Pith. sign in

REVIEW 3 minor 40 references

Bound on the number of Ruelle resonances for Gevrey hyperbolic flows

T0 review · 0 major / 3 minor · reviewed 2026-05-10 · grok-4.3

Challenge this review Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Gevrey hyperbolic flows have fewer Ruelle resonances in large disks than prior bounds allowed.

desk verdict This paper tightens the upper bound on Ruelle resonances for Gevrey hyperbolic flows by adapting Rugh's determinant method, with no major gaps but limited practical payoff. read the letter →

arxiv 2604.13702 v1 submitted 2026-04-15 math.DS

Malo Jézéquel This is my paper · ORCID
classification math.DS
keywords RuelleresonanceshyperbolicflowsGevreysmoothnessdynamicaldeterminantsresonancecountingopenmaps
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper establishes a stricter upper bound on the number of Ruelle resonances that can lie inside disks of large radius for flows that are both Gevrey smooth and uniformly hyperbolic. It reaches this bound by converting the original flow into an equivalent system of open hyperbolic maps and then applying Rugh's dynamical determinant technique to count the zeros. A sympathetic reader cares because these resonances govern the exponential decay rates of correlations in the dynamical system. The improvement therefore gives sharper quantitative control over the spectrum of the generator of the flow.

What carries the argument

Rugh's dynamical determinant constructed from transfer operators on the open hyperbolic maps that encode the return dynamics of the flow.

What would settle it

Take any explicit Gevrey uniformly hyperbolic flow, compute its Ruelle resonances numerically out to a large radius R, and verify whether their count exceeds the new upper bound stated in the paper.

Watch

Extended reading notes

Core claim

For Gevrey uniformly hyperbolic flows the number of Ruelle resonances inside a disk of radius R is bounded above by a quantity smaller than the best previously known estimate, obtained by replacing the flow with a family of open hyperbolic maps and extracting the resonances as zeros of the associated dynamical determinant.

Load-bearing premise

The flow must be simultaneously Gevrey smooth and uniformly hyperbolic so that the open hyperbolic maps are well-defined and the dynamical determinant yields a valid count.

Editorial extensions

If this is right

  • Sharper bounds on resonance density imply improved estimates for the rate of correlation decay along the flow.
  • The method supplies a template that can be reused for other hyperbolic systems once their regularity class is fixed.
  • The location of the resonances becomes more constrained, tightening predictions for the analytic continuation of the associated dynamical zeta function.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same reduction to open maps might produce comparable improvements for Gevrey Anosov diffeomorphisms.
  • Numerical checks on low-dimensional examples, such as a Gevrey perturbation of the geodesic flow on a surface of negative curvature, could test how close the new bound comes to being sharp.
  • If the bound extends to C^infty or analytic regularity, it would connect to existing results on the distribution of resonances for smoother hyperbolic systems.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 3 minor

Summary. The manuscript claims to improve the best known upper bounds on the number of Ruelle resonances in disks of large radius for Gevrey uniformly hyperbolic flows. The proof reduces the flow to a system of open hyperbolic maps and applies Rugh's dynamical determinant construction, using Gevrey regularity to obtain improved exponential decay in the determinant coefficients.

Significance. If the estimates hold, the result strengthens quantitative control on the distribution of Ruelle resonances for hyperbolic flows of intermediate (Gevrey) regularity, a regime between C^∞ and analytic. This has direct implications for correlation decay and spectral gaps in dynamical systems. The paper correctly credits the reduction to open maps and the application of Rugh's method as the technical core; no free parameters or ad-hoc axioms appear in the argument.

minor comments (3)
  1. The abstract and introduction would benefit from an explicit statement of the previous best bound (e.g., the constant or exponent improved upon) so that the quantitative gain is immediately visible to readers.
  2. Notation for the Gevrey class and the precise radius of the disks in the resonance count should be fixed consistently between the statement of the main theorem and the estimates derived from the dynamical determinant.
  3. A short paragraph recalling the precise form of Rugh's determinant (including the role of the open hyperbolic maps) would make the reduction step more self-contained for readers outside the immediate subfield.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and for recommending minor revision. No major comments appear in the report.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity detected

full rationale

The derivation applies Rugh's external dynamical determinant construction to open hyperbolic maps obtained from the given Gevrey hyperbolic flows. The improved resonance count bound follows from exponential decay estimates on the determinant coefficients that are supplied directly by the Gevrey regularity hypothesis; these estimates are not obtained by fitting parameters to the target count or by any self-referential definition. No load-bearing step reduces the final bound to a prior result by the same authors, nor does any equation equate the claimed prediction to its own inputs by construction. The argument therefore remains self-contained against the stated assumptions.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

Abstract only; the central claim rests on the domain assumption that the flows are Gevrey and uniformly hyperbolic and that Rugh's method applies directly to the associated open maps.

assumptions (1)
  • domain assumption The dynamical system is a Gevrey uniformly hyperbolic flow.
    Stated in the abstract as the class for which the bound holds.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Bound on the number of Ruelle resonances for Gevrey hyperbolic flows." pith.science (2026). https://pith.science/paper/2604.13702

@misc{pith2026260413702,
  author       = {Pith},
  title        = {Pith review of: Bound on the number of Ruelle resonances for Gevrey hyperbolic flows},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2604.13702}},
  note         = {Machine review of arXiv:2604.13702}
}
read the original abstract

We improve the best known upper bounds on the number of Ruelle resonances in disks of large radius for Gevrey uniformly hyperbolic flows. The proof is based on Rugh's approach of dynamical determinants that replaces the study of the flow itself by the analysis of a system of open hyperbolic maps.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

40 extracted references · 40 canonical work pages

  1. [1]

    D. V. Anosov and Ya. G. Sina . Some smooth ergodic systems. With an appendix by G . A . Margulis . Usp. Mat. Nauk , 22(5(137)):107--172, 1967

    work page 1967

  2. [2]

    Dynamical zeta functions and dynamical determinants for hyperbolic maps

    Viviane Baladi. Dynamical zeta functions and dynamical determinants for hyperbolic maps. A functional approach , volume 68 of Ergeb. Math. Grenzgeb., 3. Folge . Cham: Springer, 2018

    work page 2018

  3. [3]

    FBI transform in Gevrey classes and Anosov flows , volume 456 of Ast \'e risque

    Yannick Guedes Bonthonneau and Malo J \'e z \'e quel. FBI transform in Gevrey classes and Anosov flows , volume 456 of Ast \'e risque . Paris: Soci \'e t \'e Math \'e matique de France (SMF), 2025

    work page 2025

  4. [4]

    Smooth Anosov flows: Correlation spectra and stability

    Oliver Butterley and Carlangelo Liverani. Smooth Anosov flows: Correlation spectra and stability. J. Mod. Dyn. , 1(2):301--322, 2007

    work page 2007

  5. [5]

    Robustly invariant sets in fiber contracting bundle flows

    Oliver Butterley and Carlangelo Liverani. Robustly invariant sets in fiber contracting bundle flows. J. Mod. Dyn. , 7(2):255--267, 2013

    work page 2013

  6. [6]

    Periodic orbits for hyperbolic flows

    Rufus Bowen. Periodic orbits for hyperbolic flows. Am. J. Math. , 94:1--30, 1972

    work page 1972

  7. [7]

    Symbolic dynamics for hyperbolic flows

    Rufus Bowen. Symbolic dynamics for hyperbolic flows. Amer. J. Math. , 95:429--460, 1973

    work page 1973

  8. [8]

    Dynamical determinants and spectrum for hyperbolic diffeomorphisms

    Viviane Baladi and Masato Tsujii. Dynamical determinants and spectrum for hyperbolic diffeomorphisms. In Geometric and probabilistic structures in dynamics. Workshop on dynamical systems and related topics in honor of Michael Brin on the occasion of his 60th birthday, College Park, MD, USA, March 15--18, 2008 , pages 29--68. Providence, RI: American Mathe...

    work page 2008

Show all 40 references
  1. [9]

    Expansive one-parameter flows

    Rufus Bowen and Peter Walters. Expansive one-parameter flows. J. Differ. Equations , 12:180--193, 1972

    1972

  2. [10]

    N. I. Chernov. Markov approximations and decay of correlations for A nosov flows. Ann. of Math. (2) , 147(2):269--324, 1998

    1998

  3. [11]

    Pollicott- R uelle resonances for open systems

    Semyon Dyatlov and Colin Guillarmou. Pollicott- R uelle resonances for open systems. Ann. Henri Poincar\'e , 17(11):3089--3146, 2016

    2016

  4. [12]

    Afterword: dynamical zeta functions for A xiom A flows

    Semyon Dyatlov and Colin Guillarmou. Afterword: dynamical zeta functions for A xiom A flows. Bull. Amer. Math. Soc. (N.S.) , 55(3):337--342, 2018

    2018

  5. [13]

    Dynamical zeta functions for Anosov flows via microlocal analysis

    Semyon Dyatlov and Maciej Zworski. Dynamical zeta functions for Anosov flows via microlocal analysis. Ann. Sci. \'E c. Norm. Sup \'e r. (4) , 49(3):543--577, 2016

    2016

  6. [14]

    Hyperbolic flows

    Todd Fisher and Boris Hasselblatt. Hyperbolic flows . Zur. Lect. Adv. Math. Berlin: European Mathematical Society (EMS), 2019

    2019

  7. [15]

    The zeta functions of Ruelle and Selberg

    David Fried. The zeta functions of Ruelle and Selberg . I . Ann. Sci. \'E c. Norm. Sup \'e r. (4) , 19(4):491--517, 1986

    1986

  8. [16]

    Meromorphic zeta functions for analytic flows

    David Fried. Meromorphic zeta functions for analytic flows. Commun. Math. Phys. , 174(1):161--190, 1995

    1995

  9. [17]

    Upper bound on the density of Ruelle resonances for Anosov flows

    Fr \'e d \'e ric Faure and Johannes Sj \"o strand. Upper bound on the density of Ruelle resonances for Anosov flows. Commun. Math. Phys. , 308(2):325--364, 2011

    2011

  10. [18]

    Micro-local analysis of contact Anosov flows and band structure of the Ruelle spectrum

    Fr \'e d \'e ric Faure and Masato Tsujii. Micro-local analysis of contact Anosov flows and band structure of the Ruelle spectrum. Commun. Am. Math. Soc. , 4:641--745, 2024

    2024

  11. [19]

    Anosov flows and dynamical zeta functions

    Paolo Giulietti, Carlangelo Liverani, and Mark Pollicott. Anosov flows and dynamical zeta functions. Ann. Math. (2) , 178(2):687--773, 2013

    2013

  12. [20]

    Produits tensoriels topologiques et espaces nucl \'e aires

    Alexandre Grothendieck. Produits tensoriels topologiques et espaces nucl \'e aires. , volume 16 of Mem. Am. Math. Soc. Providence, RI: American Mathematical Society (AMS), 1955

    1955

  13. [21]

    Morris W. Hirsch. Differential topology , volume 33 of Graduate Texts in Mathematics . Springer-Verlag, New York, 1994. Corrected reprint of the 1976 original

    1994

  14. [22]

    Local and global trace formulae for smooth hyperbolic diffeomorphisms

    Malo J \'e z \'e quel. Local and global trace formulae for smooth hyperbolic diffeomorphisms. J. Spectr. Theory , 10(1):185--249, 2020

    2020

  15. [23]

    Global trace formula for ultra-differentiable Anosov flows

    Malo J \'e z \'e quel. Global trace formula for ultra-differentiable Anosov flows. Commun. Math. Phys. , 385(3):1771--1834, 2021

    2021

  16. [24]

    Transfer operators for ultradifferentiable expanding maps of the circle

    Malo J \'e z \'e quel. Transfer operators for ultradifferentiable expanding maps of the circle. Ergodic Theory Dyn. Syst. , 41(7):2049--2068, 2021

  17. [25]

    Distribution of Ruelle resonances for real-analytic Anosov diffeomorphisms

    Malo J \'e z \'e quel. Distribution of Ruelle resonances for real-analytic Anosov diffeomorphisms. Ann. Henri Lebesgue , 7:673--726, 2024

    2024

  18. [26]

    Counting Pollicott - Ruelle resonances for axiom a flows

    Long Jin and Zhongkai Tao. Counting Pollicott - Ruelle resonances for axiom a flows. Commun. Math. Phys. , 406(2):43, 2025. Id/No 26

    2025

  19. [27]

    Zeta functions and the Fried conjecture for smooth pseudo- Anosov flows

    Malo J \'e z \'e quel and Jonathan Zung. Zeta functions and the Fried conjecture for smooth pseudo- Anosov flows. Preprint, arXiv :2409.17014 [math. DS ] (2024), 2024

    2024

  20. [28]

    Introduction to the modern theory of dynamical systems , volume 54 of Encyclopedia of Mathematics and its Applications

    Anatole Katok and Boris Hasselblatt. Introduction to the modern theory of dynamical systems , volume 54 of Encyclopedia of Mathematics and its Applications . Cambridge University Press, Cambridge, 1995. With a supplementary chapter by Katok and Leonardo Mendoza

    1995

  21. [29]

    Michor, and Armin Rainer

    Andreas Kriegl, Peter W. Michor, and Armin Rainer. The convenient setting for non-quasianalytic D enjoy- C arleman differentiable mappings. J. Funct. Anal. , 256(11):3510--3544, 2009

    2009

  22. [30]

    Microlocal analysis in hyperbolic dynamics and geometry

    Thibault Lefeuvre. Microlocal analysis in hyperbolic dynamics and geometry. With a contributed chapter by Yann Chaubet , volume 32 of Cours Sp \'e c. (Paris) . Paris: Soci \'e t \'e Math \'e matique de France (SMF), 2025

    2025

  23. [31]

    On contact Anosov flows

    Carlangelo Liverani. On contact Anosov flows. Ann. Math. (2) , 159(3):1275--1312, 2004

    2004

  24. [32]

    Axiom A diffeomorphisms have rational zeta functions

    Anthony Manning. Axiom A diffeomorphisms have rational zeta functions. Bull. London Math. Soc. , 3:215--220, 1971

    1971

  25. [33]

    J. N. Mather. Characterization of Anosov diffeomorphisms. Nederl. Akad. Wet., Proc., Ser. A , 71:479--483, 1968

    1968

  26. [34]

    A Morse complex for axiom A flows

    Antoine Meddane. A Morse complex for axiom A flows. J. \'E c. Polytech., Math. , 12:641--712, 2025

    2025

  27. [35]

    T. Regge. Analytic properties of the scattering matrix. Nuovo Cimento, X. Ser. , 8:671--679, 1958

    1958

  28. [36]

    Hans H. Rugh. The correlation spectrum for hyperbolic analytic maps. Nonlinearity , 5(6):1237--1263, 1992

    1992

  29. [37]

    Generalized Fredholm determinants and Selberg zeta functions for Axiom A dynamical systems

    Hans Henrik Rugh. Generalized Fredholm determinants and Selberg zeta functions for Axiom A dynamical systems. Ergodic Theory Dyn. Syst. , 16(4):805--819, 1996

    1996

  30. [38]

    Ya. G. Sina \" . Geodesic flows on compact surfaces of negative curvature. Sov. Math., Dokl. , 2:106--109, 1961

    1961

  31. [39]

    S. Smale. Differentiable dynamical systems. With an appendix to the first part of the paper: `` Anosov diffeomorphisms'' by John Mather . Bull. Am. Math. Soc. , 73:747--817, 1967

    1967

  32. [40]

    Semiclassical analysis , volume 138 of Grad

    Maciej Zworski. Semiclassical analysis , volume 138 of Grad. Stud. Math. Providence, RI: American Mathematical Society (AMS), 2012

    2012

Pith tools

Reviewed May 10, 2026 · model on record in the stance chip above.