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arxiv: 2604.08262 · v1 · submitted 2026-04-09 · 🧮 math.DS · math.DG

Marked magnetic action rigidity

Louis-Brahim Beaufort , Sebasti\'an Mu\~noz-Thon , Sean Richardson This is my paper

Pith reviewed 2026-05-10 17:53 UTC · model grok-4.3

classification 🧮 math.DS math.DG
keywords magnetic systemsAnosov flowmarked spectrumaction rigidityexact magnetic systemsdynamical rigidityRiemannian geometry
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The pith

The marked magnetic action spectrum determines the metric and magnetic 1-form up to natural obstruction for Anosov flows in local and conformal cases.

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

This paper studies whether the marked magnetic action spectrum uniquely identifies an exact magnetic system consisting of a Riemannian metric and a magnetic 1-form on a closed manifold, when the associated magnetic flow is Anosov. It generalizes the marked length spectrum rigidity question by replacing geodesic length with the magnetic action functional that combines length and the integral of the 1-form along curves. The authors establish that the spectrum recovers the metric and 1-form in two regimes: when the systems are sufficiently close to each other, and when the metrics lie in the same conformal class. A reader would care because this supplies a dynamical invariant built only from periodic orbit data that can reconstruct both geometry and the magnetic field strength without direct access to the manifold's embedding or coordinates.

Core claim

For exact magnetic systems (g, α) on closed manifolds whose magnetic flow is Anosov, the marked magnetic action spectrum determines the pair (g, α) up to the natural obstruction. This holds first in the local setting, where metrics and 1-forms are close in the C^∞ topology, and second in the global conformal setting where the metrics differ by a positive conformal factor. The spectrum is the collection, over all free homotopy classes of closed curves, of the infima of the magnetic action functional.

What carries the argument

The marked magnetic action spectrum, defined as the map from free homotopy classes to the infimum of the magnetic action ∫ (length + α) along closed curves in each class, serves as the complete invariant that encodes all periodic orbit actions and permits recovery of the underlying metric and 1-form.

If this is right

  • Locally close systems with the same spectrum must coincide, so infinitesimal deformations of metric or 1-form are detected by changes in action values.
  • Within one conformal class the spectrum fixes both the conformal factor and the 1-form completely.
  • The result transfers classical marked-length rigidity techniques to magnetic flows while preserving the Anosov hypothesis.
  • Periodic orbit data alone suffice to reconstruct the geometry without solving the full inverse problem on the entire manifold.
  • The natural obstruction corresponds to a diffeomorphism that preserves the magnetic flow and leaves the action spectrum invariant.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Numerical orbit-tracking experiments could test whether the local rigidity persists under small random perturbations of the 1-form.
  • If the Anosov assumption is weakened to a weaker hyperbolicity condition, similar spectrum rigidity might hold for non-exact magnetic systems.
  • The conformal result suggests that action spectra could be used to distinguish conformal classes in higher-dimensional magnetic inverse problems.
  • Extending the marked spectrum to include multiplicity or stability information might remove the need for the local or conformal restrictions.

Load-bearing premise

The magnetic flow must be Anosov, and the statements are proved only for exact systems that are either locally close or share the same conformal class of metrics.

What would settle it

Exhibit two exact magnetic systems with Anosov flows that have identical marked magnetic action spectra yet are not related by any diffeomorphism preserving the flow or by the natural equivalence in the local or conformal settings.

read the original abstract

An exact magnetic system over a closed manifold $M$ consists of a pair $(g,\alpha)$, where $g$ is a Riemannian metric and $\alpha$ is a 1-form encoding a magnetic field. In this context, we consider a generalization of the marked length rigidity conjecture: does the marked magnetic action spectrum of magnetic systems with Anosov magnetic flow determine the metric and the 1-form, up to a natural obstruction? In this article we answer this question in two settings: 1) locally for systems with close metrics and 1-forms and 2) for metrics in the same conformal class.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper proves a rigidity result for exact magnetic systems (g, α) on closed manifolds with Anosov magnetic flows: the marked magnetic action spectrum determines the metric g and 1-form α up to the addition of an exact 1-form. The result is established in two settings: (i) locally, for systems with sufficiently close metrics and 1-forms, and (ii) globally, for metrics lying in the same conformal class.

Significance. This constitutes a meaningful extension of the marked length spectrum rigidity conjecture to the magnetic setting. The Anosov hypothesis supplies dense periodic orbits and hyperbolicity, permitting recovery of the underlying data from the action values; the explicit identification of the exact 1-form obstruction clarifies the precise sense in which the spectrum is a complete invariant. The local and conformal restrictions are technically natural and the proofs appear to rest on standard perturbation and conformal deformation techniques.

minor comments (3)
  1. The abstract and introduction should state the precise definition of the marked magnetic action spectrum (including the marking by free homotopy classes) at the outset rather than deferring it to a later section.
  2. Notation for the magnetic 2-form and the action functional should be introduced uniformly; occasional shifts between ω and dα are distracting.
  3. A brief remark on the necessity of the Anosov condition (or a reference to a counter-example when it fails) would help readers assess the scope of the result.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, recognition of the significance of the result as an extension of marked length spectrum rigidity to the magnetic setting, and recommendation for minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper establishes local rigidity for sufficiently close exact magnetic systems and conformal-class rigidity on closed manifolds, both under the Anosov magnetic flow hypothesis. The marked action spectrum is treated as an external invariant whose values on closed curves are used to recover the metric and 1-form up to the explicit natural obstruction of adding an exact 1-form. No equation or definition reduces the claimed determination to a fitted parameter, self-referential construction, or load-bearing self-citation; the Anosov assumption supplies independent dynamical input (dense periodic orbits and hyperbolicity) that is not derived from the spectrum itself. The argument therefore remains self-contained with independent mathematical content.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the Anosov property of the magnetic flow and the definition of exact magnetic systems; no free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption The magnetic flow is Anosov
    Explicitly required in the abstract for the systems under consideration.
  • domain assumption The manifold is closed and the system is exact
    Stated as the setting in which the magnetic system is defined.

pith-pipeline@v0.9.0 · 5395 in / 1199 out tokens · 55814 ms · 2026-05-10T17:53:46.279505+00:00 · methodology

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Reference graph

Works this paper leans on

2 extracted references · 2 canonical work pages

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    [Ain15] Gareth Ainsworth,The magnetic ray transform on anosov surfaces, Discrete Con- tin

    [AdSMRT24] Valerio Assenza, Jacopo de Simoi, James Marshall Reber, and Ivo Terek, Marked length spectrum rigidity for anosov magnetic surfaces, arXiv preprint arXiv:2409.20545 (2024). [Ain15] Gareth Ainsworth,The magnetic ray transform on anosov surfaces, Discrete Con- tin. Dyn. Syst.35(2015), no. 5, 1801–1816. MR3294225 [Ass24] Valerio Assenza,Magnetic c...

    work page arXiv 2024

  2. [2]

    MR5022124 [LT05] A. O. Lopes and Ph. Thieullen,Sub-actions for Anosov flows, Ergodic Theory Dynam. Systems25(2005), no. 2, 605–628. MR2129112 [Ma˜ n97] Ricardo Ma˜ n´ e,Lagrangian flows: the dynamics of globally minimizing orbits, Bol. Soc. Brasil. Mat. (N.S.)28(1997), no. 2, 141–153. MR1479499 [MTR25] Sebasti´ an Mu˜ noz Thon and Sean Richardon,Guillarmo...

    work page arXiv 2005