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Paper metadata

Original submission
September 3, 2026
Claimed MSC
37E30, 37C15, 37D25, 37A35, 54F15, 54C10
arXiv categories
math.DS

Abstract

Let gg be a C∞C^\infty diffeomorphism in the isotopy class of a pseudo-Anosov homeomorphism ff such that gg and ff have the same topological entropy. In 1988, Handel proved that this implies the existence of a semiconjugacy ππ from gg to ff. He stated that, in general, there is at least one point xx such that π−1(x)π^{-1}(x) is disconnected. We show that this is not the case: for every xx, the set π−1(x)π^{-1}(x) is the intersection of a nested sequence of closed topological disks, and hence is connected. We also prove that there is a unique gg-invariant probability measure projecting to the measure of maximal entropy of ff. This measure is entropy-maximizing, hyperbolic, and Bernoulli, and the semiconjugacy induces a metric isomorphism between the corresponding measure-preserving systems.

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Audited against arXiv v1

Not a correctness certificate. A green “no issues found” result may include yellow suspected typos or minor formal issues 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.

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Detailed mathematical audit

Generated October 24, 2026
01Statements2 reported findingsSuspected unsupported statements

The existence, uniqueness, hyperbolicity and Bernoulli assertions for the lift of the pseudo-Anosov maximal-entropy measure are supported. A precise global-holonomy measure-class obligation remains in the proof that the particular semiconjugacy is a metric isomorphism; the cellular-fibre theorem subsequently depends on that step. No counterexample to either conclusion is established. The review covers all thirty-seven pages.

Theorem B: invariant measure and Bernoulli assertionsNo issue found

The large measure is uniquely determined and Bernoulli

PDF pp. 2, 12–23 · Lemmas 2.13–2.14, Propositions 3.2, 3.4 and 3.10–3.11 · arXiv:2609.03390v1

A continuous surjective factor has an invariant lift of μMME\mu_{\mathrm{MME}}: a measurable section followed by Cesaro averaging supplies one. Since μMME\mu_{\mathrm{MME}} is ergodic, almost every ergodic component of any such lift still projects to it. The factor inequality and equality of topological entropies force these components to be maximal-entropy measures. Forward and inverse Ruelle inequalities give opposite-sign exponents. An interval collapsed along a typical unstable leaf would, by recurrence and a periodic stable intersection, put the factor measure on the stable set of a periodic orbit, contradicting positive entropy. Expansion and recurrence consequently give unbounded images of the global invariant leaves. The bounded-distance lift of the semiconjugacy then supplies topological crossings of these leaves. The dynamical Sard argument upgrades them to transverse crossings, placing all large maximal-entropy components in the same homoclinic class. The homoclinic uniqueness theorem gives one component and hence uniqueness of the entire lift. The argument with every small class, rather than merely the orbit class, removes the cyclic period and yields Bernoulli. These conclusions precede, and do not require, the global injectivity argument in Section 4.

Theorem B: the particular factor map; Theorem ANot independently verified

Global injectivity and cellularity depend on an unresolved extension of the cited local holonomy theorem

PDF pp. 2, 26–30 and 32–35 · Proposition 4.5, Corollary 4.7, Proposition 4.10 and Section 5.1 · arXiv:2609.03390v1

Under the stated C∞C^\infty, isotopy and entropy-equality assumptions, the claimed conclusions are that π\pi itself is a metric isomorphism and every π−1(p)\pi^{-1}(p) is cellular. Proposition 4.2 does establish injectivity on a conull set intersected with each typical global stable or unstable leaf. To strengthen this to absence of any other point in the fibre on such a leaf, Proposition 4.5 constructs a stable holonomy from a positive-conditional-measure subset of one branch of a fold to another branch, potentially outside the conull set. Its asserted preservation of positive conditional measure is the precise unverified step described in Part 2. Mere local product structure on almost-everywhere defined rectangles does not, without its appropriate extension, identify the conditional measure on this other branch. If that extension holds, the subsequent contradiction, Corollary 4.7 and the monotonicity and polygon arguments have the stated implications. No proof that the extension fails, and no counterexample to the central claims, is supplied here.

02Proofs3 reported findingsFurther proof verification needed

The entropy, invariant-leaf, homoclinic and planar arguments were checked, including the lifted geometry and the displayed fold diagram. The remaining issue is the precise domain and conditional-measure identification in the cited holonomy application, not a deduction of theorem falsity.

Proposition 4.5: positive mass on the second fold branchNot independently verified

The cited primary theorem is local to admissible Markov rectangles

PDF pp. 27–28 · paragraph citing [Ova23, Theorems 5.7 and 6.1], displayed positive conditional mass and Figure 1 · arXiv:2609.03390v1

The argument has established a transverse global stable holonomy h−h^- on a positive unstable-conditional-measure set B′⊂J1+B'\subset J_1^+, with image in J2+J_2^+, another part of the same global unstable leaf. It then asserts μh−(w)ξ+(h−(B′))>0\mu^{\xi^+}_{h^-(w)}(h^-(B'))>0. The exact primary Ben Ovadia paper, Theorem 5.7(3), proves equivalence for the stable holonomy between the two coded plaques of chains sharing a common cylinder [R][R], on that Markov rectangle. Theorem 6.1 identifies the constructed leaf measures with equilibrium-state conditionals almost everywhere. Neither statement, as given there, directly covers arbitrary transverse global return branches on a typical global unstable manifold. In the present construction the target branch is not already shown to be one of those plaques carrying the corresponding conditional; indeed the desired contradiction is that its holonomy image lies outside the conull set. The concrete obligation is to decompose a positive-measure subset of this return holonomy into admissible rectangle holonomies, and prove that their target leaf measures coincide in measure class with the conditional on the common unstable atom after the backward iterate on p. 28. Invariance alone transports already identified measure classes; it does not establish the required initial identification on the target branch. A global homoclinic extension of the leaf-measure construction may supply this, but no fully verified such extension is supplied here. Plausible repair only. This is kept as Not able to verify rather than called an incorrect theorem or a formally false holonomy assertion.

Sections 2–4: checked entropy and homoclinic inputsNo issue found in the reviewed proof

The preceding measure and leaf arguments have the required scope

PDF pp. 10–26 · Theorem 2.9, Proposition 2.10, Sections 3.1–3.5 and Proposition 4.2 · arXiv:2609.03390v1

The primary Handel 1988 theorem, Theorem 0.1, supplies the global factor at the stated entropy equality. The global-shadowing uniqueness used in Proposition 2.10 follows from Handel 1985, Theorem 1(i) and property (1.4): the stable and unstable pseudodistances expand in opposite time directions. Deck equivariance makes the lifted factor a bounded distance from the identity, so equality of factor points is exactly bounded two-sided separation of the chosen lifted orbits. The BCS dynamical Sard theorem, Theorem 4.2, bounds the transverse dimension of leaves with a nontransverse intersection by 1/r1/r under Lipschitz holonomy. Applying it on the surface Pesin blocks and their countably many iterates, with arbitrarily large finite rr, gives the zero-dimensional exception used here; positive-entropy transverse conditionals do not charge it. The one-dimensional holonomy regularity needed on these blocks is checked in HHTU, pp. 47–48. The closures in Lemma 3.9 were checked visually: its assertion is concentration on W±(p)‾\overline{W^\pm(p)}, not on the unclosed periodic leaf. For Proposition 4.2, zero relative entropy makes ζ=ξ∨Pπ\zeta=\xi\vee\mathcal P_\pi invariant under g−1g^{-1}; the generating property of the increasing unstable partition then gives ζ\zeta equal to the point partition. Recurrence expands the positive internal radius in Lemma 4.3 to infinity, proving surjectivity on almost every entire invariant leaf.

Section 5.1, conditional on Proposition 4.5No issue found in the reviewed proof

The adapted polygon construction has the stated topological consequence

PDF pp. 30–35 · Claim 4.9, Proposition 4.10 and Lemmas 5.1–5.5 · arXiv:2609.03390v1

The remaining implication was checked with its dependence explicit. Once Proposition 4.5 and Corollary 4.7 hold, full factor-leaf images and uniqueness on the typical leaves identify lifted intersections and imply leafwise monotonicity. Product measure class for the pseudo-Anosov maximal-entropy measure allows polygon vertices to be selected in the conull image even near a pronged singularity. Their unique lifts determine stable/unstable sides; adjacent sides have only their intended vertex in common and nonadjacent sides have disjoint images, so the lifted boundary is Jordan. Its degree gives an interior point of the desired fibre. A point of that same fibre outside the polygon would be separated from this interior point by one of the extended supporting invariant leaves. Under all forward and backward iterates their factor images remain at bounded distance, forcing the common factor point onto that supporting pseudo-Anosov leaf, contrary to its location strictly inside the factor polygon. Thus the entire fibre is trapped. Disjoint factor polygons have disjoint lifted trapping disks by the order argument using a complete typical leaf. Nested factor polygons therefore give nested trapping disks whose intersection is exactly the lifted fibre. Compactness and deck equivariance permit sufficiently small such disks to project embeddedly to the surface. This verifies the topological conclusion conditional on the unresolved earlier holonomy step; it does not independently settle that step.

03Novelty0 reported findingsNo novelty concerns found

No non-novelty findings.

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Paper
arXiv:2609.03390v1
Authors listed
Pablo D. Carrasco, Federico Rodriguez Hertz, Jana Rodriguez Hertz, Raul Ures
Audit date
October 24, 2026
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