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arxiv: 2509.13986 · v3 · submitted 2025-09-17 · 🧮 math.DS

Rigidity of strong and weak foliations

Boris Kalinin , Victoria Sadovskaya This is my paper

Pith reviewed 2026-05-18 16:39 UTC · model grok-4.3

classification 🧮 math.DS
keywords foliation rigidityhyperbolic toral automorphismsconjugacy smoothnessstable foliationsholonomynormal formssymplectic rigidityperturbations
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The pith

If the conjugacy maps the strong foliation to the linear one, it becomes smooth along the weak foliation for perturbations of hyperbolic toral automorphisms.

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

The paper establishes that for a small perturbation f of a hyperbolic toral automorphism L, a homeomorphism conjugacy h that sends the strong stable foliation of f to the corresponding linear foliation must be smooth along the weak stable foliation. It further shows that the combined strong and unstable foliations become regular under this condition. A parallel result holds when the weak foliation satisfies a regularity assumption, yielding smoothness of h along the strong foliation and regularity of the joint weak-unstable foliation. When both conditions are present, h is smooth along the full stable foliation. These conclusions, which also produce a symplectic rigidity statement, rest on a new relation between holonomes and normal forms that unifies the arguments.

Core claim

If the strong foliation is mapped to the linear one by the conjugacy h between f and L, we obtain smoothness of h along the weak foliation and regularity of the joint foliation of the strong and unstable foliations. We also establish a similar global result. If the weak foliation is sufficiently regular, we obtain smoothness of the conjugacy along the strong foliation and regularity of the joint foliation of the weak and unstable foliations. If both conditions hold then we get smoothness of h along the stable foliation. We also deduce a rigidity result for the symplectic case. The main theorems are obtained in a unified way using our new result on relation between holonomes and normal forms.

What carries the argument

The relation between holonomes and normal forms, which provides a unified method to prove the smoothness and regularity statements for the strong and weak foliations under the given mapping conditions on the conjugacy.

If this is right

  • The conjugacy h is smooth along the weak foliation when it maps the strong foliation to the linear one.
  • The joint foliation formed by the strong and unstable directions is regular under the same mapping condition.
  • Symmetrically, sufficient regularity of the weak foliation implies smoothness of h along the strong foliation.
  • When both the strong and weak conditions hold, h is smooth along the entire stable foliation.
  • A rigidity conclusion holds in the symplectic case.

Where Pith is reading between the lines

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

  • The holonomy-normal form relation may serve as a tool for studying conjugacy regularity in other hyperbolic systems that admit continuous conjugacies to linear models.
  • Foliation mapping conditions could be checked numerically in concrete examples to test the predicted smoothness of the conjugacy.
  • The results suggest that partial regularity of one foliation can be leveraged to upgrade regularity of the conjugacy in the transverse direction.

Load-bearing premise

The perturbation f is small enough that a homeomorphism conjugacy h to the linear automorphism L exists.

What would settle it

An explicit small perturbation f of L where the conjugacy h maps the strong foliation to the linear foliation but h fails to be differentiable along some leaf of the weak foliation.

read the original abstract

We consider a perturbation $f$ of a hyperbolic toral automorphism $L$. We study rigidity related to exceptional properties of the strong and weak stable foliations for $f$. If the strong foliation is mapped to the linear one by the conjugacy $h$ between $f$ and $L$, we obtain smoothness of $h$ along the weak foliation and regularity of the joint foliation of the strong and unstable foliations. We also establish a similar global result. If the weak foliation is sufficiently regular, we obtain smoothness of the conjugacy along the strong foliation and regularity of the joint foliation of the weak and unstable foliations. If both conditions hold then we get smoothness of $h$ along the stable foliation. We also deduce a rigidity result for the symplectic case. The main theorems are obtained in a unified way using our new result on relation between holonomes and normal forms.

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

2 major / 2 minor

Summary. The manuscript considers a sufficiently small perturbation f of a hyperbolic toral automorphism L, with homeomorphism conjugacy h between them. It proves conditional rigidity results: if h maps the strong stable foliation of f to the linear one, then h is smooth along the weak stable foliation and the joint strong-unstable foliation has specified regularity; a similar global result holds. If the weak foliation is sufficiently regular, then h is smooth along the strong foliation with regularity of the joint weak-unstable foliation. If both conditions hold, h is smooth along the stable foliation. A symplectic rigidity result is deduced. All results are obtained uniformly via a new relation between holonomes and normal forms.

Significance. If the central claims and the supporting holonomy-normal form relation hold, the work supplies new conditional criteria for smoothness of conjugacies along foliations in Anosov systems. This could strengthen the toolkit for studying rigidity phenomena in perturbations of hyperbolic toral automorphisms and related symplectic cases, particularly by linking foliation mapping properties to regularity of h.

major comments (2)
  1. The abstract states that the main theorems rely on a newly introduced relation between holonomes and normal forms, but the manuscript must explicitly state this relation (including its hypotheses and conclusion) and verify that it applies directly to the small-perturbation setting without additional regularity assumptions on f; this relation is load-bearing for the unification claim.
  2. The setup assumes a homeomorphism conjugacy h exists for sufficiently small perturbations, which is standard, but the manuscript should clarify in the introduction or §2 how the 'sufficiently small' condition is quantified (e.g., in C^1 or C^r topology) to ensure the foliation-mapping hypotheses are well-posed.
minor comments (2)
  1. Notation for the strong and weak stable foliations should be introduced consistently (e.g., W^s_strong and W^s_weak) and used uniformly when stating the mapping assumptions on h.
  2. The symplectic rigidity deduction at the end would benefit from a brief paragraph recalling the relevant symplectic structure on the torus and how the foliation conditions specialize in that case.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation of the significance of our conditional rigidity results and for the constructive major comments. We address each point below and will incorporate the suggested clarifications in the revised manuscript.

read point-by-point responses

  1. Referee: The abstract states that the main theorems rely on a newly introduced relation between holonomes and normal forms, but the manuscript must explicitly state this relation (including its hypotheses and conclusion) and verify that it applies directly to the small-perturbation setting without additional regularity assumptions on f; this relation is load-bearing for the unification claim.

    Authors: We agree that the holonomy-normal form relation is central to the unified proofs and that its explicit formulation strengthens the manuscript. In the revision we will insert a precise statement of this relation (as a theorem in Section 3, for example) that lists all hypotheses on the foliation mapping properties and the resulting regularity conclusions for the conjugacy h. We will also add a short verification paragraph showing that the relation applies verbatim in the C^1-small perturbation regime: the standard hyperbolic estimates for toral automorphisms guarantee that the required holonomy and normal-form data exist under precisely the hypotheses already stated, without any extra regularity on f. This change improves readability while preserving the original arguments. revision: yes

  2. Referee: The setup assumes a homeomorphism conjugacy h exists for sufficiently small perturbations, which is standard, but the manuscript should clarify in the introduction or §2 how the 'sufficiently small' condition is quantified (e.g., in C^1 or C^r topology) to ensure the foliation-mapping hypotheses are well-posed.

    Authors: We thank the referee for noting this point of clarity. In the revised introduction and in §2 we will explicitly quantify the smallness condition by stating that there exists δ>0 such that whenever the C^1 distance between f and L is less than δ, the structural stability theorem supplies a unique homeomorphism conjugacy h, and the foliation-mapping hypotheses (e.g., h sending the strong stable foliation of f onto that of L) become well-defined. This quantification is already implicit in the standard references we cite, but spelling it out removes any ambiguity about the topology in which the perturbation is taken. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via new independent result

full rationale

The paper establishes its main rigidity theorems for perturbations of hyperbolic toral automorphisms by introducing and applying a new result relating holonomes to normal forms. This relation is presented as a fresh contribution within the manuscript rather than a redefinition or fit of the target regularity properties. The setup relies on a standard homeomorphism conjugacy between the perturbed map f and the linear model L, with no equations or steps shown to reduce by construction to the claimed smoothness or regularity outputs. No self-citation chains or ansatzes imported from prior author work are load-bearing for the central claims. The derivation therefore remains independent of its conclusions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard background facts from hyperbolic dynamics and introduces one new technical relation; no free parameters or invented entities appear in the abstract.

axioms (2)
  • domain assumption A homeomorphism conjugacy h exists between the perturbation f and the linear toral automorphism L.
    This is the standard setup for studying rigidity of perturbations of hyperbolic toral automorphisms.
  • domain assumption The strong and weak foliations of f are well-defined and invariant under f.
    Invoked throughout the statements about mapping properties and regularity.

pith-pipeline@v0.9.0 · 5680 in / 1409 out tokens · 46094 ms · 2026-05-18T16:39:50.791387+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

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