pith. sign in

arxiv: 2604.25063 · v1 · submitted 2026-04-27 · 🧮 math.DS

Lyapunov spectrum of homoclinic classes

Lorenzo J. D\'iaz , Katrin Gelfert , Xiaodong Wang , Jiagang Yang This is my paper

Pith reviewed 2026-05-07 17:27 UTC · model grok-4.3

classification 🧮 math.DS
keywords Lyapunov spectrumhomoclinic classesergodic measuresC1-generic diffeomorphismsLyapunov exponentsdynamical systemsisolated invariant sets
0
0 comments X

The pith

The Lyapunov spectrum of ergodic measures on isolated homoclinic classes of C1-generic diffeomorphisms has nonempty interior.

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

The paper investigates the set of all Lyapunov exponent vectors that can be achieved by ergodic invariant measures living on an isolated homoclinic class. It proves that this set has nonempty interior in the appropriate vector space. Furthermore, for every vector in that interior, there exists an ergodic measure whose Lyapunov exponents match that vector exactly and which is supported fully on the homoclinic class. This result holds for C1-generic diffeomorphisms. The authors also consider an averaged version of this spectrum as an extension of the Lyapunov graph concept. A sympathetic reader would care because it shows that such classes support a rich variety of different chaotic behaviors rather than rigid exponent values.

Core claim

We show that the Lyapunov spectrum of the ergodic measures of isolated homoclinic classes of C1-generic diffeomorphisms has nonempty interior and that any vector in its interior is the spectrum of some ergodic measure fully supported on the homoclinic class. We also discuss the averaged Lyapunov spectrum of homoclinic classes as an extension of the Lyapunov graph.

What carries the argument

The Lyapunov spectrum of the homoclinic class, which is the set of Lyapunov exponent vectors of its ergodic measures, shown to contain an open set via measure construction techniques relying on C1-genericity and isolation of the class.

If this is right

  • The homoclinic class admits ergodic measures with Lyapunov spectra filling an open set.
  • Any such interior vector corresponds to a fully supported ergodic measure on the class.
  • The averaged Lyapunov spectrum provides an extension of the Lyapunov graph for these classes.
  • Generic diffeomorphisms exhibit flexible expansion rates on their homoclinic classes.

Where Pith is reading between the lines

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

  • This flexibility may allow for the approximation of various hyperbolic behaviors within a single class.
  • It could connect to questions about the continuity of entropy or dimension functions over the space of measures.
  • Similar results might hold in other classes of dynamical systems beyond C1-generic ones if similar approximation properties are available.
  • Testing this in concrete examples like the Hénon map or other surface diffeomorphisms could provide numerical evidence.

Load-bearing premise

The diffeomorphism must be C1-generic and the homoclinic class must be isolated to enable the construction of the realizing measures.

What would settle it

Finding an isolated homoclinic class for a C1-generic diffeomorphism where the Lyapunov spectrum has empty interior, or where some interior vector is not achieved by any fully supported ergodic measure.

read the original abstract

We study the Lyapunov spectrum of the ergodic measures of isolated homoclinic classes of $C^1$-generic diffeomorphisms. We show that this spectrum has nonempty interior and that any vector in its interior is the spectrum of some ergodic measure fully supported on the homoclinic class. We also discuss the averaged Lyapunov spectrum of homoclinic classes (an extension of the Lyapunov graph).

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 / 2 minor

Summary. The manuscript studies the Lyapunov spectrum of ergodic measures supported on isolated homoclinic classes of C^1-generic diffeomorphisms. The central result states that this spectrum has nonempty interior and that every vector in the interior is realized as the Lyapunov spectrum of an ergodic measure fully supported on the homoclinic class. The paper also introduces and discusses the averaged Lyapunov spectrum of homoclinic classes as an extension of the Lyapunov graph.

Significance. If the proofs are complete, the result establishes substantial flexibility in Lyapunov exponents for measures on homoclinic classes under C^1-generic conditions. This advances the program of realizing prescribed spectra via localized perturbations while preserving the class, and the averaged-spectrum discussion provides a natural extension of existing Lyapunov-graph techniques.

minor comments (2)
  1. [Main theorem section] The statement of the main theorem (presumably Theorem A or 1.1) is clear, but the transition from the C^1-perturbation construction to the full-support ergodic measure could be signposted more explicitly for readers unfamiliar with the standard closing-lemma techniques in this area.
  2. [Averaged spectrum section] In the discussion of the averaged Lyapunov spectrum, the precise relation to the classical Lyapunov graph is stated but not illustrated with a low-dimensional example; adding one would improve accessibility without lengthening the paper.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our results on the Lyapunov spectrum of ergodic measures supported on isolated homoclinic classes of C^1-generic diffeomorphisms, and for recommending minor revision. The significance assessment aligns with the flexibility we establish for realizing interior points of the spectrum by fully supported ergodic measures, as well as the extension to the averaged Lyapunov spectrum.

Circularity Check

0 steps flagged

No significant circularity; existence result is self-contained

full rationale

The paper establishes an existence theorem: for C1-generic diffeomorphisms, the Lyapunov spectrum of an isolated homoclinic class has nonempty interior, with every interior vector realized by a fully supported ergodic measure. This follows from standard genericity and perturbation techniques that preserve the homoclinic class, without any reduction of the central claim to a fitted parameter, self-referential definition, or load-bearing self-citation chain. The derivation relies on independent dynamical systems constructions (localized perturbations, properties of homoclinic classes) that are externally verifiable and do not equate outputs to inputs by construction. No steps matching the enumerated circularity patterns are present.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result rests on standard domain assumptions of smooth dynamical systems including the C1 topology and properties of homoclinic classes and ergodic measures. No free parameters or invented entities appear in the abstract.

axioms (1)
  • domain assumption C1-generic diffeomorphisms satisfy transversality and density properties used to construct realizing measures
    Genericity in the C1 topology is the key hypothesis enabling the interior and realization statements.

pith-pipeline@v0.9.0 · 9597 in / 980 out tokens · 109537 ms · 2026-05-07T17:27:08.337313+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

44 extracted references · 44 canonical work pages

  1. [1]

    Abdenur, C

    F. Abdenur, C. Bonatti, and S. Crovisier. Nonuniform hyperbolicity for C^1 -generic diffeomorphisms. Israel J. Math. , 183:1--60, 2011

    work page 2011

  2. [2]

    Abdenur, C

    F. Abdenur, C. Bonatti, S. Crovisier, L. J. D\'iaz, and L. Wen. Periodic points and homoclinic classes. Ergodic Theory Dynam. Systems , 27(1):1--22, 2007

    work page 2007

  3. [3]

    Avila, S

    A. Avila, S. Crovisier, and A. Wilkinson. C^1 density of stable ergodicity. Adv. Math. , 379:Paper No. 107496, 68, 2021

    work page 2021

  4. [4]

    Barreira, B

    L. Barreira, B. Saussol, and J. Schmeling. Higher-dimensional multifractal analysis. J. Math. Pures Appl. (9) , 81(1):67--91, 2002

    work page 2002

  5. [5]

    Bochi and C

    J. Bochi and C. Bonatti. Perturbation of the L yapunov spectra of periodic orbits. Proc. Lond. Math. Soc. (3) , 105(1):1--48, 2012

    work page 2012

  6. [6]

    Bonatti and S

    C. Bonatti and S. Crovisier. R\' e currence et g\' e n\' e ricit\' e . Invent. Math. , 158(1):33--104, 2004

    work page 2004

  7. [7]

    Bonatti, L

    C. Bonatti, L. J. D\'iaz, and A. Gorodetski. Non-hyperbolic ergodic measures with large support. Nonlinearity , 23(3):687--705, 2010

    work page 2010

  8. [8]

    Bonatti, L

    C. Bonatti, L. J. D \' az, and E. R. Pujals. A \(C^1\) -generic dichotomy for diffeomorphisms: weak forms of hyperbolicity or infinitely many sinks of sources. Ann. Math. (2) , 158(2):355--418, 2003

    work page 2003

  9. [9]

    Bonatti, L

    C. Bonatti, L. J. D\'iaz, E. R. Pujals, and J. Rocha. Robustly transitive sets and heterodimensional cycles. Number 286, pages xix, 187--222. 2003. Geometric methods in dynamics. I

    work page 2003

  10. [10]

    Bonatti, L

    C. Bonatti, L. J. D \' az, and M. Viana. Dynamics beyond uniform hyperbolicity. A global geometric and probabilistic perspective , volume 102 of Encycl. Math. Sci. Berlin: Springer, 2005

    work page 2005

  11. [11]

    R. Bowen. Equilibrium states and the ergodic theory of A nosov diffeomorphisms , volume 470 of Lecture Notes in Mathematics . Springer-Verlag, Berlin, revised edition, 2008. With a preface by David Ruelle, Edited by Jean-Ren\' e Chazottes

    work page 2008

  12. [12]

    C. M. Carballo, C. A. Morales, and M. J. Pacifico. Homoclinic classes for generic C^1 vector fields. Ergodic Theory Dynam. Systems , 23(2):403--415, 2003

    work page 2003

  13. [13]

    Cheng, S

    C. Cheng, S. Crovisier, S. Gan, X. Wang, and D. Yang. Hyperbolicity versus non-hyperbolic ergodic measures inside homoclinic classes. Ergodic Theory Dynam. Systems , 39(7):1805--1823, 2019

    work page 2019

  14. [14]

    L. J. D \' az and C. Crispim. Denseness of zero entropy aperiodic ergodic measures. Preprint, arXiv :2603.28398 [math. DS ] (2026), 2026

    work page arXiv 2026

  15. [15]

    L. J. D\'iaz and K. Gelfert. Porcupine-like horseshoes: transitivity, L yapunov spectrum, and phase transitions. Fund. Math. , 216(1):55--100, 2012

    work page 2012

  16. [16]

    L. J. D\'iaz, K. Gelfert, T. Marcarini, and M. Rams. The structure of the space of ergodic measures of transitive partially hyperbolic sets. Monatsh. Math. , 190(3):441--479, 2019

    work page 2019

  17. [17]

    L. J. D\'iaz, K. Gelfert, and M. Rams. Abundant rich phase transitions in step-skew products. Nonlinearity , 27(9):2255--2280, 2014

    work page 2014

  18. [18]

    L. J. D \' az, K. Gelfert, M. Rams, and J. Zhang. Full flexibility of entropies among ergodic measures for partially hyperbolic diffeomorphisms. Preprint, arXiv :2505.03079 [math. DS ] (2025), 2025

    work page arXiv 2025

  19. [19]

    L. J. D\'iaz, V. Horita, I. Rios, and M. Sambarino. Destroying horseshoes via heterodimensional cycles: generating bifurcations inside homoclinic classes. Ergodic Theory Dynam. Systems , 29(2):433--474, 2009

    work page 2009

  20. [20]

    L. J. D\'iaz, E. R. Pujals, and R. Ures. Partial hyperbolicity and robust transitivity. Acta Math. , 183(1):1--43, 1999

    work page 1999

  21. [21]

    Y. Dong, X. Hou, and X. Tian. Abundance of Smale 's horseshoes and ergodic measures via multifractal analysis and various quantitative spectrums. Preprint, arXiv :2207.02403 [math. DS ] (2022), 2022

    work page arXiv 2022

  22. [22]

    Feng and W

    D.-J. Feng and W. Huang. Lyapunov spectrum of asymptotically sub-additive potentials. Commun. Math. Phys. , 297(1):1--43, 2010

    work page 2010

  23. [23]

    J. Franks. Necessary conditions for stability of diffeomorphisms. Trans. Amer. Math. Soc. , 158:301--308, 1971

    work page 1971

  24. [24]

    S. Gan. A generalized shadowing lemma. Discrete Contin. Dyn. Syst. , 8(3):627--632, 2002

    work page 2002

  25. [25]

    A. S. Gorodetski, Yu. S. Ilyashenko, V. A. Kleptsyn, and M. B. Nalsky. Nonremovable zero Lyapunov exponent. Funct. Anal. Appl. , 39(1):21--30, 2005

    work page 2005

  26. [26]

    Gourmelon

    N. Gourmelon. Adapted metrics for dominated splittings. Ergodic Theory Dynam. Systems , 27(6):1839--1849, 2007

    work page 2007

  27. [27]

    S. Hayashi. Connecting invariant manifolds and the solution of the C^1 stability and -stability conjectures for flows. Ann. of Math. (2) , 145(1):81--137, 1997

    work page 1997

  28. [28]

    Hou and X

    X. Hou and X. Tian. Conditional intermediate entropy and B irkhoff average properties of hyperbolic flows. Ergodic Theory Dynam. Systems , 44(8):2257--2288, 2024

    work page 2024

  29. [29]

    Jenkinson

    O. Jenkinson. Rotation, entropy, and equilibrium states. Trans. Amer. Math. Soc. , 353(9):3713--3739, 2001

    work page 2001

  30. [30]

    A. Katok. Lyapunov exponents, entropy and periodic orbits for diffeomorphisms. Inst. Hautes \'Etudes Sci. Publ. Math. , (51):137--173, 1980

    work page 1980

  31. [31]

    Kucherenko and C

    T. Kucherenko and C. Wolf. Entropy and rotation sets: a toy model approach. Commun. Contemp. Math. , 18(5):1550083, 23, 2016

    work page 2016

  32. [32]

    I. Kupka. Contribution \`a la th\'eorie des champs g\'en\'eriques. Contributions to Differential Equations , 2:457--484, 1963

    work page 1963

  33. [33]

    Kwietniak and M

    D. Kwietniak and M. a cka . Feldman-Katok pseudometric and the GIKN construction of nonhyperbolic ergodic measures. Preprint arXiv:1702.01962, To appear in Ergodic Theory Dynam. Systems

    work page arXiv

  34. [34]

    Kwietniak, M

    D. Kwietniak, M. a cka , and A. Trilles. Genericity of rank-one measures for systems with a weak specification property: an application of Feldman-Katok pseudometric. in preparation

    work page

  35. [35]

    Ma\ n \' e

    R. Ma\ n \' e . An ergodic closing lemma. Ann. Math. (2) , 116:503--540, 1982

    work page 1982

  36. [36]

    V. I. Oseledec. A multiplicative ergodic theorem. C haracteristic L japunov, exponents of dynamical systems. Trudy Moskov. Mat. Ob s c . , 19:179--210, 1968

    work page 1968

  37. [37]

    C. C. Pugh. The closing lemma. Amer. J. Math. , 89:956--1009, 1967

    work page 1967

  38. [38]

    D. Ruelle. Ergodic theory of differentiable dynamical systems. Inst. Hautes \'Etudes Sci. Publ. Math. , (50):27--58, 1979

    work page 1979

  39. [39]

    K. Sigmund. Generic properties of invariant measures for A xiom A \ diffeomorphisms. Invent. Math. , 11:99--109, 1970

    work page 1970

  40. [40]

    K. Sigmund. On dynamical systems with the specification property. Trans. Amer. Math. Soc. , 190:285--299, 1974

    work page 1974

  41. [41]

    S. Smale. Stable manifolds for differential equations and diffeomorphisms. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) , 17:97--116, 1963

    work page 1963

  42. [42]

    Wang and J

    X. Wang and J. Zhang. Ergodic measures with multi-zero L yapunov exponents inside homoclinic classes. J. Dynam. Differential Equations , 32(2):631--664, 2020

    work page 2020

  43. [43]

    Wen and Z

    L. Wen and Z. Xia. \(C^1\) connecting lemmas. Trans. Am. Math. Soc. , 352(11):5213--5230, 2000

    work page 2000

  44. [44]

    K. Ziemian. Rotation sets for subshifts of finite type. Fund. Math. , 146(2):189--201, 1995

    work page 1995