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Margulis Measures on Expanding Foliations: Construction and Rigidity

T0 review · 3 major / 6 minor · reviewed 2026-08-02 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2607.13556 v1 pith:VDS5FKOH submitted 2026-07-15 math.DS

Jérôme Buzzi , Yi Shi , Fan Yang , Jiagang Yang This is my paper · ORCID
classification math.DS MSC 37D2037D2537A3537C40
keywords expandingfoliationsMargulismeasuresofmaximalentropypartialhyperbolicitycohomologicalequationhomogeneousexponentialgrowthgeodesicflowsAnosovdiffeomorphisms
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 6 minor

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.

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 (3)
  1. [§3.2, Lemma 3.7] 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.
  2. [§6.2.2, Proposition 6.4 and Remark 6.5] 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.
  3. [§7.2, Theorem 7.12] 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.
minor comments (6)
  1. [§2.2, Lemma 2.9] 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.
  2. [§7.1, Lemma 7.4] Equation (7.1) compares |f^n(D1)|_u with itself; the right-hand side should be |f^n(D2)|_u.
  3. [§2.2, after Lemma 2.9] The text refers to 'Lemma 2.7' and 'Proposition 2.9' when it means Lemma 2.9; the numbering is inconsistent.
  4. [§4.1, definition of ϕ] In the displayed definition of ϕ(x), the limit should be as n→∞, not n→0.
  5. [§4.2, Lemma 4.3] The notation ildeϕ and ilde eϕ is used inconsistently in the proof; the same symbol should denote the same function throughout.
  6. [§7.2, proof of Theorem 7.12] The phrase 'the last inequality is due to Theorem 7.2' should read 'the last equality'.

Circularity Check

1 steps flagged · score 4.0 of 10

Self-admitted circular reasoning in the 3-nilmanifold application (Remark 6.5); the central weak-Margulis construction is not circular.

  1. other [Section 6.2.2, Remark 6.5 (justifying Proposition 6.4 and Theorem 6.6)]
    "Here the proof is in fact a circular reasoning. In [14] it was first proven that F^u_f is quasi isometric, and therefore f has a global product structure. The semi-conjugacy h:M→T^2 was built using this fact."

    Proposition 6.4 uses the semi-conjugacy h (from [37, Section 8]/[14, Corollary 1.2]) to prove that h is a quasi-isometry on unstable leaves; Theorem 6.6 then derives homogeneous exponential growth for F^u from this quasi-isometry. But the quoted remark admits h itself was constructed using the quasi-isometry of F^u proved in [14]. Hence the proof of Proposition 6.4 assumes the quasi-isometry it is supposed to establish, and the HEG verification for 3-nilmanifolds inherits this circular dependency. This is an application-level circularity, not a defect in the conditional construction of Sections 2–4.

full rationale

The weak Margulis construction (Section 2) is not circular: Γ(D) is defined as weak-* limits of pushed-forward leafwise Lebesgue measures, and the Gibbs/conformal properties are direct consequences of the homogeneous-growth assumption (Definition 2.1, eq. (2.2)); no fitted parameter is relabeled as a prediction. Theorem 3.2 is a substantive conditional statement: assuming an ergodic F-MME exists, it derives equivalence of its conditional measures to Γ from entropy comparisons. The proof contains a serious unsupported step in Lemma 3.7: the constancy of d f_*µ_x/dµ_{f(x)} is asserted from 'uniqueness of the disintegration' although F_A and f(F_A) are different partitions, so the density is a cocycle and need not be constant. I flag this as a non-circular correctness gap: it is load-bearing for the f-invariance of B∞ and hence for the contradiction in Lemma 3.8, but it is not a reduction of the conclusion to the assumptions. The explicit circularity in the paper is the admitted Remark 6.5: the proof of Proposition 6.4 relies on the very quasi-isometry from [14] that was used to build the semi-conjugacy h, and Theorem 6.6 depends on Proposition 6.4. Because this affects an application (HEG for 3-nilmanifolds) rather than the central conditional theorems, and because it is openly acknowledged, the overall circularity score is moderate (4), not severe.

Assumptions & free parameters 2 free parameters · 6 assumptions · 1 invented entities

The paper postulates HEG, the expanding foliation condition, and regularity assumptions, then imports several external theorems. The main fragile input is HEG; the explicit circular proof for the 3-nilmanifold application and the apparently inconsistent Lyapunov assumption in Theorem 7.12 add further burden.

free parameters (2)
  • H_F = equal to h_top^F(f) by Lemma 2.3
    Introduced in Definition 2.1 as part of the HEG hypothesis; identified with the foliation topological entropy, not chosen to fit a target result.
  • C_G = >1, arbitrary
    Uniformity constant in HEG; its value enters the constants L, L_0, L_H but not the qualitative conclusions.
assumptions (6)
  • domain assumption Homogeneous exponential growth (Definition 2.1)
    The construction of weak Margulis measures and all Gibbs/conformal estimates rely on Eq. (2.2).
  • domain assumption Expanding foliation condition (2.1): 1<λ<m(Df|TF)≤||Df|TF||<K_f
    Uniform expansion along F is assumed throughout and is needed for n0 estimates and distortion control.
  • domain assumption C^{1+α} regularity of f (Remark 3.3)
    Lemma 3.7 requires distortion control that fails for merely C^1 maps.
  • domain assumption Existence of measures of maximal F-entropy
    Existence of F-MMEs is imported from [40, Theorem A] and [19, Proposition 2.15].
  • domain assumption Transversal foliation eF is not expanding (Definition 5.2)
    Theorem 5.3 on holonomy absolute continuity requires this non-expansion condition.
  • ad hoc to paper Lyapunov coincidence λ_s(m)=λ_u(m)=h_top(f) in Theorem 7.12
    As written this is inconsistent with volume preservation near time-one geodesic flows: λ_s is negative while h_top is positive, so the equality cannot hold.
invented entities (1)
  • Weak Margulis measures Γ(D)
    purpose: Reference conditional measures on F-disks whose Jacobian is controlled by HEG
    Constructed as weak-* limit points of pushed leaf volumes; existence is automatic by compactness, but the Gibbs and conformal properties are the content of Theorem 2.5 and depend on HEG.

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Pith. "Pith review of Margulis Measures on Expanding Foliations: Construction and Rigidity." pith.science (2026). https://pith.science/paper/VDS5FKOH

@misc{pith2026260713556,
  author       = {Pith},
  title        = {Pith review of: Margulis Measures on Expanding Foliations: Construction and Rigidity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VDS5FKOH}},
  note         = {Machine review of arXiv:2607.13556}
}
abstract

Given a diffeomorphism preserving a one-dimensional expanding foliation $\mathcal F$ with homogeneous exponential growth, we construct a family of reference measures on each leaf of the foliation with controlled Jacobian and a Gibbs property. We then prove that for any measure of maximal $u$-entropy, its conditional measures on each leaf must be equivalent to the reference measures. When the measure of maximal $u$-entropy is a Gibbs $\mathcal F$-state (i.e., when the reference measures are equivalent to the leafwise Lebesgue measure), we prove that the log-Jacobian of $f$ must be cohomologous to a constant via a measurable function. We provide several applications, including the strong and center foliations of Anosov diffeomorphisms, factor over Anosov diffeomorphisms, and perturbations of the time-one map of geodesic flows on surfaces with negative curvature.

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