional expectation onto a factor intertwines the Koopman actions and preserves integrals, so a finite generating family upstairs would give one downstairs in both $L^1$ and $L^1_0$ . The proof also checks essential freeness of Bernoulli shifts for both finite- and infinite-order group elements and uses positive base entropy, verifying every hypothesis of Theorem 1.1.

02Proofs3 reported findingsCorrect

The proof is correct and complete. The factor, dimension, Følner, and rigidity estimates are used with compatible quantifiers.

Sections 3–4Correct and complete

The finite-dimensional contradiction closes

Pages 4–8 · Sections 3–4 · arXiv:2607.11549v1

The approximation is first made on finitely many coordinates and only then is the Følner set selected. The resulting span has the dimension required by the rigidity lemma, factor maps preserve the needed $L^1$ approximation, and the constants can be chosen in the announced order. No unproved limiting interchange is used.

Theorem 2.1 and Lemma 2.3Correct and complete

The rigidity and Følner inputs have compatible constants

Pages 2–3 · Malykhin rigidity and Følner packing · arXiv:2607.11549v1

The proof fixes the dimension deficit before invoking the rigidity constant, then selects the finite approximation set $K$ , and only afterward chooses $F$ with $|FK|\leq(1+\delta)|F|$ . This order makes the real dimension bound strict and keeps the approximation error below the fixed rigidity lower bound.

Proof of Theorem 1.1Correct and complete

The Bernoulli-factor contradiction works in both ambient spaces

Pages 5–6 · proof of Theorem 1.1 · arXiv:2607.11549v1

Seward's theorem is applied to a finite-entropy $(2m+1)$ -fold binary base whose entropy is chosen below the Rokhlin entropy. The factor monotonicity lemma then transfers a hypothetical finite multiplicity to the Bernoulli factor. Every centered coordinate already lies in $L^1_0$ , so the same contradiction proves the mean-zero assertion without an extra approximation step.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2607.11549v1
Authors listed: Zongrui Hu, Leiye Xu, Shuhao Zhang
Audit date: August 18, 2026

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Positive Rokhlin Entropy Implies Infinite L1L^1L1-Orbit Multiplicity: A Negative Answer to Thouvenot's Question

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Positive entropy forces infinite orbit multiplicity

A Bernoulli shift with enough independent coordinates has multiplicity larger than mmm

Multiplicity descends to factors and applies to every nontrivial Bernoulli shift

The finite-dimensional contradiction closes

The rigidity and Følner inputs have compatible constants

The Bernoulli-factor contradiction works in both ambient spaces

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Positive Rokhlin Entropy Implies Infinite $L^1$ -Orbit Multiplicity: A Negative Answer to Thouvenot's Question

A Bernoulli shift with enough independent coordinates has multiplicity larger than $m$