arXiv:2309.07282v3

Centralizer Rigidity near Elements of the Weyl Chamber Flow

Zhijing Wendy Wang

math.DS37C7937D3037A17

Paper metadata

Original submission: September 13, 2023
Latest revision: June 3, 2026 · version 3
Claimed MSC: 37C79, 37D30, 37A17
arXiv categories: math.DS
Paper license: CC BY 4.0 ↗
Comments: 39 pages, 3 figures

Abstract

In this paper, we prove centralizer rigidity near an element of the Weyl chamber flow on a semisimple Lie group. We show that a volume preserving perturbation of an element of the Weyl chamber flow on a quotient $G/Γ$ of an $\mathbb{R}$ -split, simple Lie group $G$ either has centralizer of dimension $0$ or $1$ , or is smoothly conjugate to an element of the Weyl chamber flow. We also acquire a general condition for the centralizer of a partially hyperbolic diffeomorphism to be a Lie group.

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Audited against arXiv v3

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.

Current report

Detailed mathematical audit

Generated August 18, 2026

01Statements3 reported findingsCorrect

The centralizer dichotomy near generic Weyl-chamber elements, its semisimple refinement, and the center-fixing Lie-group theorem are supported under the paper's bunching, accessibility, and compact-quotient hypotheses.

Theorems 1.1 and 1.2Correct

Local centralizer rigidity near Weyl-chamber elements

Pages 3–4 and 23–48 · Sections 4–7 and Appendix A · arXiv:2309.07282v3

The finite outer-automorphism quotient reduces the dimension question to the center-fixing centralizer. If all center translations lie in the cone of the perturbed generator, the free Lie action has the asserted dimension bound. Otherwise a second direction produces a genuinely higher-rank topological restriction. The Lyapunov-cycle functional is trivialized by the higher-rank comparison and property $(T)$ , yielding a Hölder conjugacy; Theorem 7.11 and the normal-form argument then upgrade it to a smooth conjugacy. The componentwise dimension calculation gives $k-\min_i k_i+1$ in the semisimple case.

Theorem 3.1Correct

The center-fixing centralizer is a Lie group

Pages 17–23 · Section 3 · arXiv:2309.07282v3

Accessibility makes global stable/unstable holonomies transitive on a generic center leaf. Commutation with those holonomies forces a center-fixing map to be determined by one value, proving freeness. Center bunching and the homogeneous-graph regularity theorem upgrade the leafwise action to $C^r$ . Properness follows from compactness of the possible basepoint images and continuous dependence of bounded holonomy paths. The resulting locally compact effective diffeomorphism group is Lie by the cited Hilbert–Smith theorem.

Theorems 3.3 and 3.4Correct

Applications to discretized Anosov flows

Pages 18–19 · arXiv:2309.07282v3

In one center dimension, a Lie group acting freely and properly on a center leaf is virtually $\mathbb Z$ or $\mathbb R$ . The cited finite quotient by the center-fixing subgroup then gives precisely the virtually trivial versus flow-embedding alternative. The required narrow-band, bunching, and global-holonomy conditions are checked for both applications.

02Proofs3 reported findingsCorrect

The center-action, higher-rank cocycle, partial-hyperbolicity, and smoothness arguments are correct and complete. A few continuity steps are compressed but follow from the uniform holonomy estimates already established and require no formal change.

Section 3Correct and complete

Properness and regularity of the center action

Pages 17–23 · arXiv:2309.07282v3

For properness, the representation $g_n(z)=h_{x\to z}(g_n(x))$ and compact finite holonomy charts upgrade convergence of $g_n(x)$ to uniform convergence on the compact manifold; applying the same argument to inverses gives a homeomorphic limit. The proof also establishes freeness from accessibility and obtains leafwise $C^r$ regularity from the homogeneous graph theorem before invoking the stated Hilbert–Smith input.

Proposition 6.1Correct and complete

Construction of a genuinely higher-rank topological action

Pages 31–38 · Sections 6.1–6.5 · arXiv:2309.07282v3

The second centralizer generator is chosen outside a narrow cone about the perturbed Weyl-chamber element in every simple factor. Leaf conjugacy produces the fine topological foliations, while the bi-Hölder estimates give exponential contraction or expansion away from the corresponding root walls. The projective comparison in Section 6.4 supplies an element safely inside every Weyl chamber, and the center derivative estimate is subexponential at the required scale. These ingredients verify each clause of the defined topological perturbation and exclude a rank-one factor.

Theorem 7.11 and Appendix ACorrect and complete

Partial hyperbolicity and smoothness upgrade

Pages 43–47 · Section 7.4 and Appendix A · arXiv:2309.07282v3

The bi-Hölder conjugacy identifies the topological stable and unstable leaves with the Pesin leaves for every ergodic measure. Normal-form coordinates then recover continuous invariant bundles, and Lemma 7.12 converts the measurewise Lyapunov bounds into uniform contraction and expansion, with zero center exponent. For smoothness, limiting isometries act transitively on each coarse Lyapunov leaf; their conjugates are bounded-degree normal-form polynomials, so the transfer map is smooth leafwise, and the bracket-generating regularity theorem quoted as Theorem A.1 gives global smoothness.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2309.07282v3
Authors listed: Zhijing Wendy Wang
Audit date: August 18, 2026

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Centralizer Rigidity near Elements of the Weyl Chamber Flow

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Local centralizer rigidity near Weyl-chamber elements

The center-fixing centralizer is a Lie group

Applications to discretized Anosov flows

Properness and regularity of the center action

Construction of a genuinely higher-rank topological action

Partial hyperbolicity and smoothness upgrade

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