arXiv:2608.05649v1

Annealed Ruelle-Pollicott Resonances

Sakshi Jain, Maxence Phalempin

math.DS37A3037D2037E1037H05

Paper metadata

Original submission: August 6, 2026
Claimed MSC: 37A30, 37D20, 37E10, 37H05
arXiv categories: math.DS
Paper license: arXiv non-exclusive distribution license ↗
Comments: 24 pages, 1 figure, 3 images

Abstract

We adapt the theory of Ruelle-Pollicott resonances to annealed random dynamical systems generated by independent and identically distributed families of maps. Introducing annealed transfer and Koopman operators, we define resonances as elements of the point spectrum of the associated operators and establish a decorrelation formula relating these resonances to the asymptotic decay of annealed correlations. We then study several classes of systems for which the theory can be made explicit. First, we consider a family of piecewise expanding Markov maps of the interval, we construct Banach spaces adapted to the dynamics and obtain a complete description of the annealed resonance spectrum. Then we investigate an inverse spectral problem, proving realisability results for prescribed collections of complex numbers as resonances of suitably constructed annealed dynamical systems.

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Audit summary

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.

Current report

Detailed mathematical audit

Generated August 18, 2026

01Statements3 reported findingsContains wrong statements

The Blaschke-product spectrum and the full-resonance conclusion for the interval families are correct. The general Jordan expansion is false when its index is identified with geometric multiplicity, and the claimed mixing corollary is false under transitivity alone.

Theorems A and 11Correct

The annealed Blaschke spectrum is correct

Pages 8 and 24–27 · Theorems A and 11 · arXiv:2608.05649v1

The common Hilbert basis makes the averaged Koopman operator compact and triangular. Its nonzero diagonal entries are $1$ , $\int(-\omega)^n\,d\mathbb P(\omega)$ , and $\int(-\overline\omega)^n\,d\mathbb P(\omega)$ . Compactness excludes any further nonzero spectrum, and the row structure gives the stated algebraic and geometric multiplicities after the proof repair recorded below. The deterministic matrix formula used before averaging agrees with the cited analytic-Anosov calculation.

Slipantschuk–Bandtlow–Just, Complete spectral data for analytic Anosov maps of the torus ↗

Proposition 2.2(2) and Lemma A.2Incorrect

Geometric multiplicity does not determine the polynomial powers in a Jordan expansion

Pages 8–9 and 30–32 · Proposition 2.2(2), Lemma A.2 and equations (54)–(64) · arXiv:2608.05649v1

The statements set $m_i$ equal to geometric multiplicity and truncate the terms $n^k\lambda_i^n$ at $k<m_i$ . Take the constant Banach scale $W_r=\mathbb C^2$ and $L=\begin{pmatrix}\lambda&1\\0&\lambda\end{pmatrix},\qquad 0<|\lambda|<1.$ All compactness, spectral-gap, inclusion, density, and generalized-eigenspace hypotheses hold, but the geometric multiplicity is one while $L^n=\begin{pmatrix}\lambda^n&n\lambda^{n-1}\\0&\lambda^n\end{pmatrix}.$ The required $n\lambda^{n-1}$ contribution is absent from the printed formula. Replace geometric multiplicity by the maximal Jordan-block size, or sum separately over the blocks; with that correction, the intended decorrelation expansion is valid.

Theorem B, Theorem 5, and Corollary 7Incorrect mixing conclusion; resonance conclusion correct

Transitivity does not imply mixing

Pages 9, 12, and 20–21 · Theorem B, Theorem 5 and Corollary 7 · arXiv:2608.05649v1

Use a deterministic one-point base and the four Markov intervals $I_j=((j-1)/4,j/4)$ . Define $T$ with slope $2$ so that each of $I_1,I_2$ maps onto $(1/2,1)$ and each of $I_3,I_4$ maps onto $(0,1/2)$ . This is a transitive piecewise-linear expanding Markov map satisfying every displayed hypothesis, but its irreducible transition matrix has period two. Lebesgue measure is invariant and $h=\mathbf1_{(0,1/2)}-\mathbf1_{(1/2,1)}$ satisfies $h\circ T=-h$ , so $\int h(h\circ T^n)=(-1)^n$ and the system is not mixing. Equivalently, the transfer operator has eigenvalue $-1$ . The full decorrelation formula remains correct because that peripheral resonance can be included; only the asserted mixing and spectral-gap consequence fails.

02Proofs5 reported findingsContains incorrect or incomplete proofs

The proofs contain a false Perron–Frobenius inference, an incorrect Jordan-multiplicity argument, and an unproved orthogonality step in the averaged abstract decomposition. The core compactness and interval quasi-compactness arguments are otherwise repairable.

Corollary 7Incorrect as written; no repair at the stated scope

Irreducibility is incorrectly strengthened to primitivity

Pages 20–21 · proof of Corollary 7 · arXiv:2608.05649v1

Perron–Frobenius gives simplicity of the eigenvalue $1$ for an irreducible stochastic matrix, but it does not put every other eigenvalue strictly inside the unit circle. An irreducible matrix of period $q>1$ has peripheral $q$ th roots of unity. The explicit period-two map in the Statements finding realizes exactly this obstruction. A repair requires an aperiodicity or primitivity hypothesis; it cannot be obtained from the stated transitivity assumption.

Lemma A.2Incorrect as written; verified repair

The Jordan calculation uses the wrong multiplicity

Pages 30–31 · proof of Lemma A.2 · equations (55)–(59) · arXiv:2608.05649v1

A nilpotent part does not satisfy $Q_i^{m_i}=0$ when $m_i$ is merely the geometric multiplicity. Its nilpotency order is the largest Jordan-block size. Replacing $m_i$ by that order and expanding $(\lambda_i I+N_i)^n=\sum_{k=0}^{s_i-1}\binom nk\lambda_i^{n-k}N_i^k$ gives the required finite polynomial-exponential expansion. This repair propagates through Proposition 2.2 and Theorem 5 without changing the existence of full resonances.

Proof of Theorem 11Incorrect as written; verified repair

The displayed basis vectors are not eigenvectors

Page 27 · proof of Theorem 11 · arXiv:2608.05649v1

The proof says that $e_{(0,k)}$ is an eigenvector, but equation (34) gives $U_\omega e_{(0,k)}=z_2^k((z_1-\omega)/(1-\overline\omega z_1))^k$ up to normalization, which is not generally a scalar multiple of $e_{(0,k)}$ . The multiplicity conclusion is nevertheless recoverable. Each row carrying a nonzero diagonal value has no off-diagonal entry, so its coordinate functional is an eigenvector of $U^*$ . For $\lambda\ne0$ , $U-\lambda I$ is Fredholm of index zero; hence $\dim\ker(U-\lambda I)=\dim\ker(U^*-\overline\lambda I)$ . The triangular finite sections give the algebraic multiplicity as the number of occurrences, so the geometric multiplicity equals it.

Proposition 2.2(1)Incomplete as written; no repair supplied

Orthogonality does not follow by averaging the fibre decompositions

Pages 8 and 31–32 · Proposition 2.2(1) and its proof · arXiv:2608.05649v1

The proof shows only that $L$ is a compact perturbation of an operator whose norm is bounded by $\int\varepsilon_r(\omega)\,d\mathbb P(\omega)$ . It then asserts an orthogonal decomposition $L=\Pi_r+N_r$ . Pointwise relations $\Pi_{\omega,r}N_{\omega,r}=0$ do not imply $(\int\Pi_{\omega,r})(\int N_{\omega,r})=0$ , because cross terms with different $\omega$ remain. A spectral projection can give an orthogonal invariant splitting, but the proof does not establish the separately claimed one-step operator-norm bound for its remainder.

Definition 4 and Lemmas 3.2–3.4Typo

Two notation defects have unique harmless corrections

Pages 6–7 and 15–17 · Definition 4 and equations (16), (20) · arXiv:2608.05649v1

In Definition 4, replace the printed remainder $o(\varepsilon)$ by $o(\varepsilon^n)$ . In the Lasota–Yorke displays, the factor $\lambda^{-(k+r-1)}$ must multiply $\lVert h\rVert_r$ , while the bounded coefficient multiplies $\lVert h\rVert_{r-1}$ . The calculations immediately below the displays derive exactly these corrected inequalities, which are the form needed for the essential-spectral-radius argument.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2608.05649v1
Authors listed: Sakshi Jain, Maxence Phalempin
Audit date: August 18, 2026

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You are a mathematical auditor. Evaluate only the correctness of the paper's central statements, the correctness and completeness of its proofs, and evidence-backed non-novelty. Do not make editorial comments or evaluate importance, exposition, style, or suitability for publication. Keep the report neutral, precise, and professional. PART 1 — CORRECTNESS OF THE MAIN STATEMENTS Identify the paper's central statements and assess each one separately using exactly one of these outcomes: - Correct: the statement has a correct, verifiable proof; or, despite a defect in the printed proof, you can give a fully verified repair or independent proof. - Incorrect: you can give strong formal evidence that the statement as written is false, preferably a counterexample or a complete contradiction. - Not able to verify: the supplied proof does not establish the statement, and you can neither supply a fully verified proof nor disprove it. For every adverse or unverified outcome, restate the exact claim and hypotheses, identify its location in the reviewed version, and give the decisive reasoning. Do not infer that a theorem is false merely because its printed proof is defective. Assign the overall statements status by the following priority: 1. Contains wrong statements, if at least one central statement is Incorrect. 2. Contains unsupported statements, if no central statement is Incorrect but at least one is Not able to verify. 3. Correct, if every central statement is Correct, apart from any typographical or minor formal findings allowed by Part 4. PART 2 — CORRECTNESS AND COMPLETENESS OF THE PROOFS Audit the proofs of the central statements and every material internal or cited input on which they depend. For each proof or proof segment, use the narrowest supported outcome: - Correct and complete: every substantive step is verifiable and no case or crucial nontrivial step is missing. - Incorrect as written: a displayed calculation, inference, lemma, or cited application is formally false or inapplicable. - Incomplete as written: a necessary substantive step or case is missing, even though no explicit contradiction has been proved. - Not able to verify: the available paper and sources are insufficient to decide whether the step is valid. Do not flag omitted routine details, compressed arguments, or “analogously” steps when the omitted reasoning is recoverable and correct. For every defect, identify its smallest exact location, explain its downstream dependency, and classify any proposed repair as one of: - Verified repair: supply enough reasoning to verify that the repair works throughout every affected use. - Plausible repair only: describe the possible repair but do not treat it as established. - No repair supplied. Assign the overall proofs status by the following priority: 1. Contains incorrect or incomplete proofs, if at least one proof is Incorrect as written or Incomplete as written. 2. Contains unverified proofs, if there is no proved incorrectness or incompleteness but at least one material proof is Not able to verify. 3. Correct, if all audited proofs are Correct and complete, apart from any typographical or minor formal findings allowed by Part 4. PART 3 — NOVELTY OF THE MAIN STATEMENTS This is a one-sided, evidence-only check. Evaluate only central results. Report a result only when a verifiable primary source, publicly available before the earliest public version of the paper that contains the result, proves the same statement up to a change of notation, or when the result follows directly and immediately from such an earlier theorem. For each non-novelty finding, provide the earlier source, its public date, the exact theorem or proposition, and an explicit hypothesis-and-conclusion mapping. A later source cannot establish non-novelty. Do not flag a result merely because an easier proof exists or because it follows through a nontrivial argument. Do not flag a result that the paper itself explicitly presents as known, a survey item, or an alternative proof. Mention only statements that meet this non-novelty threshold. Say nothing about statements for which no such evidence was found: do not certify novelty, list unsuccessful searches, or add statement-level “no prior source found” entries. If the list is empty, the section may say only “No non-novelty findings.” PART 4 — TECHNICAL AND EVIDENTIARY RULES 1. Criticism must never be overstated. Call a statement incorrect, a proof incorrect or incomplete, or a result non-new only when strong formal evidence supports that exact judgment. Otherwise use Not able to verify where that outcome exists, or remain silent in the novelty section. 2. Review a specified paper version. Give exact page, theorem, equation, or proof locations; restate relevant quantifiers and hypotheses; and check domains, endpoints, parameter ranges, constants, and dependencies explicitly. Apply a defect only to the regimes and downstream results that actually depend on it. 3. Immediate-consequence closure: when deciding whether a conclusion is supported, treat the argument as including every immediate formal consequence of facts established earlier in the paper and available at that point. An immediate formal consequence is a local, mechanically checkable inference using only definitions already in force, elementary logic, direct substitution or renaming, weakening of an inequality or set inclusion, or a monotonicity or symmetry property explicit in a definition or already proved. 4. Such an inference must introduce no new lemma, estimate, construction, case analysis, limiting or compactness argument, external result, or new issue of uniformity or parameter dependence. Check the direction of monotonicity and the dependence of every threshold or constant explicitly. Do not exchange pointwise and uniform conclusions, change exceptional sets, interchange quantifiers or limits, or extend an endpoint under this rule unless the required fact was already established. 5. If the claimed conclusion follows after immediate-consequence closure, it is fully supported. Do not report a gap, unsupported statement, minor formal correction, or recommendation to add an explanatory sentence, and do not mention the suppressed item in the report. This suppression does not apply to a genuine typographical error as defined below, because a typo is a literal notation defect that still requires a textual correction. For example, if

P(delta)

is proved for every sufficiently small

delta>0

and

P

is immediately monotone in the required direction, then the claim that

P(delta)

holds for every

delta>0

requires no correction; the omitted monotonicity step is exposition rather than an audit finding. 6. Immediate-consequence closure permits reconstruction of an omitted inference; it does not permit silent alteration of the printed domain, hypotheses, or quantifiers of an intermediate statement later invoked outside its stated scope. If formal validity requires changing the text of that statement or its stated range, report the appropriate finding even when the repair is easy. Thus a result stated only for

t>0

and subsequently applied at

t=0

remains a formal scope mismatch unless the endpoint case was separately established before the application. 7. Apply this mandatory gate before recording any adverse, unverified, typographical, or minor formal finding: (a) identify the exact assertion or inference alleged to be defective; (b) reconstruct the strongest conclusion following from the preceding argument after all admissible immediate formal consequences are added; (c) identify the specific defect that remains — a formally demonstrated falsehood, a concrete nontrivial unresolved proof obligation, or a genuine domain, hypothesis, quantifier, or notation mismatch requiring a textual correction; and (d) state the evidence and minimal repair. If no specific defect remains, or if the only possible addition would be explanatory, suppress the finding entirely. Use Not able to verify only when a precise, nontrivial unresolved obligation can be identified. 8. A typographical error is an incorrect symbol, index, exponent, sign, variable name, or notation mismatch whose unique intended correction is mechanically determined from the surrounding text and does not alter the mathematical argument. Report it in yellow as Typo, with the printed text, correction, exact location, and reason it is harmless. It does not lower an overall status. 9. A minor formal correction is a boundary, endpoint, or displayed parameter-range defect that can be repaired consistently in every affected statement by a local change, without a new argument, without invalidating any related result or application, and without creating another substantive defect. Report it in yellow as Minor formal correction, with the same details. It does not lower an overall status. 10. Do not use either minor category if more than one materially different repair is possible; if the repair changes the substance or scope of a central result; if an excluded case is used later; if the correction fails to propagate consistently; or if a new mathematical idea is required. In such cases use the strongest substantive classification actually supported by evidence. 11. Write every mathematical expression as valid, compilable LaTeX. Use

...

for inline mathematics and

...

for displayed mathematics, with balanced delimiters and braces. Do not leave raw LaTeX commands outside math delimiters. 12. Separate theorem truth from the correctness of the printed proof. A theorem can be Correct in Part 1 because of a fully verified independent proof while its printed proof is adverse in Part 2. Conversely, a defective proof alone never licenses an Incorrect theorem classification. Organize the report under Parts 1–3 and apply Part 4 throughout. Do not add editorial commentary, scores, rankings, recommendations, or a list of suppressed observations.

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Annealed Ruelle-Pollicott Resonances

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

The annealed Blaschke spectrum is correct

Geometric multiplicity does not determine the polynomial powers in a Jordan expansion

Transitivity does not imply mixing

Irreducibility is incorrectly strengthened to primitivity

The Jordan calculation uses the wrong multiplicity

The displayed basis vectors are not eigenvectors

Orthogonality does not follow by averaging the fibre decompositions

Two notation defects have unique harmless corrections

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