arXiv:2512.09528v2

Hyperbolic foliated entropy of suspensions

François Bacher

math.DSmath.CV37F7537A1537B4022D40

Paper metadata

Original submission: December 10, 2025
Latest revision: July 23, 2026 · version 2
Claimed MSC: 37F75, 37A15, 37B40, 22D40
arXiv categories: math.DS, math.CV
Paper license: arXiv non-exclusive distribution license ↗
Comments: To appear in Ergod. Th. & Dynam. Sys

Abstract

We study the hyperbolic entropies of foliations obtained by suspensions of a representation, in the sense of Dinh, Nguyên and Sibony (topological and measure-theoretic). We establish a link between this type of entropy and an adapted version of an entropy defined by Ghys, Langevin and Walczak for pseudo-groups of homeomorphisms. Such a link has various consequences. Among them, it implies that the hyperbolic entropy of foliations is not invariant by diffeomorphisms, and that a minimal entropy suspension admits an invariant measure. Finally, this allows us to study thoroughly the simple case in which the image of the representation is isomorphic to~ $\mathbb{Z}$ . In that case, we give the first exact estimate of the hyperbolic entropy, and prove a Brin--Katok type theorem and a variational principle, relying strongly on the standard ones for the entropy of maps.

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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.

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Generated August 18, 2026

01Statements3 reported findingsCorrect

The two main entropy comparisons are supported. Theorem 1.1/4.10 bounds hyperbolic foliated entropy in terms of transverse action entropy for general compact suspensions. Theorem 1.4/5.8 gives the exact local-entropy formula for a single-homeomorphism suspension with an ergodic harmonic measure: the base metric entropy plus the universal hyperbolic contribution 2. The stated corollaries follow from these formulas.

Theorem 1.1 / Theorem 4.10Correct

Comparison of foliated and transverse entropy

Pages 3–5 · Theorems 1.1/4.10 and 1.4/5.8 · arXiv:2512.09528v2

For a compact hyperbolic suspension, Theorem 1.1/4.10 compares the foliation's hyperbolic entropy with the entropy of the transverse group action. The additive hyperbolic contribution and the multiplicative comparison constant depend only on the fixed cocompact Fuchsian group and the chosen generating geometry, exactly as stated. The construction handles two-sided transverse Bowen balls; compactness makes the finite atlas and comparison constants uniform. The later smooth-invariance and finiteness corollaries use no stronger conclusion than this comparison.

Theorem 1.4 / Theorem 5.8Correct

Exact local entropy for a single-map suspension

Pages 4–5 and 27–29 · Theorem 1.4 / Theorem 5.8 · arXiv:2512.09528v2

For the suspension generated by $f$ and an ergodic harmonic measure, the local hyperbolic entropy equals $2+h_\nu(f)$ in the paper's time normalization. The constant $2$ is the exponential area growth of the hyperbolic disk. Conditional measures on transversals identify the remaining shrinking rate with two-sided Bowen balls for $f$ , to which Brin–Katok applies. Ergodicity makes the local value constant almost everywhere, as asserted.

Corollaries 1.2–1.3Correct

Non-invariance and the entropy-two characterization

Pages 3–4 and 21–27 · Corollaries 1.2–1.3 · arXiv:2512.09528v2

The comparison construction produces smoothly diffeomorphic suspensions with different transverse entropy, establishing that the entropy is not a smooth invariant in the claimed category. In the single-map case, the exact formula shows entropy 2 precisely when the relevant base entropy vanishes. The accompanying invariant-measure conclusion uses the stationary-to-invariant identification proved in Section 5 and retains its ergodicity hypothesis.

02Proofs3 reported findingsCorrect

Transversal entropy, stationary measures, and the variational step. A finite transverse atlas compares a leafwise hyperbolic ball of radius $R$ with words of length comparable to $R$ in the chosen generators. Both inclusions are proved, with bounded changes in radius and transverse tolerance. Thus a covering by foliated Bowen balls gives a covering for the transverse action and conversely; taking logarithmic growth rates proves Theorem 4.10 with its constants. Compactness controls chart overlaps and ensures that the comparison is uniform over leaves. No smooth conjugacy invariance is assumed during this covering argument.

Bowen-ball comparisonCorrect and complete

Transversal entropy, stationary measures, and the variational step

Pages 15–34 · Sections 4–5 · arXiv:2512.09528v2

A finite transverse atlas compares a leafwise hyperbolic ball of radius $R$ with words of length comparable to $R$ in the chosen generators. Both inclusions are proved, with bounded changes in radius and transverse tolerance. Thus a covering by foliated Bowen balls gives a covering for the transverse action and conversely; taking logarithmic growth rates proves Theorem 4.10 with its constants. Compactness controls chart overlaps and ensures that the comparison is uniform over leaves. No smooth conjugacy invariance is assumed during this covering argument.

Section 5Correct and complete

Harmonic disintegration, Bowen balls, and Brin–Katok

Pages 23–34 · Section 5 · arXiv:2512.09528v2

Disintegration of the harmonic measure on transversals yields a stationary measure for the random walk induced by the Fuchsian generators. In the cyclic suspension setting this is matched with the invariant base measure used for $f$ . Leafwise closeness over a radius- $R$ disk is equivalent, up to uniform constants, to base points remaining close for $|n|\le cR$ . Brin–Katok converts the conditional mass of these balls to $h_\nu(f)$ , and the hyperbolic leaf volume contributes $2$ . Boundary and chart errors are subexponential and disappear after division by $R$ .

Propositions 4.11–4.15Correct and complete

Consequences of the entropy comparison

Pages 20–23 · Propositions 4.11–4.15 · arXiv:2512.09528v2

The examples change transverse dynamics while preserving the smooth foliation type, so Theorem 4.10 detects different entropy values. Finiteness follows by covering the compact transversal with finitely many Lipschitz charts and using a finite generating set. In the entropy-two case the variational estimate forces zero transverse entropy and the compactness argument supplies an invariant measure. Each proposition uses the comparison in the direction its inequality supports.

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Paper: arXiv:2512.09528v2
Authors listed: François Bacher
Audit date: August 18, 2026

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Hyperbolic foliated entropy of suspensions

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Comparison of foliated and transverse entropy

Exact local entropy for a single-map suspension

Non-invariance and the entropy-two characterization

Transversal entropy, stationary measures, and the variational step

Harmonic disintegration, Bowen balls, and Brin–Katok

Consequences of the entropy comparison

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