pith. sign in

arxiv: 2606.30956 · v1 · pith:TFTW567Rnew · submitted 2026-06-29 · 🧮 math.DS

Discontinuity of Lyapunov exponents vs Entropy for smooth surface diffeomorphisms

J\'er\^ome Buzzi This is my paper

Pith reviewed 2026-07-01 00:50 UTC · model grok-4.3

classification 🧮 math.DS
keywords Lyapunov exponentsentropysurface diffeomorphismscontinuitymaximal entropy measuresmooth dynamicsergodic theory
0
0 comments X

The pith

Entropy controls the continuity of Lyapunov exponents near maximal entropy measures for smooth surface diffeomorphisms.

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

The paper explains how and why entropy can be used to control Lyapunov exponents in the dynamics of smooth surface diffeomorphisms. It presents a simplified argument establishing that these exponents vary continuously for measures close to the maximal entropy measure. This matters because Lyapunov exponents capture the asymptotic infinitesimal stretching and contracting behavior along typical orbits. A reader would care as the control via entropy offers a practical handle on orbit stability in two-dimensional smooth systems.

Core claim

For smooth surface diffeomorphisms, Lyapunov exponents can be controlled using entropy, and in particular they satisfy a continuity property when the measures are taken near the measure of maximal entropy, as shown by a simplified version of the argument.

What carries the argument

A simplified argument that links entropy to the variation of Lyapunov exponents for measures near the maximal entropy one, relying on the setting of smooth surface diffeomorphisms.

If this is right

  • Measures near the maximal entropy measure inherit controlled Lyapunov exponent values from the maximal entropy measure itself.
  • Entropy provides an upper bound on possible jumps in Lyapunov exponents within that neighborhood.
  • The continuity allows the asymptotic orbit behavior to be predicted from the maximal entropy case for nearby invariant measures.
  • The simplification makes the control accessible without the full technical machinery of the original argument.

Where Pith is reading between the lines

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

  • Numerical approximation of the maximal entropy measure could then yield reliable estimates of nearby Lyapunov exponents without separate computation for each measure.
  • The same entropy-based control might be tested for other invariants such as dimension or pressure in the same surface setting.
  • If the continuity fails in higher dimensions, the surface case would highlight a special two-dimensional phenomenon tied to the entropy control.

Load-bearing premise

The background results available in the setting of smooth surface diffeomorphisms are sufficient to support the simplified continuity argument.

What would settle it

A concrete smooth surface diffeomorphism together with a sequence of measures approaching the maximal entropy measure for which the Lyapunov exponents fail to approach the value at the maximal entropy measure would falsify the continuity property.

read the original abstract

Lyapunov exponents are fundamental invariants in smooth ergodic theory describing the asymptotic infinitesimal behavior along typical orbits. This text aims to explain how and why to control Lyapunov exponents using entropy for smooth surface diffeomorphisms. It fits into the framework of our recent joint works with Sylvain CROVISIER and Omri SARIG. We will focus especially on the continuity property of exponents for measures near the maximal entropy measure, by presenting a simplified version of the original argument. Our exposition is geared towards advanced students and researchers in dynamics that are not necessarily familiar with smooth ergodic theory.

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 manuscript provides an expository simplification of results from the authors' recent joint works with Crovisier and Sarig on controlling Lyapunov exponents via entropy for smooth surface diffeomorphisms. It focuses on establishing the continuity of Lyapunov exponents for invariant measures near the measure of maximal entropy, with the exposition aimed at advanced students and researchers unfamiliar with smooth ergodic theory.

Significance. The continuity result near maximal entropy measures is of interest in smooth ergodic theory as it provides a concrete instance where entropy controls the asymptotic behavior of exponents. The paper's value lies in its simplified presentation of an already-established argument, which makes the framework more accessible without introducing new theorems or derivations.

minor comments (3)
  1. The title emphasizes 'Discontinuity' while the body and abstract focus exclusively on the continuity property near maximal entropy; a brief clarifying sentence in the introduction relating the two would improve reader orientation.
  2. Explicit cross-references to the precise statements (e.g., theorem numbers) being simplified from the Crovisier-Sarig joint works would help readers verify the simplification without consulting the original papers.
  3. Notation for Lyapunov exponents and entropy is introduced clearly but could be summarized in a short table or list for quick reference by readers outside the subfield.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive recommendation to accept the manuscript. The report accurately characterizes the work as an expository simplification of our joint results with Crovisier and Sarig, focused on making the continuity of Lyapunov exponents near maximal entropy measures accessible to readers unfamiliar with smooth ergodic theory.

Circularity Check

0 steps flagged

Expository simplification of prior results; no internal derivation chain present

full rationale

The manuscript explicitly positions itself as an exposition and simplification of an already-established continuity statement from the authors' recent joint works with Crovisier and Sarig. No new theorem, derivation, or prediction is advanced in this text whose logic could reduce to its own inputs or self-citations. The provided abstract and description contain no equations or load-bearing steps that equate outputs to fitted inputs by construction. This is the normal case of an expository note referencing external (even if co-authored) background results.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review provides no explicit free parameters, axioms, or invented entities; the central claim rests on standard assumptions of smooth ergodic theory for surface diffeomorphisms.

axioms (1)
  • domain assumption Lyapunov exponents and entropy are well-defined invariants for smooth surface diffeomorphisms
    Invoked implicitly by the focus on continuity of exponents near maximal entropy measure.

pith-pipeline@v0.9.1-grok · 5613 in / 1123 out tokens · 37285 ms · 2026-07-01T00:50:00.230154+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

45 extracted references · 6 canonical work pages

  1. [1]

    TheC 1+α hypothesis in Pesin Theory revisited.J

    Christian Bonatti, Sylvain Crovisier, and Katsutoshi Shinohara. TheC 1+α hypothesis in Pesin Theory revisited.J. Mod. Dyn., 7(4):605–618, 2013

    2013

  2. [2]

    The entropy theory of symbolic extensions.Invent

    Mike Boyle and Tomasz Downarowicz. The entropy theory of symbolic extensions.Invent. Math., 156:119–161, 2004

    2004

  3. [3]

    Residual entropy, conditional entropy and sub- shift covers.Forum Math., 14:713–757, 2002

    Mike Boyle, Doris Fiebig, and Ulrich Fiebig. Residual entropy, conditional entropy and sub- shift covers.Forum Math., 14:713–757, 2002

    2002

  4. [4]

    A direct proof of the tail variational principle and its extension to maps

    David Burguet. A direct proof of the tail variational principle and its extension to maps. Ergodic Theory and Dynamical Systems, 29(2):357–369, 2009

    2009

  5. [5]

    SRB measures forC ∞ surface diffeomorphisms.Invent

    David Burguet. SRB measures forC ∞ surface diffeomorphisms.Invent. Math., 235(3):1019– 1062, 2024

    2024

  6. [6]

    Intrinsic ergodicity of smooth interval maps.Israel J

    J´ erˆ ome Buzzi. Intrinsic ergodicity of smooth interval maps.Israel J. Math., 100:125–161, 1997

    1997

  7. [7]

    J´ erˆ ome Buzzi.Cr surface diffeomorphisms with no maximal entropy measure.Ergodic Theory and Dynamical Systems, 34:1770–1793, 2014

    2014

  8. [8]

    Continuity of Lyapunov exponents and entropy for surface diffeomorphisms.Inventiones Mathematicae, 230:767–849, 2022

    J´ erˆ ome Buzzi, Sylvain Crovisier, and Omri Sarig. Continuity of Lyapunov exponents and entropy for surface diffeomorphisms.Inventiones Mathematicae, 230:767–849, 2022

    2022

  9. [9]

    Measures of maximal entropy for surface diffeomorphisms.Annals of Math., 195:421–508, 2022

    J´ erˆ ome Buzzi, Sylvain Crovisier, and Omri Sarig. Measures of maximal entropy for surface diffeomorphisms.Annals of Math., 195:421–508, 2022

    2022

  10. [10]

    On the existence of SRB measures forC∞ surface diffeomorphisms.Inter

    J´ erˆ ome Buzzi, Sylvain Crovisier, and Omri Sarig. On the existence of SRB measures forC∞ surface diffeomorphisms.Inter. Math. Res. Notices, 24:20812––20826, 2023

    2023

  11. [11]

    Strong positive recurrence and exponential mixing for diffeomorphisms,arxiv:2501.07455, 2025

    J´ erˆ ome Buzzi, Sylvain Crovisier, and Omri Sarig. Strong positive recurrence and exponential mixing for diffeomorphisms,arxiv:2501.07455, 2025

    work page arXiv 2025

  12. [12]

    Cornfeld, S

    I.P. Cornfeld, S. V. Fomin, and Ya. G. Sina˘ ı.Ergodic theory. Springer-Verlag, New York, 1982

    1982

  13. [13]

    Spectral gap and transience for Ruelle operators on countable Markov shifts.Comm

    Van Cyr and Omri Sarig. Spectral gap and transience for Ruelle operators on countable Markov shifts.Comm. Math. Phys., 292(3):637–666, 2009

    2009

  14. [14]

    Continuity of Lyapunov exponents forC r one- dimensional maps,arxiv:2510.18804, 2025

    Alexandre Delplanque and Hengyi Li. Continuity of Lyapunov exponents forC r one- dimensional maps,arxiv:2510.18804, 2025

    work page arXiv 2025

  15. [15]

    Lecture Notes in Mathematics

    Manfred Denker, Christian Grillenberger, and Karl Sigmund.Ergodic theory on compact spaces. Lecture Notes in Mathematics. 1976

    1976

  16. [16]

    E. I. Dinaburg. On the relations among various entropy characteristics of dynamical systems. Math. USSR-Izv., 5:2:337–378, 1971

    1971

  17. [17]

    Entropy structures.J

    Tomasz Downarowicz. Entropy structures.J. Analyse Math., 96:57–116, 2005

    2005

  18. [18]

    John Wiley & Sons, Chichester, UK, 3rd edition, 2014

    Kenneth Falconer.Fractal Geometry: Mathematical Foundations and Applications. John Wiley & Sons, Chichester, UK, 3rd edition, 2014

    2014

  19. [19]

    T. N. T. Goodman. Relating topological entropy and measure entropy.Bull. Lond. Math. Soc., 3:176–180, 1971

    1971

  20. [20]

    B. M. Gurevich and S. V. Savchenko. Thermodynamic formalism for symbolic Markov chains with a countable number of states.Uspekhi Mat. Nauk, 53(2(320)):3–106, 1998

    1998

  21. [21]

    Lyapunov exponents, entropy and periodic orbits for diffeomorphisms.Pub- lications Math´ ematiques de l’IH´ES, 51:137–173, 1980

    Anatole Katok. Lyapunov exponents, entropy and periodic orbits for diffeomorphisms.Pub- lications Math´ ematiques de l’IH´ES, 51:137–173, 1980

    1980

  22. [22]

    Cambridge University Press, Cambridge, UK, 1997

    Anatole Katok and Boris Hasselblatt.Introduction to the Modern Theory of Dynamical Sys- tems, volume 54 ofEncyclopedia of Mathematics and Its Applications. Cambridge University Press, Cambridge, UK, 1997

    1997

  23. [23]

    Walter de Gruyter & Co., Berlin, 1985

    Ulrich Krengel.Ergodic theorems, volume 6 ofDe Gruyter Studies in Mathematics. Walter de Gruyter & Co., Berlin, 1985. With a supplement by Antoine Brunel. 91 92 BIBLIOGRAPHY

    1985

  24. [24]

    Ledrappier and P

    F. Ledrappier and P. Walters. A relativised variational principle for continuous transforma- tions.Journal of the London Mathematical Society, s2-16:568–576

  25. [25]

    Ledrappier and L.-S

    F. Ledrappier and L.-S. Young. The metric entropy of diffeomorphisms. I. Characterization of measures satisfying Pesin’s entropy formula.Ann. of Math. (2), 122(3):509–539, 1985

    1985

  26. [26]

    Ledrappier and L.-S

    F. Ledrappier and L.-S. Young. The metric entropy of diffeomorphisms. II. Relations between entropy, exponents and dimension.Ann. of Math. (2), 122(3):540–574, 1985

    1985

  27. [27]

    Entropy continuity of Lyapunov exponents for non-flat 1-dimensional maps, arxiv:2412.19007

    Hengyi Li. Entropy continuity of Lyapunov exponents for non-flat 1-dimensional maps, arxiv:2412.19007. 2024

    work page arXiv 2024

  28. [28]

    Ergodic measures with large entropy have long unstable manifolds forC ∞ surface diffeomorphisms,arxiv:2410.07979

    Chiyi Luo and Dawei Yang. Ergodic measures with large entropy have long unstable manifolds forC ∞ surface diffeomorphisms,arxiv:2410.07979. 2024

    work page arXiv 2024

  29. [29]

    Exponential mixing of measures of max- imal entropy for certain skew products,arxiv:2509.09908, 2025

    Karina Marin, Mauricio Poletti, and Filiphe Veiga. Exponential mixing of measures of max- imal entropy for certain skew products,arxiv:2509.09908, 2025

    work page arXiv 2025

  30. [30]

    Springer- Verlag, Berlin, Heidelberg, 1987

    Ricardo Ma˜ n´ e.Ergodic Theory and Differentiable Dynamics, volume 8 ofErgebnisse der Mathematik und ihrer Grenzgebiete (Results in Mathematics and Related Areas). Springer- Verlag, Berlin, Heidelberg, 1987

    1987

  31. [31]

    Diffeomorphism without any measure with maximal entropy.Bull

    Michal Misiurewicz. Diffeomorphism without any measure with maximal entropy.Bull. Acad. Polon. Sci. S´ er. Sci. Math. Astronom. Phys., 21:903–910, 1973

    1973

  32. [32]

    A short proof of the variational principle for az n action on a compact space.Publications des s´ eminaires de math´ ematiques et informatique de Rennes, S4:1–10, 1974

    Michal Misiurewicz. A short proof of the variational principle for az n action on a compact space.Publications des s´ eminaires de math´ ematiques et informatique de Rennes, S4:1–10, 1974

    1974

  33. [33]

    Topological conditional entropy.Studia Mathematica, 55:175–200, 1976

    Michal Misiurewicz. Topological conditional entropy.Studia Mathematica, 55:175–200, 1976

    1976

  34. [34]

    Continuity properties of entropy.Annals Math., 129:215–235, 1989

    Sheldon Newhouse. Continuity properties of entropy.Annals Math., 129:215–235, 1989

    1989

  35. [35]

    Ya. B. Pesin. Characteristic lyapunov exponents and smooth ergodic theory.Russian Math. Surveys, 32:55–114, 1977

    1977

  36. [36]

    Ergodic attractors.Trans

    Charles Pugh and Michael Shub. Ergodic attractors.Trans. Amer. Math. Soc., 312(1):1–54, 1989

    1989

  37. [37]

    V.A. Rokhlin. Lectures on the entropy theory of measure-preserving transformations.Russian Math. Surveys, 22:1–52, 1967

    1967

  38. [38]

    Cambridge University Press, Cambridge, UK, 2014

    Marcelo Viana.Lectures on Lyapunov exponents. Cambridge University Press, Cambridge, UK, 2014

    2014

  39. [39]

    Continuity of Lyapunov exponents and entropy for surface diffeomorphisms

    Marcelo Viana. Continuity of Lyapunov exponents and entropy for surface diffeomorphisms. ICM, 230:767–849, 2022

    2022

  40. [40]

    Cambridge University Press, 2016

    Marcelo Viana and Krerley Oliveira.Foundations of Ergodic Theory. Cambridge University Press, 2016

    2016

  41. [41]

    Unique ergodicity and random matrix products

    Peter Walters. Unique ergodicity and random matrix products. 1186:37–55, 1986

    1986

  42. [42]

    Y. Yomdin. Volume growth and entropy.Israel J. Math., 57(3):285–300, 1987

    1987

  43. [43]

    Dimension, entropy and Lyapunov exponents.Ergodic Theory Dynam

    Lai Sang Young. Dimension, entropy and Lyapunov exponents.Ergodic Theory Dynam. Sys- tems, 2(1):109–124, 1982

    1982

  44. [44]

    What are SRB measures, and which dynamical systems have them?J

    Lai-Sang Young. What are SRB measures, and which dynamical systems have them?J. Stat. Phys., 108:733–754, 2002

    2002

  45. [45]

    Measures of maximal entropy for

    Yuntao Zang. Measures of maximal entropy forc ∞ three-dimensional flows, arxiv:2503.21183. 2025

    work page arXiv 2025