MathAudit

Paper metadata

Original submission
October 18, 2026
Claimed MSC
37D30, 37C40, 37D25, 37A25
arXiv categories
math.DS
Comments
85 pages, 1 figure

Abstract

We study the statistical properties of partially hyperbolic diffeomorphisms with a two-dimensional center exhibiting mixed behavior, without assuming a dominated splitting inside the center. We introduce the notion of an expanding on average for partially hyperbolic diffeomorphism and, under a center jacobian assumption, we prove the existence of finitely many ergodic hyperbolic SRB measures, and show the strong positive recurrence property, which implies that each SRB measure, up to a period, satisfies exponential decay of correlations, among other strong statistical laws. We apply our results to several examples.

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Audit summary

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.

Current report

Detailed mathematical audit

Generated October 23, 2026
01Statements2 reported findingsNo statement issues found

The finiteness and full-basin theorem for hyperbolic SRB measures, continuity of the two center exponents, strong statistical properties, and robustness consequences are stated with the required two-dimensional center, regularity, bunching, average-expansion, and positive-gap assumptions. No false central statement was found.

Theorems A and BNo issue found

Average expansion and a positive exponent gap give uniform hyperbolicity and continuous center exponents

PDF pages 3–4, 11–27, and 45–58 · Theorems A–B and Sections 3–7 · arXiv:2610.25559v1

Average expansion forces the top center exponent of every unstable Gibbs measure above the optimal constant, while the determinant bound defining the gap forces the bottom exponent below its negative. Projective unstable Gibbs measures select the positive Oseledets direction uniquely; compactness under perturbation then gives continuity of the top exponent, and the determinant identity gives continuity of the bottom one. Uniform Pesin blocks yield finiteness, while effective hyperbolicity supplies existence and the full-basin argument.

Theorem CNo issue found

The strong positive recurrence conclusion follows from uniform exponent control

PDF pages 4 and 59–68 · Theorem C and Section 8 · arXiv:2610.25559v1

Measures in a fixed homoclinic class with geometric pressure tending to zero converge to unstable Gibbs measures, and their center exponents converge as well. The uniform-Pesin-block criterion therefore rules out escape from every fixed block, establishing strong positive recurrence. The cited structural theorem then gives the finite cyclic decomposition and the listed statistical laws.

02Proofs3 reported findingsNo proof issues found

The principal proof chains are mathematically coherent. One harmless reversed inequality in the choice of an auxiliary positive parameter has a uniquely determined correction; with that typographical correction, the stable-direction estimate and all downstream arguments are valid. No substantive incorrectness or incompleteness was found.

TypoSuspected typo

The admissible range for the auxiliary parameter is reversed in print

PDF page 38 · proof of Theorem 5.1 immediately after equation (5.4) · arXiv:2610.25559v1

The proof has fixed C2<C1C_2<C_1 and obtains the decay exponent C1−C2−βC_1-C_2-\beta. The printed instruction β<C2−C1\beta<C_2-C_1 is impossible because its right-hand side is negative. It must read 0<β<C1−C20<\beta<C_1-C_2. This correction is mechanically determined by the preceding parameter ranges and makes the displayed geometric-series estimate decay; it does not change the argument or any theorem statement.

Theorems 5.1 and 6.1No issue found in the reviewed proof

Stable directions and uniform Pesin blocks support the finiteness argument

PDF pages 35–49 · Sections 5–6 · arXiv:2610.25559v1

The ratio of singular values is summable at almost every point on each unstable leaf, so the most-contracted directions converge and contract at rate at least the exponent gap; every transverse direction expands. Continuity of the exponent sums over the compact unstable-Gibbs family yields a common Pesin block of arbitrarily large measure. Two limiting SRB measures charging the same sufficiently small block ball are homoclinically related and hence equal, proving finiteness.

Theorem 7.1 and proof of Theorem ANo issue found in the reviewed proof

Effective hyperbolicity yields existence and full-volume basins

PDF pages 50–58 · Section 7 · arXiv:2610.25559v1

Positive-density simultaneous Pliss times control expansion, contraction, and angles on a transverse disk, meeting the cited effective-hyperbolicity criterion and producing an SRB measure. Repeating the construction inside a hypothetical positive-volume complement of all basins yields a nonzero SRB component whose stable saturation meets that complement in positive volume, a contradiction.

03Novelty0 reported findingsNo novelty concerns found

No non-novelty findings.

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Paper
arXiv:2610.25559v1
Authors listed
Pablo D. Carrasco, Davi Obata, Radu Saghin, Jiagang Yang
Audit date
October 23, 2026
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