{"id":"ad194ea1-690e-409c-971c-9965d173348c","arxiv_id":"2607.13556","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":2,"one_line_summary":"Under homogeneous exponential growth, every measure maximizing entropy along an expanding one-dimensional foliation has conditional measures equivalent to a canonically constructed weak Margulis measure.","lead":"This paper constructs reference measures on one-dimensional expanding foliations whose growth is uniformly exponential, and shows any entropy-maximizing measure along the foliation must have conditional densities comparable to these reference measures. The construction opens a route to rigidity theorems for partially hyperbolic systems and Anosov diffeomorphisms without assuming a global hyperbolic splitting.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 3.7's proof of f-invariance of B∞ assumes an unproved proportionality of conditional measures; without it, the proof of Theorem 3.2 is incomplete.","rationale":"The reader identified HEG as the weakest assumption, but HEG is an explicit hypothesis of Theorem 3.2; the theorem is a conditional statement, so the absence of a general proof of HEG for all minimal expanding foliations is not an internal defect of the central claim. The more load-bearing concern is internal: even assuming HEG, the proof of Theorem 3.2 relies on Lemma 3.7, and the key proportionality assertion there is unsupported. The centrality is clear: Lemma 3.7 is what gives µ(B∞)=0/1, which Lemma 3.8 converts into a contradiction with the entropy bound; without it, the proof of absolute continuity µ_x≺ν fails at the first step. This is more fundamental than the circularity in the 3-nilmanifold application (Remark 6.5) or the sign error in Theorem 7.12, which affect applications rather than the abstract construction. The paper's weak-Margulis construction (Theorem 2.5) may well be correct and valuable, so a flat rejection is not warranted; the verdict should remain CONDITIONAL, but the condition should now include a rigorous proof or replacement for Lemma 3.7.","tokens_in":31510,"tokens_out":21875,"duration_ms":243007,"concrete_test":"Take a C^{1+α} expanding circle map f, e.g. f(x)=2x+0.05 sin(2πx) mod 1, whose unique MME µ can be approximated numerically (Ulam's method or periodic orbit). Fix a finite partition η=F_A into equal subintervals of length 0.1. For a µ-typical x, compute R(y)=d(f_*µ_x)/dµ_{f(x)}(y) at two distinct points y1,y2 in η(f(x))∩f(η(x)) by evaluating both measures on small intervals around y_i. If |R(y1)-R(y2)| > 10^{-2}, the proportionality asserted in Lemma 3.7 is false in a one-dimensional HEG setting, confirming the gap. If R is constant, the lemma needs an explicit proof of that constancy.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In Section 3.2, Lemma 3.7 aims to prove that the 'bad set' B∞ is f-invariant, which is then combined with ergodicity to conclude µ(B∞)=0. To pass from x∈B∞ to f(x)∈B∞, the proof needs a lower bound on µ_{f(x)}(f(B)) in terms of µ_x(B). The authors assert: 'By the uniqueness of the disintegration, for almost every y∈F_A(x)∩f^{-1}(F_A(f(x))), one has d f_*(µ_x)/dµ_{f(x)}(y)=c_1'. This assertion does not follow from Rokhlin's uniqueness theorem: F_A is not f-invariant, so f_*µ_x and µ_{f(x)} are conditional measures of µ with respect to two different partitions — f(F_A) and F_A. For a general invariant measure, the density on the overlap is a cocycle that need not be constant on the atom; constancy is a Markov/Parry-type property. The proof invokes no property of F-MME or HEG at this step, so the gap is internal to the central theorem. Equation (3.5), and thus the invariance of B∞, depends on this unjustified proportionality. If B∞ is not known to be f-invariant, the contradiction in Lemma 3.8 cannot be obtained, and the proof of µ_x≺ν in §3.2 — and hence Theorem 3.2 — collapses.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a condition called homogeneous exponential growth (HEG) for a one-dimensional expanding foliation F preserved by a C^1 diffeomorphism f, and constructs a family of 'weak Margulis measures' Γ(D) on F-disks as weak-* limit points of pullbacks of normalized leaf volume. Theorem 2.5 establishes their invariance, Gibbs bounds, uniform equivalence on overlaps, and conformal Jacobian estimates. The central Theorem 3.2 asserts that for any ergodic measure of maximal F-entropy, the conditional measures on the atoms of a measurable foliation partition are boundedly equivalent to the corresponding weak Margulis measures. Under the additional assumption that the F-MME is a Gibbs F-state, Theorem 4.1 derives a measurable solution to the cohomological equation for the leaf Jacobian. The paper also proves a transversal holonomy theorem and applies the framework to codimension-one Anosov diffeomorphisms, derived-from-Anosov diffeomorphisms on T^3, partially hyperbolic diffeomorphisms on 3-nilmanifolds, and perturbations of the time-one map of geodesic flows, culminating in rigidity results including Theorem 7.12.","tokens_in":31939,"tokens_out":10741,"duration_ms":109633,"significance":"If the central results are correct, the paper gives a canonical reference measure class for all measures of maximal F-entropy in the HEG setting, with explicit uniform Radon-Nikodym bounds. This is a natural and potentially very useful extension of the classical Margulis construction beyond uniform hyperbolicity. The HEG condition is transparent, the geometric construction of Γ(D) is elegant, and many estimates are explicit. The applications are broad and ambitious, and the paper is generally well structured. However, the proof of the main theorem has a serious gap, and two of the applications contain load-bearing problems, one of which is explicitly admitted to be circular. These issues must be repaired before the claims can be regarded as established.","major_comments":[{"comment":"The step 'By the uniqueness of the disintegration, for almost every y∈F_A(x)∩f^{-1}(F_A(f(x))), one has d f_*(µ_x)/dµ_{f(x)}(y)=c_1' is not justified. The two conditional measures correspond to two different measurable partitions, f(F_A) and F_A. On the overlap F_A(f(x))∩f(F_A(x)), the Radon-Nikodym derivative is generally a cocycle and need not be constant; constancy is a Markov/Parry-type property that is not proved and is not shown to follow from the F-MME or HEG hypotheses at this point. Equation (3.5), and therefore the conclusion that f(x)∈B∞, depends on this constancy. Since Lemma 3.8 and the contradiction with Lemma 3.4 rely on the f-invariance of B∞, this gap affects the proof of µ_x≺ν and hence Theorem 3.2. This is the central mechanism of the paper and needs a complete proof.","section":"§3.2, Lemma 3.7"},{"comment":"The authors explicitly state that the proof of Proposition 6.4 is circular, because the semi-conjugacy h:M→T^2 was constructed using the quasi-isometry result from [14], which is exactly what Proposition 6.4 is supposed to establish. Since Proposition 6.4 is then used to prove HEG for the unstable foliation of partially hyperbolic diffeomorphisms on 3-nilmanifolds, Theorem 6.6 is not established as written. If [14] already contains a quasi-isometry theorem for the unstable foliation, the paper should cite it directly and derive HEG from it; otherwise an independent proof is needed. As written, this application rests on an acknowledged circularity.","section":"§6.2.2, Proposition 6.4 and Remark 6.5"},{"comment":"Assumptions (1) and (2) are impossible as stated. For any partially hyperbolic diffeomorphism in a small C^1 neighborhood of g^1, the strong stable Lyapunov exponent is negative and the strong unstable exponent is positive for every invariant measure, while h_top(f)>0. Thus λ_s(m)=λ_u(m)=h_top(f) cannot hold for volume-preserving m, and λ_s(µ_MME)=λ_u(µ_MME)=h_top(f) cannot hold for any invariant measure. The theorem is therefore vacuous. If the intended hypothesis is λ_u(m)=h_top(f) and λ_c(m)=0 (or the analogue for µ_MME), the statement must be corrected, and the role of λ_s in the proof of absolute continuity of F^cs must be reexamined.","section":"§7.2, Theorem 7.12"}],"minor_comments":[{"comment":"In the proof, the numerator uses e^{(n-n0(D))H_F} but should use e^{(n-n0(I))H_F}; the denominator uses e^{-(n-n0(I))H_F} but should use e^{-(n-n0(D))H_F}. The final estimate is correct, but the intermediate lines are misleading.","section":"§2.2, Lemma 2.9"},{"comment":"Equation (7.1) compares |f^n(D1)|_u with itself; the right-hand side should be |f^n(D2)|_u.","section":"§7.1, Lemma 7.4"},{"comment":"The text refers to 'Lemma 2.7' and 'Proposition 2.9' when it means Lemma 2.9; the numbering is inconsistent.","section":"§2.2, after Lemma 2.9"},{"comment":"In the displayed definition of ϕ(x), the limit should be as n→∞, not n→0.","section":"§4.1, definition of ϕ"},{"comment":"The notation \tildeϕ and \tilde eϕ is used inconsistently in the proof; the same symbol should denote the same function throughout.","section":"§4.2, Lemma 4.3"},{"comment":"The phrase 'the last inequality is due to Theorem 7.2' should read 'the last equality'.","section":"§7.2, proof of Theorem 7.12"}],"recommendation":"major_revision","confidential_remarks":"This is an ambitious and potentially important paper, but the current version cannot be accepted: the gap in Lemma 3.7 is load-bearing for the main theorem, and the applications in §6.2 and §7.2 contain serious issues, including an explicitly admitted circular proof. I would need to see a complete repair of these points before recommending publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The genuinely new thing here is the construction of weak Margulis measures for arbitrary expanding foliations under homogeneous exponential growth, using a geometric limit-point argument instead of functional-analytic fixed points. Theorem 2.5’s Gibbs and conformal properties are elementary and look correct once the typos in Lemma 2.9 are fixed (the indices n0(I) and n0(D) are swapped in the intermediate lines, but the final formula is right). That part is worth taking seriously.\n\nThe soft spots are real. The stress-test note about Lemma 3.7 lands. The assertion that d f_*(μ_x)/dμ_{f(x)} is constant on the overlap does not follow from Rokhlin’s uniqueness theorem: f_*(μ_x) and μ_{f(x)} are conditionals with respect to two different partitions, and the density on the intersection is generally a cocycle, not a constant. Without that, the f-invariance of B_∞ and hence μ(B_∞)=0 are not established. That is load-bearing for Theorem 3.2. It might be fixable with a separate argument, but as written the proof of the main equivalence theorem is incomplete.\n\nThe applications section has its own problems. Remark 6.5 openly admits that Proposition 6.4 is circular: the semi-conjugacy to T^2 was built using the quasi-isometry result it purports to prove. So the 3-nilmanifold HEG verification is not independent. And Theorem 7.12 assumes λ_s(m)=λ_u(m)=h_top(f); for a volume-preserving partially hyperbolic diffeomorphism the stable exponent is strictly negative, so this condition is impossible as stated. The same objection applies to the μ_MME version unless μ_MME is not supported on the partially hyperbolic splitting, which is not argued. Lemma 7.4’s displayed (7.1) compares a disk with itself; that is a harmless typo but indicative of careless proofreading.\n\nWhat the paper does well is the core idea: a geometric, uniformity-based construction of reference measures on leaves, with a clean Gibbs property, and the high-level strategy of using those measures to constrain F-MMEs. The authors are also honest about at least one circularity, which is more than many papers manage.\n\nA serious referee should be assigned, because the construction and the F-MME equivalence are potentially important and do not look irreparable. But the referee should be told to focus on Lemma 3.7 first: either the proportionality claim gets a real proof, or Theorem 3.2 is downgraded to a conditional result. The applications need correction or removal. After that, this could be a solid paper.","headline":"The weak Margulis construction is a plausible new tool, but the proof of the central Theorem 3.2 has a real gap at Lemma 3.7, and the applications section has an admitted circularity and an impossible Lyapunov assumption.","tokens_in":32362,"tokens_out":2886,"would_cite":false,"duration_ms":31519,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37D20","37D25","37A35","37C40"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper builds leaf-by-leaf reference measures for any expanding foliation with homogeneous growth, and proves every measure of maximal F-entropy must be conditionally equivalent to them on almost every leaf.","keywords":["expanding foliations","Margulis measures","measures of maximal entropy","partial hyperbolicity","cohomological equation","homogeneous exponential growth","geodesic flows","Anosov diffeomorphisms"],"falsifier":"Construct an expanding foliation that is minimal but lacks homogeneous exponential growth — for instance, a skew product over a base where expansion rates vary from leaf to leaf — and compute Γ(D) directly: if the Gibbs inequalities fail or two ergodic F-MMEs have non-equivalent conditional measures, the main theorem is false. Alternatively, in the 3-nilmanifold example, independently verify the quasi-isometry of the unstable foliation, removing the circularity noted in Remark 6.5.","tokens_in":31432,"feed_emoji":"📏","tokens_out":3575,"duration_ms":40318,"temperature":0.7,"pith_summary":"The paper tries to show that a single family of leafwise measures, constructed purely geometrically from the dynamics and requiring no partial hyperbolicity, governs all measures of maximal entropy along an expanding foliation. Under a uniformity assumption called homogeneous exponential growth, it proves that the conditional measures of any ergodic measure of maximal F-entropy are equivalent to these reference measures, with Radon-Nikodym derivatives bounded independently of the base point. It then shows that if such a measure is also a Gibbs F-state, the log-Jacobian of the diffeomorphism along the foliation is cohomologous to a constant. If correct, this gives a canonical leafwise structure for all F-MMEs and yields rigidity statements for Anosov diffeomorphisms, partially hyperbolic diffeomorphisms on nilmanifolds, and perturbations of time-one maps of geodesic flows.","feed_headline":"Leafwise reference measures pin down all entropy-maximizing states","feed_subtitle":"A geometric construction without hyperbolicity gives canonical conditional measures on expanding foliations and a Jacobian rigidity theorem.","key_machinery":"The central object is the family Γ(D) of weak Margulis measures on an F-disk D, defined as limit points of (f^{-n})_*(vol_F|_{f^n(D)}). The load-bearing identity is the Gibbs property ν(I) ≍ L e^{-n0(I)H_F} / e^{-n0(D)H_F}, which relates the measure of any subdisk to the time n0 it takes to grow to unit length. The homogeneous exponential growth condition, |f^n(D)|_F ≍ C_G e^{nH_F} for all unit-size disks, is what makes this comparison uniform across leaves and scales; measurability of the function e^{-n0(·)H_F} replaces any measurable selection of sections.","core_discovery":"The central claim is Theorem 3.2: for a C^{1+α} diffeomorphism preserving an expanding foliation F with homogeneous exponential growth, any ergodic measure of maximal F-entropy has conditional measures equivalent to the reference measures Γ(F_A(x)) on almost every leaf, with density ratio bounded by a constant independent of x. The reference measures are weak-* limit points of pullbacks of normalized leafwise Lebesgue measure under backward iteration, and they satisfy a Gibbs property controlled by the time n0 at which a leaf disk first reaches length one. The paper also proves a rigidity theorem: when a measure of maximal F-entropy is a Gibbs F-state, the log-Jacobian along the leaves solve","pith_inferences":["If homogeneous exponential growth turns out to be automatic for every minimal expanding foliation, as the paper suspects, then the equivalence theorem would give a canonical conditional-measure structure for all such foliations without any extra hypothesis; checking this could be done by verifying quasi-isometry of leaf embeddings in candidate examples.","The measurable solution to the cohomological equation might be promotable to a continuous solution whenever the F-MME is fully supported, following existing techniques for partially hyperbolic cohomological equations; this would strengthen the rigidity conclusion in the geodesic-flow perturbation setting.","The construction relies essentially on one-dimensional leaves, where ordering and the integer n0 are natural; a higher-dimensional version would need a different normalisation, perhaps replacing lengths by leafwise volumes and n0 by a covering-time, and the current proof does not indicate how that would work.","In the 3-nilmanifold application, the paper's Remark 6.5 flags that the verification of HEG uses a quasi-isometry whose proof is circular; an independent proof of quasi-isometry for those unstable foliations would make that part of the theorem unconditional."],"forward_implications":["All ergodic measures of maximal F-entropy in the homogeneous-growth setting have leafwise conditional measures that are uniformly equivalent to a single canonical reference class, so their supports consist of entire F-leaves.","If such a measure is also a Gibbs F-state, the log-Jacobian along F is cohomologous to the constant H_F, giving concrete periodic-point Jacobian constraints.","The construction applies to co-dimension one Anosov diffeomorphisms, derived-from-Anosov diffeomorphisms on the 3-torus, partially hyperbolic diffeomorphisms on 3-nilmanifolds, and C^1-perturbations of time-one maps of geodesic flows on negative-curvature surfaces.","For C∞ volume-preserving perturbations of time-one geodesic-flow maps whose stable and unstable Lyapunov exponents equal the topological entropy, the diffeomorphism is the time-one map of a smooth flow, smoothly conjugate to an algebraic flow."],"fun_headline_variants":["Expanding foliations get canonical leaf measures","Entropy-maximizing states forced to match reference measures","Reference measures dictate all u-entropy states","New leaf measures rigidify expanding foliations","Leaf measures settle entropy maximizers"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The construction and theorem collapse unless every leaf disk of length between 1 and K_f grows in length exactly like e^{nH_F} up to a uniform constant, for all iterates, with no exceptional leaves or scales.","fun_headline_variants_meta":{"raw":{"variants":["Expanding foliations get canonical leaf measures","Entropy-maximizing states forced to match reference measures","Reference measures dictate all u-entropy states","New leaf measures rigidify expanding foliations","Leaf measures settle entropy maximizers"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000779,"raw_usage":{"total_tokens":3252,"prompt_tokens":688,"completion_tokens":2564,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":432,"completion_tokens_details":{"reasoning_tokens":2497}},"tokens_in":432,"tokens_out":2564,"duration_ms":17480,"temperature":1.0,"reasoning_tokens":2497,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T04:50:51.490588+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Construct an expanding foliation that is minimal but lacks homogeneous exponential growth — for instance, a skew product over a base where expansion rates vary from leaf to leaf — and compute Γ(D) directly: if the Gibbs inequalities fail or two ergodic F-MMEs have non-equivalent conditional measures, the main theorem is false. Alternatively, in the 3-nilmanifold example, independently verify the quasi-isometry of the unstable foliation, removing the circularity noted in Remark 6.5.","supporting_citations":[],"review_version":1}