arXiv:2303.12900v2

Anti-classification results for weakly mixing diffeomorphisms

Philipp Kunde

math.DS37A3537A0537A2037C4003E15

Paper metadata

Original submission: March 22, 2023
Latest revision: August 4, 2026 · version 2
Claimed MSC: 37A35, 37A05, 37A20, 37C40, 03E15
arXiv categories: math.DS
Paper license: CC BY 4.0 ↗
Comments: 53 pages, 2 figures

Abstract

We extend anti-classification results in ergodic theory to the collection of weakly mixing systems by proving that the isomorphism relation as well as the Kakutani equivalence relation of weakly mixing invertible measure-preserving transformations are not Borel sets. This shows in a precise way that classification of weakly mixing systems up to isomorphism or Kakutani equivalence is impossible in terms of computable invariants, even with a very inclusive understanding of ``computability''. We even obtain these anti-classification results for weakly mixing area-preserving smooth diffeomorphisms on compact surfaces admitting a non-trivial circle action as well as real-analytic diffeomorphisms on the $2$ -torus.

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Audit summary

Audited against arXiv v2

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.

Current report

Detailed mathematical audit

Generated August 18, 2026

01Statements5 reported findingsContains wrong statements

Theorem A is false in its assertion that every equivalence relation intermediate between isomorphism and Kakutani equivalence gives a complete analytic set: arbitrary intermediate relations need not be analytic. The reduction does verify the universal non-Borel conclusion, and it gives complete analyticity for the specifically analytic relations of isomorphism and Kakutani equivalence. The analogous unrestricted clauses in Theorems B and C and the asserted injectivity in Theorem 3 are not verified.

Theorem AIncorrect

An arbitrary intermediate equivalence relation need not be analytic

Page 3 and pages 5-6 · Theorem A and the reduction argument · arXiv:2303.12900v2

The exact claim is that for every equivalence relation $R$ with $R_{\mathrm{iso}}\subseteq R\subseteq R_{\mathrm{Kak}}$ , the restriction of $R$ to pairs of weakly mixing transformations is complete analytic. This is false because no definability hypothesis is imposed on $R$ . For a formal counterexample, choose a continuum family $(B_h)_{0<h<\log 2}$ of pairwise nonisomorphic Bernoulli shifts indexed injectively by entropy. Every $B_h$ is mixing, and all finite-positive-entropy Bernoulli shifts are Kakutani equivalent. Fix $h_0$ . For every set $A\subseteq(0,\log 2)$ containing $h_0$ , let $R_A$ merge precisely the isomorphism classes of the systems $B_h$ with $h\in A$ into one class and leave every other isomorphism class unchanged. Then $R_{\mathrm{iso}}\subseteq R_A\subseteq R_{\mathrm{Kak}}$ , and the restrictions of the $R_A$ to weakly mixing pairs are distinct for $2^{\mathfrak c}$ choices of $A$ . A Polish space has only $\mathfrak c$ analytic subsets, so at least one such restriction is not analytic and therefore cannot be complete analytic. The paper's reduction nevertheless proves that every one of these restrictions is $\Sigma^1_1$ -hard and hence non-Borel.

Ornstein-Rudolph-Weiss, Equivalence of measure preserving transformations ↗

Theorems A-C for isomorphism and Kakutani equivalenceCorrect

The named classification relations are complete analytic on the constructed classes

Pages 3-6 and pages 50-52 · Theorems A-C and Theorem 3 · arXiv:2303.12900v2

For $R=R_{\mathrm{iso}}$ and $R=R_{\mathrm{Kak}}$ , the relevant pair relations are analytic. The constructed continuous map sends a tree $\mathcal T$ to a weakly mixing transformation $T=\Phi(\mathcal T)$ , and Sections 6.2-6.3 prove that $\mathcal T$ is ill-founded exactly when $T$ and $T^{-1}$ are $R$ -equivalent. Composing with $T\mapsto(T,T^{-1})$ reduces the complete analytic set of ill-founded trees to each pair relation. Analyticity plus this reduction proves complete analyticity; the same reasoning applies after the smooth and real-analytic realization maps used in Theorems B and C.

Full paper, version 2 ↗

Theorems B and C for arbitrary intermediate relationsNot able to verify

The complete-analytic clauses lack an analyticity hypothesis

Pages 3-6 and page 52 · Theorems B-C and their proofs · arXiv:2303.12900v2

As stated, these theorems quantify over every equivalence relation $R$ between the restrictions of isomorphism and Kakutani equivalence, without requiring $R$ to be analytic. The proofs establish a continuous reduction from ill-founded trees and therefore establish $\Sigma^1_1$ -hardness and non-Borelness. They do not establish that the target is analytic, which is an independent requirement in the paper's own definition of complete analytic. The abstract counterexample above proves the analogous clause of Theorem A false, but the present audit does not supply the additional smooth or real-analytic realization family needed to turn it into a counterexample in each of these two narrower ambient spaces. Thus their complete-analytic clauses are not verified; their non-Borel conclusions are verified.

Full paper, version 2 ↗

Theorem 3Correct

The reduction properties are correct

Pages 5-6 and pages 50-52 · Theorem 3 and Section 6 · arXiv:2303.12900v2

Apart from the separate injectivity clause below, the asserted properties are established. Lemma 64 proves continuity; Proposition 31 together with the construction requirements proves weak mixing; Lemmas 65-66 turn an infinite branch into an isomorphism with the inverse; and Section 6.3 transfers the spacer-ignored separation estimates to prove that Kakutani equivalence with the inverse forces an infinite branch. These implications are sufficient for the continuous reduction used in Theorems A-C.

Full paper, version 2 ↗

Theorem 3, injectivity clauseNot able to verify

The claim that $\Phi$ is one-to-one is not established

Pages 5-6 and pages 50-52 · Theorem 3 and Section 6 · arXiv:2303.12900v2

Theorem 3 calls $\Phi:\mathrm{Trees}\to\mathrm{Diff}^{\infty}_{\lambda}(M)$ one-to-one. Section 6 proves continuity and the three dynamical implications needed for the reduction, but it contains no argument that two distinct trees produce distinct diffeomorphisms. The prefix dependence used in Lemma 64 proves continuity only; it does not give a converse separation statement. A verification would require a quantitative or exact construction-level argument showing that the first differing tree datum survives all subsequent conjugations and the realization map. No such argument or independent proof was found. Injectivity is not needed for the reduction or the non-Borel conclusions.

Full paper, version 2 ↗

02Proofs6 reported findingsContains incorrect or incomplete proofs

The proofs of Theorems A-C establish hardness and non-Borelness but incorrectly treat hardness as complete analyticity without proving that an arbitrary intermediate relation is analytic. The proof of Theorem 3 omits its injectivity clause. The central weak-mixing, realization, branch-isomorphism, and non-Kakutani chains are otherwise correct. Three groups of local notation defects have unique harmless corrections.

Proofs of Theorems A-CIncorrect as written

A reduction proves analytic hardness, not analytic membership

Pages 5-6 and page 52 · deductions from Theorem 3 · arXiv:2303.12900v2

The proof constructs a continuous reduction of the complete analytic set of ill-founded trees to the target pair set. This proves that the target is $\Sigma^1_1$ -hard. It does not prove that the target itself is analytic, whereas Definition 2 requires both analyticity and universal reducibility for the label complete analytic. For arbitrary intermediate $R$ , analyticity does not follow from $R_{\mathrm{iso}}\subseteq R\subseteq R_{\mathrm{Kak}}$ ; Theorem A is formally contradicted by the family in Part 1. Downstream dependency: this is the sole step claimed to establish complete analyticity in Theorems A-C. Verified repair: for unrestricted $R$ , replace complete analytic by $\Sigma^1_1$ -hard and retain the proved non-Borel conclusion; alternatively, add the hypothesis that the relevant restriction of $R$ is analytic, under which the existing reduction does prove complete analyticity.

Full paper, version 2 ↗

Proof of Theorem 3Incomplete as written

Injectivity of the tree-to-diffeomorphism map is not proved

Pages 50-52 · Section 6 · arXiv:2303.12900v2

Section 6 announces verification of all properties required by Theorem 3, but Lemma 64 proves only continuity and the remaining subsections prove weak mixing and the two branch/Kakutani implications. None proves that $\Phi(\mathcal T)=\Phi(\mathcal S)$ forces $\mathcal T=\mathcal S$ . Downstream dependency: this leaves the literal one-to-one clause of Theorem 3 unsupported, although no later reduction needs injectivity. Repair classification: No repair supplied; a new separation argument for distinct construction prefixes is required.

Full paper, version 2 ↗

Central construction and dynamical implicationsCorrect and complete

The substantive reduction chain is correct and complete

Pages 18-23 and pages 30-52 · Propositions 28 and 31; Sections 4.3, 5, and 6 · arXiv:2303.12900v2

Requirement (R4) and the counting in Proposition 31 verify the weak-mixing criterion at the stated mixing times. The substitution construction enforces the required uniformity and separation, and the symbolic systems are transferred by the smooth realization theorem and its real-analytic counterpart. For an infinite branch, the compatible odd-parity actions in Lemmas 65-66 give an isomorphism with the reversed system. For a well-founded tree, Lemmas 62-63 supply the same $\bar f$ separation estimates used in the cited Gerber-Kunde argument; that argument explicitly ignores the spacer symbols, so the change from the circular to the twisting operator preserves the non-Kakutani conclusion. No substantive defect was found in these chains.

Gerber-Kunde, Non-classifiability of Ergodic Flows up to Time Change ↗

PreliminariesTypo

The join of two partitions repeats the wrong partition name

Page 8 · definition of $P\vee Q$ · arXiv:2303.12900v2

Printed: $P\vee Q=\{c\cap d:c\in P, d\in P\}$ . Correction: replace the second $P$ by $Q$ . The surrounding sentence defines the join of $P$ and $Q$ , so the intended correction is unique and no later argument changes.

Full paper, version 2 ↗

Definition 20Typo

A twisted system is said to come from the wrong construction type

Page 14 · Definition 20 · arXiv:2303.12900v2

Printed: a system built from a circular construction sequence is called a twisted system. Correction: replace circular by twisted. Definition 19 has just defined twisted construction sequences, and every subsequent use of $K^{\mathrm{twist}}$ uses that construction, so this is a unique harmless terminology correction.

Full paper, version 2 ↗

Requirements (R3) and proof of Proposition 31Typo

Two terminal indices lie one step beyond their declared range

Pages 21-23 · requirement (R3) and the case distinction in the proof of Proposition 31 · arXiv:2303.12900v2

The tuples $b_n(i,s)$ are defined only for $0\leq i<2^{n+2}q_n$ . Requirement (R3) nevertheless ends its displayed list at $b_n(2^{n+2}q_n,s)$ ; the explanatory sentence immediately below ends it correctly at $2^{n+2}q_n-1$ . Later, the proof of Proposition 31 treats the boundary case $i_2=2^{n+2}q_n$ , although its standing range is $0\leq i_2<2^{n+2}q_n$ . In both places the unique correction is to replace the terminal index by $2^{n+2}q_n-1$ . The earlier rotation formula (4.16), the cyclic successor, and the subsequent counting all use that corrected endpoint, so the argument is unchanged.

Full paper, version 2 ↗

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2303.12900v2
Authors listed: Philipp Kunde
Audit date: August 18, 2026

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Anti-classification results for weakly mixing diffeomorphisms

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

An arbitrary intermediate equivalence relation need not be analytic

The named classification relations are complete analytic on the constructed classes

The complete-analytic clauses lack an analyticity hypothesis

The reduction properties are correct

The claim that Φ\PhiΦ is one-to-one is not established

A reduction proves analytic hardness, not analytic membership

Injectivity of the tree-to-diffeomorphism map is not proved

The substantive reduction chain is correct and complete

The join of two partitions repeats the wrong partition name

A twisted system is said to come from the wrong construction type

Two terminal indices lie one step beyond their declared range

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The claim that $\Phi$ is one-to-one is not established