arXiv:2606.10007v1

On automorphic measures, Lyapunov exponents and instability of rational maps

Makienko Peter, Carlos Cabrera

math.DS37F1037F1537A25

Paper metadata

Original submission: June 8, 2026
Claimed MSC: 37F10, 37F15, 37A25
arXiv categories: math.DS
Paper license: arXiv non-exclusive distribution license ↗

Abstract

To construct obstructions to the stability of rational maps with non-summable critical points in their Julia sets, we introduce automorphic measures with complex eigenvalues for rational maps on the Riemann sphere. In particular, these measures extend the classical notions of quasi-invariant and conformal measures by allowing the respective Radon--Nikodym derivative to be complex-valued and proportional to a multiplicative cocycle. $j_{(s,t)}(R) = |R'|^{s} \left(\frac{|R'|}{R'}\right)^{t},$ which plays the role of a generalized automorphy factor in the sense of group actions. The existence of such measures reveals a close connection between geometric and dynamical properties of rational maps. We show that the existence of certain automorphic measures, particularly unimodular measures and their associated vector fields, implies instability of the corresponding rational map. Specifically, for a weakly dissipative rational map admitting a $( -1, 1)$ -unimodular measure, there exists an integer $q \ge 1$ such that the map is $q$ -unstable. This result generalizes earlier instability criteria involving pseudoconformal measures and connects the presence of such measures to the failure of structural stability. Furthermore, we establish ergodic and combinatorial conditions ensuring the existence of unimodular measures or vector fields -- most notably, through bounded recurrence, bounded velocity of arguments, and relations with the Milnor--Thurston kneading theory. These criteria provide a unified framework linking automorphic measures, Lyapunov spectra, and the geometric deformation spaces of rational maps.

AI-generated audit

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

01Statements4 reported findingsContains unsupported statements

The definitions and several elementary measure/cocycle observations are correct, but the main hyperbolicity and instability conclusions are not able to be verified. Essential compactness and deformation-space steps are asserted without hypotheses or proofs, and the construction of a global unimodular vector field from a positive-measure level set does not establish the required invariance.

Theorem 2Not able to verify

The characterization by negative pseudoconformal measures is not established

Pages 6 and 24–30 · Theorem 2, Lemmas 26–28, and Theorem 27 · arXiv:2606.10007v1

The converse direction uses Theorem 27(5), which is supposed to pass conformal measures on degenerating cycles of maps $R_i$ to a pseudoconformal measure for the limit map $R$ . For negative $s$ , the proof introduces Radon–Nikodym factors $\varphi_i$ and then says `by assumption' that they converge uniformly on compact subsets away from the critical values. No such assumption appears in the definition of a degenerated sequence, and weak convergence of the atomic cycle measures does not imply uniform convergence of those derivatives. Without this step, the conformal identity cannot be passed to the limit, so the deduction from non-hyperbolicity to existence of the measure remains unsupported.

Theorems 4 and 5Not able to verify

The q-stability and unimodular-measure instability chain lacks the hypotheses used in its proof

Pages 7–8 and 31–34 · Theorems 4 and 5, Lemma 29, Corollary 30, and Proposition 24 · arXiv:2606.10007v1

Lemma 29 assumes every critical orbit is infinite and that there are no critical relations. Corollary 30 drops those assumptions, and the proof of Theorem 4(2) invokes Lemma 29 for general hyperbolic maps, although hyperbolic maps may have periodic critical points and critical relations. Theorem 5 then uses Corollary 30 and Proposition 24, the latter again requiring no critical relations, without deriving those conditions from q-stability or from weak dissipativity. These are substantive missing cases, not notation changes, and the paper supplies no alternative deformation-space argument for them.

Theorem 11Not able to verify

The proof does not construct a globally valid unimodular vector field

Pages 11–12 and 35–38 · bounded-recurrence theorem · arXiv:2606.10007v1

From a Fourier coefficient of an eigenfunction, the proof obtains a phase $\tau$ satisfying $\tau^N=\lambda$ and chooses a positive-measure level set $S_j=\{\tau=j\}$ . It then declares the restriction to $S_j$ a unimodular vector field. The required equation, however, must hold almost everywhere for a measure whose support is dynamically compatible; the proof does not show that $S_j$ is invariant or nonsingular under $R$ , so extending the restriction by zero need not satisfy the automorphy relation. The earlier analogous selection of a positive-measure level set has the same issue.

Basic automorphic-measure identitiesCorrect

The variation and cocycle calculations are correct where their hypotheses hold

Pages 2–10 and Section 2 · definitions, transfer identities, and Propositions 7–10 · arXiv:2606.10007v1

Taking variation of a unimodular automorphic measure yields the stated pseudoconformal measure, and its Radon–Nikodym derivative satisfies the displayed unit-vector cocycle. The transfer-operator inequalities and the construction from an already existing invariant line field are algebraically correct. These valid preliminary facts do not resolve the missing existence and deformation arguments above.

02Proofs4 reported findingsContains incorrect or incomplete proofs

Three central proof chains are incomplete: conformal identities are passed through degenerating maps using an unstated uniform convergence assumption; q-stability arguments apply lemmas outside their hypotheses; and positive-measure level sets are used without proving invariance. These gaps propagate to Theorems 2, 4, 5, and 11 and their instability corollaries.

Theorem 27(5)Incomplete as written

Weak convergence of cycle measures does not give uniform convergence of Radon–Nikodym factors

Pages 28–29 · degenerated-sequence case · arXiv:2606.10007v1

The definition of a degenerated sequence controls only the cycle multipliers and local uniform convergence of the rational maps away from poles. It gives no compact-uniform control of $d(\rho_{s,X_i}\circ R_i^{-1})/d\rho_{s,X_i}$ on the varying atomic supports. The asserted convergence of $\varphi_i$ is therefore not an available assumption, and critical-point concentration can make negative powers singular. A repair needs explicit tightness away from critical values and a uniform-integrability argument strong enough to pass the weighted pullback identity.

Lemma 29, Corollary 30, and proof of Theorem 4Incomplete as written

The no-critical-relations hypothesis disappears at the point it is needed

Page 31 · deformation-space argument · arXiv:2606.10007v1

Lemma 29 states its dimension criterion only when every critical point has infinite orbit and no critical relations occur. Corollary 30 restates the criterion for `a rational map' without retaining either condition. The proof of Theorem 4(2) then applies it to arbitrary hyperbolic structurally stable maps. Since hyperbolicity does not exclude periodic critical points, the cited lemma cannot justify that step. The missing critical-relation cases must be analyzed separately or the theorem restricted.

Proof of Theorem 5Incomplete as written

The transfer-operator cancellation invokes an inapplicable proposition

Pages 32–34 · generalized Cauchy-transform argument · arXiv:2606.10007v1

The proof uses Proposition 24 to expand the Ruelle transform and then uses q-stability to select invariant Beltrami differentials forcing all critical-value coefficients to vanish. Proposition 24 assumes normalization and absence of critical relations, while Theorem 5 assumes neither. Corollary 30, used for the selection, inherits the unresolved hypothesis problem above. Consequently the eigenfunction relation needed for Lemma 31 and the contradiction with weak dissipativity has not been obtained in the full stated generality.

Proof of Theorem 11Incomplete as written

A positive-measure cocycle level set is not shown to support an automorphic measure

Pages 36–38 · essential-values and Fourier-coefficient argument · arXiv:2606.10007v1

Selecting $S_j$ with positive measure only proves that one root value occurs frequently. To turn a phase on $S_j$ into a vector field whose product with the pseudoconformal measure is automorphic, one must prove $R^{-1}S_j=S_j$ modulo that measure or construct an invariant component carrying the same cocycle equation. Neither is done. The missing invariance cannot be supplied by merely restricting the vector field, because the automorphy identity compares values at $x$ and $R(x)$ .

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

Detailed audit reportFull reasoning, manuscript locations, and references.

Open report PDF ↗

Author response

Challenge an audit finding

Local workflow preview

A listed author may submit formal evidence that an audit is inaccurate. The response would be considered in a fresh AI re-evaluation; it would not edit the audit automatically.

Paper: arXiv:2606.10007v1
Authors listed: Makienko Peter, Carlos Cabrera
Audit date: August 18, 2026

01Establish identityMatch an authenticated scholarly identity to this paper.
02Submit evidenceIdentify the finding and give a formal mathematical response.
03Re-evaluateA separate agent checks the response and records a disposition.

Email fallback for papers without a linked ORCID

A production fallback could send a one-time link only when the submitted address matches an independently maintained author-contact allowlist for this paper. MathAudit must return the same message for every address so the form cannot reveal which contacts are on that list.

Structured response preview

This form remains unavailable until production identity verification succeeds. Nothing entered here is submitted.

Audit finding challengedWhy the evaluation is inaccurateFormal evidence or repairVerifiable sources, if anyI am submitting evidence, not instructions for the reviewing agent.

This panel never establishes authorship in the local prototype. A production result should be described narrowly as an authenticated ORCID match or control of a separately allowlisted author-contact mailbox.

Audit controls

Status: Report ready
Source: arXiv v1
Prompt: Standard prompt · locked

The report is available below and in the public summary.

Review prompt

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.

This panel is visibly marked as owner-only for the concept prototype. Authentication is not enabled locally.

On automorphic measures, Lyapunov exponents and instability of rational maps

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

The characterization by negative pseudoconformal measures is not established

The q-stability and unimodular-measure instability chain lacks the hypotheses used in its proof

The proof does not construct a globally valid unimodular vector field

The variation and cocycle calculations are correct where their hypotheses hold

Weak convergence of cycle measures does not give uniform convergence of Radon–Nikodym factors

The no-critical-relations hypothesis disappears at the point it is needed

The transfer-operator cancellation invokes an inapplicable proposition

A positive-measure cocycle level set is not shown to support an automorphic measure

Challenge an audit finding

Authenticate with ORCID, then require an exact arXiv match