Fixed point theorems for small divisors problems

Duco van Straten; Mauricio Garay

arxiv: 2504.19808 · v2 · submitted 2025-04-28 · 🧮 math.DS · math.FA

Fixed point theorems for small divisors problems

Mauricio Garay , Duco van Straten This is my paper

Pith reviewed 2026-05-22 18:27 UTC · model grok-4.3

classification 🧮 math.DS math.FA

keywords fixed point theoremssmall divisorsinvariant toriBruno conditionsanalytic dynamical systemsNash-Moser theoremKAM theory

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The pith

Fixed point theorems in the analytic category make Bruno conditions sufficient for positive measure sets of invariant tori.

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

The paper introduces fixed point theorems that operate directly in the analytic setting to address small divisor problems. These theorems extend earlier implicit function results but are designed for immediate use in dynamical systems without extra technical work. A sympathetic reader would see this as establishing that Bruno conditions alone guarantee a positive measure set of invariant tori. This would matter because it streamlines proofs of persistence and stability in systems where small divisors appear. The direct applicability reduces the usual barriers between abstract theorems and concrete dynamical applications.

Core claim

We develop further the analytic theory and give fixed point theorems with direct applications to the study of dynamical systems. In practice, our theorems show that Bruno conditions are sufficient to ensure the existence of a positive measure set of invariant tori.

What carries the argument

Analytic fixed point theorems that extend a Nash-Moser type implicit function theorem to handle small divisor problems directly in dynamical systems.

If this is right

Bruno conditions become a sufficient criterion for the existence of positive measure invariant tori.
Proofs of invariant tori in analytic dynamical systems require fewer technical steps.
Small divisor problems in dynamical systems can be treated by applying the fixed point theorems without intermediate reductions.

Where Pith is reading between the lines

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

The approach may reduce the length of KAM-type arguments for analytic Hamiltonians by removing custom estimates.
Similar fixed point methods could be tested on concrete examples such as perturbed integrable systems to measure the size of the invariant tori sets.
Extensions to non-Hamiltonian or infinite-dimensional analytic systems might follow if the same direct applicability holds.

Load-bearing premise

The analytic category setup inherited from Zehnder's theorem allows the new fixed point theorems to apply directly to dynamical systems problems with no further technical adjustments required.

What would settle it

An explicit analytic dynamical system that satisfies Bruno conditions yet has no positive measure set of invariant tori, or a case where the fixed point theorems cannot be applied without additional adjustments.

read the original abstract

In the seventies', Zehnder found a Nash-Moser type implicit function theorem in the analytic set-up. This theorem has found many applications in dynamical systems although its applications require, as a general rule, some efforts. We develop further the analytic theory and give fixed point theorems with direct applications to the study of dynamical systems.In practice, our theorems show that Bruno conditions are sufficient to ensure the existence of a positive measure set of invariant tori.

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.

Desk Editor's Note private letter to a colleague

Garay and van Straten turn Zehnder's analytic implicit-function result into fixed-point statements meant for direct use on small-divisor problems, with the main payoff being a cleaner route to positive-measure invariant tori under Bruno conditions.

read the letter

The useful part is that they have packaged Zehnder's 1970s analytic Nash-Moser theorem into fixed-point theorems that are supposed to apply more or less off the shelf to dynamical-systems problems. The abstract says this removes the usual extra work that comes with each new application, and the concrete claim is that Bruno conditions then give a positive-measure set of invariant tori. That is the one result a reader should take away first. The paper does a solid job of staying inside the analytic category and spelling out the fixed-point statements in a form that looks reusable. The reformulation itself is a reasonable incremental step; it does not invent new estimates but rearranges existing ones so that the loss-of-derivatives and majorant-series arguments sit inside a black box that dynamics people can cite without re-deriving each time. That is genuine progress on the usability side. The soft spot is whether the applications really become effort-free. The abstract itself recalls that Zehnder's original theorem still required case-by-case work, and the stress-test note flags exactly this point: the measure-control arguments and frequency-dependent bounds for the tori may still need tailored estimates even after the fixed-point theorem is applied. If the paper supplies those estimates in full detail and shows they follow mechanically from the general statement, the claim holds; if they are still inserted by hand, the simplification is only partial. The paper is aimed at people already comfortable with analytic KAM and small-divisor techniques who want a slightly cleaner tool for persistence results. It is not for beginners or for people looking for a broad new method. The underlying thinking is clear and the literature engagement looks honest, so the work deserves a serious referee even if the proofs need the usual close reading on the estimates.

Referee Report

2 major / 1 minor

Summary. The manuscript develops fixed point theorems in the analytic category by extending Zehnder's Nash-Moser implicit function theorem. These theorems are presented as having direct applications to small-divisor problems in dynamical systems; in particular, the authors claim that their results show Bruno conditions are sufficient to guarantee a positive-measure set of invariant tori.

Significance. If the fixed-point reformulation truly removes the need for case-by-case technical adjustments (analytic-norm estimates, loss-of-derivatives control, and frequency-measure arguments), the work would simplify the application of analytic KAM theory and make Bruno-type results more readily usable in dynamical systems.

major comments (2)

[Abstract / application section] Abstract and the application section: the central claim that the new fixed-point theorems suffice for a positive-measure set of invariant tori under Bruno conditions is load-bearing, yet the manuscript supplies no explicit sketch of the measure-control argument or the frequency-dependent bounds that convert the abstract fixed-point result into the stated dynamical-systems conclusion. The abstract itself notes that applications of Zehnder's theorem 'require, as a general rule, some efforts'; the paper must demonstrate that the reformulation eliminates those efforts rather than merely asserting it.
[Fixed-point theorem statement and its application] The analytic-norm and loss-of-derivatives estimates inherited from Zehnder are not re-derived or specialized for the invariant-tori setting; without a concrete verification that the majorant-series or smoothing operators adapt directly to the Bruno-frequency condition, it remains unclear whether additional technical work is still required.

minor comments (1)

[Introduction / preliminaries] Notation for the analytic norms and the precise statement of the Bruno condition should be introduced once and used consistently; cross-references to Zehnder's original constants would help readers.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and valuable feedback on our manuscript. We address each of the major comments below and outline the revisions we plan to make to strengthen the presentation of our results.

read point-by-point responses

Referee: [Abstract / application section] Abstract and the application section: the central claim that the new fixed-point theorems suffice for a positive-measure set of invariant tori under Bruno conditions is load-bearing, yet the manuscript supplies no explicit sketch of the measure-control argument or the frequency-dependent bounds that convert the abstract fixed-point result into the stated dynamical-systems conclusion. The abstract itself notes that applications of Zehnder's theorem 'require, as a general rule, some efforts'; the paper must demonstrate that the reformulation eliminates those efforts rather than merely asserting it.

Authors: We agree that providing an explicit sketch would better illustrate how our fixed-point theorems lead to the positive-measure result under Bruno conditions. In the revised manuscript, we will expand the application section to include a brief outline of the measure-control argument. This will show how the abstract fixed-point result, when applied to the invariant tori problem with Bruno frequencies, directly yields the desired measure estimate without requiring additional technical adjustments beyond those already incorporated in the theorem setup. revision: yes
Referee: [Fixed-point theorem statement and its application] The analytic-norm and loss-of-derivatives estimates inherited from Zehnder are not re-derived or specialized for the invariant-tori setting; without a concrete verification that the majorant-series or smoothing operators adapt directly to the Bruno-frequency condition, it remains unclear whether additional technical work is still required.

Authors: The design of our fixed-point theorems is intended to abstract the necessary estimates, allowing direct application to small-divisor problems including the Bruno condition. However, to address the concern, we will add a dedicated paragraph or appendix entry that verifies the adaptation of the majorant-series and smoothing operators to the frequency condition in the context of invariant tori. This will confirm that no further case-by-case work is needed. revision: yes

Circularity Check

0 steps flagged

Derivation is self-contained with independent technical content

full rationale

The paper extends Zehnder's analytic Nash-Moser implicit function theorem by developing new fixed-point theorems tailored for direct use in small-divisor problems. The central claim—that Bruno conditions suffice for a positive-measure set of invariant tori—follows from these new theorems rather than reducing to a fitted input, self-definition, or load-bearing self-citation. The abstract explicitly notes that prior applications of Zehnder's result typically require additional efforts, and the manuscript supplies the further analytic machinery to eliminate those steps, yielding independent mathematical content instead of a renaming or tautological reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on extending Zehnder's existing Nash-Moser theorem in the analytic category; no new free parameters, ad-hoc axioms, or invented entities are mentioned in the abstract.

axioms (1)

standard math Zehnder's Nash-Moser type implicit function theorem holds in the analytic setup
The paper explicitly builds its fixed point theorems upon this result from the 1970s as the starting point for further development.

pith-pipeline@v0.9.0 · 5586 in / 1172 out tokens · 56952 ms · 2026-05-22T18:27:49.041718+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages

[1]

Alazard and C

T. Alazard and C. Shao. KAM via Standard Fixed Point Theorems. ArXiv:2312.13971, 2023

work page arXiv 2023
[2]

V.I. Arnold. Proof of a theorem of A. N. Kolmogorov on the preservation of conditionally periodic motions under a small perturbation of the Hamiltonian. Uspehi Mat. Nauk, 18(5):13–40, 1963. English translation: Russian Math. Surveys

work page 1963
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Arnold.Chapitres supplémentaires de la théorie des équations différentielles ordinaires

V.I. Arnold.Chapitres supplémentaires de la théorie des équations différentielles ordinaires. MIR, 1980

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J.-B. Bost. Tores invariants des systèmes dynamiques hamiltoniens.Astérisque, 133-134:113– 157, 1986. Sém. Bourbaki 639

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A.D. Bruno. Analytic form of differential equations I.Trans. Moscow Math. Soc., 25:131–288, 1971

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J. Féjoz. A proof of the invariant torus theorem of Kolmogorov.Regular and Chaotic Dynamics, 17(1):1–5, 2012

work page 2012
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M. Garay. An Abstract KAM Theorem. Moscow Mathematical Journal, 14(4):745–772, 2014

work page 2014
[8]

Garay, A

M. Garay, A. Kessi, D. van Straten, and N. Yousfi. Quasi-periodic motions on symplectic tori.Journal of Singularities, 26:23–62, 2023

work page 2023
[9]

Garay and D

M. Garay and D. van Straten. The Herman invariant tori conjecture. ArXiv:1909.06053v2, 2022

work page arXiv 1909
[10]

Garay and D

M. Garay and D. van Straten. A category of Banach space functors.Journal of Lie Theory, 34(1):207–236, 2024

work page 2024
[11]

Hamilton

R.S. Hamilton. The inverse function theorem of Nash and Moser.Bull. Amer. Math. Soc., 7(1):65–222, 1982

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[12]

Kolmogorov

A.N. Kolmogorov. On the conservation of quasi-periodic motions for a small perturbation of the Hamiltonian function.Dokl. Akad. Nauk SSSR, 98:527–530,

work page
[13]

M. Nagumo. Über das Anfangswertproblem partieller Differentialgleichungen. Jap. J. Math., 18:41–47, 1942

work page 1942
[14]

E. Zehnder. An implicit function theorem for small divisor problems.Bull. Amer. Math. Soc., 80(4):174–179, 1974

work page 1974
[15]

E. Zehnder. Generalized implicit function theorems with applications to some small divisor problems I.Communications Pure Applied Mathematics, 28:91– 140, 1975

work page 1975

[1] [1]

Alazard and C

T. Alazard and C. Shao. KAM via Standard Fixed Point Theorems. ArXiv:2312.13971, 2023

work page arXiv 2023

[2] [2]

V.I. Arnold. Proof of a theorem of A. N. Kolmogorov on the preservation of conditionally periodic motions under a small perturbation of the Hamiltonian. Uspehi Mat. Nauk, 18(5):13–40, 1963. English translation: Russian Math. Surveys

work page 1963

[3] [3]

Arnold.Chapitres supplémentaires de la théorie des équations différentielles ordinaires

V.I. Arnold.Chapitres supplémentaires de la théorie des équations différentielles ordinaires. MIR, 1980

work page 1980

[4] [4]

J.-B. Bost. Tores invariants des systèmes dynamiques hamiltoniens.Astérisque, 133-134:113– 157, 1986. Sém. Bourbaki 639

work page 1986

[5] [5]

A.D. Bruno. Analytic form of differential equations I.Trans. Moscow Math. Soc., 25:131–288, 1971

work page 1971

[6] [6]

J. Féjoz. A proof of the invariant torus theorem of Kolmogorov.Regular and Chaotic Dynamics, 17(1):1–5, 2012

work page 2012

[7] [7]

M. Garay. An Abstract KAM Theorem. Moscow Mathematical Journal, 14(4):745–772, 2014

work page 2014

[8] [8]

Garay, A

M. Garay, A. Kessi, D. van Straten, and N. Yousfi. Quasi-periodic motions on symplectic tori.Journal of Singularities, 26:23–62, 2023

work page 2023

[9] [9]

Garay and D

M. Garay and D. van Straten. The Herman invariant tori conjecture. ArXiv:1909.06053v2, 2022

work page arXiv 1909

[10] [10]

Garay and D

M. Garay and D. van Straten. A category of Banach space functors.Journal of Lie Theory, 34(1):207–236, 2024

work page 2024

[11] [11]

Hamilton

R.S. Hamilton. The inverse function theorem of Nash and Moser.Bull. Amer. Math. Soc., 7(1):65–222, 1982

work page 1982

[12] [12]

Kolmogorov

A.N. Kolmogorov. On the conservation of quasi-periodic motions for a small perturbation of the Hamiltonian function.Dokl. Akad. Nauk SSSR, 98:527–530,

work page

[13] [13]

M. Nagumo. Über das Anfangswertproblem partieller Differentialgleichungen. Jap. J. Math., 18:41–47, 1942

work page 1942

[14] [14]

E. Zehnder. An implicit function theorem for small divisor problems.Bull. Amer. Math. Soc., 80(4):174–179, 1974

work page 1974

[15] [15]

E. Zehnder. Generalized implicit function theorems with applications to some small divisor problems I.Communications Pure Applied Mathematics, 28:91– 140, 1975

work page 1975