pith. sign in

arxiv: 2602.10033 · v2 · submitted 2026-02-10 · 🧮 math.DS

Entropy formula for surface diffeomorphisms

Yuntao Zang This is my paper

Pith reviewed 2026-05-16 02:25 UTC · model grok-4.3

classification 🧮 math.DS
keywords topological entropydiffeomorphismssurface dynamicsLyapunov exponentsvolume growthentropy formulas
0
0 comments X

The pith

Surface diffeomorphisms satisfy topological entropy equal to the growth rate of the integrated derivative norm.

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

The paper proves that for any C^r diffeomorphism on a compact surface satisfying a lower bound on entropy relative to its maximal Lyapunov exponent, the topological entropy equals the limit of one over n times the log of the integral over the surface of the norm of the nth derivative. This gives a new expression for entropy in terms of average local stretching rather than direct orbit counting. The result also equates entropy to the exponential volume growth of typical curves. Applications follow for computing or bounding entropy in concrete surface systems.

Core claim

Under the hypothesis that h_top(f) is at least lambda plus of f divided by r, for a C^r diffeomorphism f of a compact surface M, the topological entropy equals the limit as n tends to infinity of one over n times the logarithm of the integral over M of the norm of the nth iterate of the derivative at x.

What carries the argument

The integral over M of the norm of Df^n_x, whose per-iterate logarithmic growth rate equals the topological entropy.

If this is right

  • Entropy admits computation via integration of local expansion rates rather than orbit enumeration.
  • Topological entropy is equivalent to the exponential volume growth rate of typical curves.
  • New estimates and bounds for entropy become available in specific families of surface diffeomorphisms.

Where Pith is reading between the lines

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

  • Numerical approximation of entropy could proceed by discretizing and integrating derivative norms on a grid.
  • Similar integral formulas might apply in higher dimensions once continuity of Lyapunov exponents is established there.
  • The formula offers a bridge between entropy and average distortion that could simplify proofs of continuity or semicontinuity results.

Load-bearing premise

That the topological entropy is at least the maximal Lyapunov exponent divided by the degree of differentiability r.

What would settle it

A C^r surface diffeomorphism obeying the entropy lower bound relative to its maximal Lyapunov exponent for which the limit of one over n log of the integral of ||Df^n_x|| differs from h_top(f).

Figures

Figures reproduced from arXiv: 2602.10033 by Yuntao Zang.

Figure 1
Figure 1. Figure 1: The stable holonomy map πs Let us first briefly recall the argument in [36] (see also [13], [27] for similar ideas) estab￾lishing h(f ,ν) ≤ liminf n→+∞ 1 n logVol¡ f n ¡ W u loc(y) ¢¢. (6) The fact that νy (Λ) > 0 is what ensures sufficient volume growth. Given any small ε > 0 and any large n ∈ N, since νy (Λ) > 0, by a fiber-version of Katok’s entropy formula (see [36, Theorem A (1)]), there is a maximal … view at source ↗
Figure 2
Figure 2. Figure 2: Growth at geometric and neutral times • Neutral times (with proportion 1−α): during which the curve remains confined inside a single dynamical ball and develops complexity only through local folding. During geometric times, one only needs to count the number of dynamical balls that the curve visits, which contributes an entropy-type growth, while no significant local complexity is generated inside each dyn… view at source ↗
Figure 3
Figure 3. Figure 3: Local volume growth for different value of [PITH_FULL_IMAGE:figures/full_fig_p051_3.png] view at source ↗
read the original abstract

Let $f$ be a $C^r$ ($r>1$) diffeomorphism on a compact surface $M$ with $h_{\rm top}(f)\geq\frac{\lambda^{+}(f)}{r}$ where $\lambda^{+}(f):=\lim_{n\to+\infty}\frac{1}{n}\max_{x\in M}\log \left\|Df^{n}_{x}\right\|$. We establish an equivalent formula for the topological entropy: $$h_{\rm top}(f)=\lim_{n\to+\infty}\frac{1}{n}\log\int_{M}\left\|Df^{n}_{x}\right\|\,dx.$$ We also characterize the topological entropy via the volume growth of curves and several applications are presented. Our approach builds on the key ideas developed in the works of Buzzi-Crovisier-Sarig (\emph{Invent. Math.}, 2022) and Burguet (\emph{Ann. Henri Poincar\'e}, 2024) concerning the continuity of the Lyapunov exponents.

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 establishes that for a C^r (r>1) diffeomorphism f on a compact surface M satisfying the entropy lower bound h_top(f) ≥ λ⁺(f)/r, where λ⁺(f) is the maximal Lyapunov exponent, the topological entropy admits the equivalent formula h_top(f) = lim_{n→∞} (1/n) log ∫_M ||Df^n_x|| dx. It further characterizes h_top via volume growth of curves and presents applications, relying on continuity of Lyapunov exponents from Buzzi-Crovisier-Sarig (Invent. Math. 2022) and Burguet (Ann. Henri Poincaré 2024).

Significance. If the central equality holds, the integral formula supplies a new, potentially computable expression for topological entropy on surfaces that directly incorporates derivative growth, complementing existing variational and volume-growth characterizations. The approach leverages established continuity results for Lyapunov exponents, which is a strength when the entropy lower bound is satisfied; this could enable further work on entropy computation or rigidity questions under the stated hypothesis.

major comments (2)
  1. [main theorem and its proof] The lower bound for lim (1/n) log ∫ ||Df^n_x|| dx requires a quantitative control on the measure of the set where ||Df^n_x|| is close to its maximum (to prevent exponential decay due to deteriorating modulus of continuity). The manuscript invokes continuity of Lyapunov exponents under the assumption h_top(f) ≥ λ⁺(f)/r, but it is not shown explicitly that this entropy threshold suffices to guarantee the needed measure lower bound; see the argument following the statement of the main theorem.
  2. [proof of the integral formula] The crude upper bound ∫ ||Df^n_x|| dx ≤ vol(M) · max ||Df^n|| always yields lim sup (1/n) log ∫ ≤ λ⁺(f). The paper must therefore establish a matching lower bound equaling h_top rather than λ⁺(f). The entropy assumption is used to apply the cited continuity results, but the precise way the assumption prevents the integral from collapsing to the maximal exponent (rather than to h_top) needs a self-contained estimate in the relevant section.
minor comments (2)
  1. [introduction] The notation λ⁺(f) is defined in the abstract but should be restated at the beginning of the main theorem statement for clarity.
  2. [applications] The volume-growth characterization of h_top is mentioned but its relation to the integral formula is not compared in detail; a short remark would help readers.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and valuable comments, which help clarify the proof structure. We address the two major comments below and will revise the manuscript to incorporate explicit estimates as suggested.

read point-by-point responses

  1. Referee: The lower bound for lim (1/n) log ∫ ||Df^n_x|| dx requires a quantitative control on the measure of the set where ||Df^n_x|| is close to its maximum (to prevent exponential decay due to deteriorating modulus of continuity). The manuscript invokes continuity of Lyapunov exponents under the assumption h_top(f) ≥ λ⁺(f)/r, but it is not shown explicitly that this entropy threshold suffices to guarantee the needed measure lower bound; see the argument following the statement of the main theorem.

    Authors: We agree that an explicit quantitative estimate is needed for the measure lower bound. In the revision we will add a new Lemma 3.2 immediately after the statement of the main theorem. The lemma combines the entropy hypothesis h_top(f) ≥ λ⁺(f)/r with the continuity of Lyapunov exponents (Buzzi-Crovisier-Sarig, Invent. Math. 2022) to produce the bound meas({x : (1/n) log ||Df^n_x|| > λ⁺(f) - ε}) ≥ exp(-C r ε n) for a constant C depending only on the surface. This lower bound on the measure is derived by lifting the entropy to a hyperbolic measure whose Lyapunov exponents are controlled by the threshold, ensuring the integral cannot decay faster than exp(n h_top(f)). The argument is now fully self-contained in the revised Section 3. revision: yes

  2. Referee: The crude upper bound ∫ ||Df^n_x|| dx ≤ vol(M) · max ||Df^n|| always yields lim sup (1/n) log ∫ ≤ λ⁺(f). The paper must therefore establish a matching lower bound equaling h_top rather than λ⁺(f). The entropy assumption is used to apply the cited continuity results, but the precise way the assumption prevents the integral from collapsing to the maximal exponent (rather than to h_top) needs a self-contained estimate in the relevant section.

    Authors: We accept the observation that the upper bound is always λ⁺(f) and that the lower bound must be shown to equal h_top(f) rather than λ⁺(f). The entropy threshold is used precisely to invoke the continuity results in a regime where the average derivative growth matches the entropy. In the revised manuscript we will insert a short subsection (3.3) containing a self-contained estimate: under h_top(f) ≥ λ⁺(f)/r the continuity theorem supplies a sequence of measures μ_n whose Lyapunov exponents satisfy ∫ log ||Df^n|| dμ_n ≥ n h_top(f) - o(n), and the integral ∫_M ||Df^n_x|| dx is then bounded from below by integrating over a neighborhood of the support of μ_n whose size is controlled by the entropy. This prevents collapse to λ⁺(f) and yields the matching lower bound lim inf (1/n) log ∫ ≥ h_top(f). revision: yes

Circularity Check

0 steps flagged

No circularity; equivalence derived from external continuity results under explicit hypothesis

full rationale

The claimed formula is obtained by applying the continuity of Lyapunov exponents (from Buzzi-Crovisier-Sarig 2022 and Burguet 2024) to control the measure of regions where ||Df^n|| is close to its maximum, under the stated hypothesis h_top(f) ≥ λ⁺(f)/r. These cited works are independent, by different authors, and supply the required modulus-of-continuity estimates without reference to the present paper. The upper bound lim (1/n) log ∫ ||Df^n|| dx ≤ λ⁺(f) is immediate from the definition of λ⁺ and does not rely on any fitted quantity or self-reference. The matching lower bound is obtained from the external continuity theorems once the entropy lower bound is assumed; no step equates the target equality to an input by construction, renames a known pattern, or imports a uniqueness claim from the same authors. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on the key assumption h_top(f) ≥ λ⁺(f)/r together with standard background results on Lyapunov exponents and topological entropy for C^r diffeomorphisms.

axioms (2)
  • domain assumption f is a C^r diffeomorphism (r>1) on a compact surface M
    Standard setup for the class of maps under consideration.
  • ad hoc to paper h_top(f) ≥ λ⁺(f)/r
    This inequality is the explicit hypothesis required for the equivalence to hold, as stated in the abstract.

pith-pipeline@v0.9.0 · 5466 in / 1317 out tokens · 43675 ms · 2026-05-16T02:25:38.627399+00:00 · methodology

discussion (0)

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

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

38 extracted references · 38 canonical work pages

  1. [1]

    Barreira and Y

    L. Barreira and Y. Pesin, Nonuniform hyperbolicity: Dynamics of systems with nonzero Lyapunov exponents. Encycl. Math. Appl.Cambridge University Press, Cam- bridge, 2007. 7, 8

    work page 2007

  2. [2]

    Ben Ovadia and D

    S. Ben Ovadia and D. Burguet, Generalizedu-Gibbs measures forC ∞ diffeomor- phisms.arXiv:2506.18238, (2025). 3

    work page arXiv 2025

  3. [3]

    Burguet, Periodic expansiveness of smooth surface diffeomorphisms and applica- tions.J

    D. Burguet, Periodic expansiveness of smooth surface diffeomorphisms and applica- tions.J. Eur . Math. Soc.,22(2020), 413–454. 10

    work page 2020

  4. [4]

    Burguet, Maximal measure and entropic continuity of Lyapunov exponents forC r surface diffeomorphisms with large entropy.Ann

    D. Burguet, Maximal measure and entropic continuity of Lyapunov exponents forC r surface diffeomorphisms with large entropy.Ann. Henri Poincaré,25(2024), 1485–

    work page 2024

  5. [5]

    3, 4, 7, 11, 13, 18, 19, 25

    work page

  6. [6]

    Burguet, SRB measures forC ∞ surface diffeomorphisms.Invent

    D. Burguet, SRB measures forC ∞ surface diffeomorphisms.Invent. Math.,235(2024), 1019–1062. 3, 6, 19

    work page 2024

  7. [7]

    Burguet, C

    D. Burguet, C. Luo and D. Yang, Effective SPR property for surface diffeomorphisms and three-dimensional vector fields.arXiv:2512.03515v1, (2025). 3, 13

    work page arXiv 2025

  8. [8]

    Buzzi, Entropy, volume growth and Lyapunov exponents.unpublished, 1996

    J. Buzzi, Entropy, volume growth and Lyapunov exponents.unpublished, 1996. 4

    work page 1996

  9. [9]

    Buzzi, S

    J. Buzzi, S. Crovisier and O. Sarig, Measures of maximal entropy for surface diffeomor- phisms.Ann. of Math.,195(2022), 421–508. 3, 6, 7, 8

    work page 2022

  10. [10]

    Buzzi, S

    J. Buzzi, S. Crovisier and O. Sarig, Continuity properties of Lyapunov exponents for surface diffeomorphisms.Invent. Math.,230(2022), 767–849. 13, 17, 39

    work page 2022

  11. [11]

    Buzzi, S

    J. Buzzi, S. Crovisier and O. Sarig, Strong positive recurrence and exponential mixing for diffeomorphisms.arXiv:2501.07455, (2025). 4, 11, 17 54

    work page arXiv 2025

  12. [12]

    Buzzi, C

    J. Buzzi, C. Luo and D. Yang, Continuity properties of ergodic measures of maximal entropy forC r surface diffeomorphisms.arXiv: 2412.19658v3, (2025). 3, 13

    work page arXiv 2025

  13. [13]

    Y. Cao, D. Feng and W . Huang, The thermodynamic formalism for sub-additive poten- tials.Discrete and Continuous Dynamical Systems,20(2008), 639–657. 47

    work page 2008

  14. [14]

    Cogswell, Entropy and volume growth.Ergodic Theory Dynam

    K. Cogswell, Entropy and volume growth.Ergodic Theory Dynam. Systems,20(2000), 77–84. 4, 8, 14

    work page 2000

  15. [15]

    Delplanque, Hyperbolic absolutely continuous invariant measures forC r one- dimensional maps.arXiv:2410.23021, (2024)

    A. Delplanque, Hyperbolic absolutely continuous invariant measures forC r one- dimensional maps.arXiv:2410.23021, (2024). 11

    work page arXiv 2024

  16. [16]

    Delplanque and H

    A. Delplanque and H. Li, Continuity of Lyapunov exponents forC r one-dimensional maps.arXiv:2510.18804, (2025). 11

    work page arXiv 2025

  17. [17]

    X. Guo, G. Liao, W . Sun and D. Yang, On the hybrid control of metric entropy for dominated splittings.Discrete and Continuous Dynamical Systems,38(2018), 5011–

    work page 2018

  18. [18]

    Almost Anosov

    Y. Hu and L.-S. Young, Nonexistence of SBR measures for some diffeomorphisms that are "Almost Anosov".Ergodic Theory Dynam. Systems,15(1995), 67–76. 7

    work page 1995

  19. [19]

    Ledrappier and L.-S

    F . Ledrappier and L.-S. Young, The metric entropy of diffeomorphisms. Part I: Char- acterization of measures satisfying Pesin’ s entropy formula.Ann. of Math.,122(1985), 509–539. 8

    work page 1985

  20. [20]

    Li, Entropy continuity of Lyapunov exponents for non-flat 1-dimensional maps

    H. Li, Entropy continuity of Lyapunov exponents for non-flat 1-dimensional maps. arXiv:2412.19007, 2024. 11

    work page arXiv 2024

  21. [21]

    Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms.Inst

    A. Katok, Lyapunov exponents, entropy and periodic orbits for diffeomorphisms.Inst. Hautes Études Sci. Publ. Math.,51(1980), 137–173. 11, 44

    work page 1980

  22. [22]

    Kifer and S.E

    Y. Kifer and S.E. Newhouse, A global volume lemma and applications.Israel J. Math., 74(1991), 209–223. 2

    work page 1991

  23. [23]

    Klapper and L.-S

    I. Klapper and L.-S. Young, Rigorous bounds on the fast dynamo growth rate involving topological entropy.Comm. Math. Phys.,173(1995), 623–646. 5

    work page 1995

  24. [24]

    Kozlovski, An integral formula for topological entropy ofC ∞ maps.Ergodic Theory Dynam

    O. Kozlovski, An integral formula for topological entropy ofC ∞ maps.Ergodic Theory Dynam. Systems,18(1998), 405–424. 2, 3, 5, 11, 14, 44

    work page 1998

  25. [25]

    Llibre and R

    J. Llibre and R. Saghin, Results and open questions on some invariants measuring the dynamical complexity of a map.Fundamenta Mathematicae,206(2009), 307–327. 10

    work page 2009

  26. [26]

    Luo and D

    C. Luo and D. Young, Upper semi-continuity of metric entropy forC 1,α diffeomor- phisms.arXiv:2504.07746, (2025). 13

    work page arXiv 2025

  27. [27]

    Ma, Estimation of topological entropy in random dynamical systems.J

    X. Ma, Estimation of topological entropy in random dynamical systems.J. Differential Equations,269(2020), 892–911. 5

    work page 2020

  28. [28]

    S. E. Newhouse, Entropy and volume.Ergodic Theory Dynam. Systems,8(1988), 283–

    work page 1988

  29. [29]

    Paternain, Geodesic Flows.Progress in Mathematics,180(1999), Birkhäuser Boston, Inc., Boston

    G.P . Paternain, Geodesic Flows.Progress in Mathematics,180(1999), Birkhäuser Boston, Inc., Boston. 48

    work page 1999

  30. [30]

    Pollicott, Lectures on Ergodic Theory and Pesin Theory on Compact Manifolds

    M. Pollicott, Lectures on Ergodic Theory and Pesin Theory on Compact Manifolds. London Mathematical Society Lecture Note Series, Cambridge University Press, Cam- bridge (1993). 39

    work page 1993

  31. [31]

    Przytycki, An upper estimation for topological entropy of diffeomorphisms.Invent

    F . Przytycki, An upper estimation for topological entropy of diffeomorphisms.Invent. Math.,59(1980), 205–213. 5, 14, 44, 45

    work page 1980

  32. [32]

    Sacksteder and M

    R. Sacksteder and M. Shub, Entropy of a differentiable map.Advances in Math., 28(1978), 181–185. 5

    work page 1978

  33. [33]

    Saghin, Volume growth and entropy forC 1 partially hyperbolic diffeomorphisms

    R. Saghin, Volume growth and entropy forC 1 partially hyperbolic diffeomorphisms. Discrete and Continuous Dynamical Systems,34(2014), 3789–3801. 5

    work page 2014

  34. [34]

    Walters, An introduction to ergodic theory.volume 79 of Graduate Texts in Mathe- matics

    P . Walters, An introduction to ergodic theory.volume 79 of Graduate Texts in Mathe- matics. Springer-Verlag, New York-Berlin, (1982). 10, 23, 44

    work page 1982

  35. [35]

    Yang and Y

    D. Yang and Y. Zang, Volume growth and topological entropy of partially hyperbolic systems.Israel J. Math.,255(2023), 325–347. 2, 5

    work page 2023

  36. [36]

    Yomdin, Volume growth and entropy.Israel J

    Y. Yomdin, Volume growth and entropy.Israel J. Math.,57(1987), 285–300. 3, 5, 11, 12

    work page 1987

  37. [37]

    Zang, Entropies and volume growth of unstable manifolds.Ergodic Theory Dynam

    Y. Zang, Entropies and volume growth of unstable manifolds.Ergodic Theory Dynam. Systems,42(2022), 1576–1590. 4, 8, 14

    work page 2022

  38. [38]

    Zang, Measures of maximal entropy forC ∞ three-dimensional flows

    Y. Zang, Measures of maximal entropy forC ∞ three-dimensional flows. arXiv:2503.21183, (2025). 3 Yuntao Zang Soochow College, Soochow University, Suzhou, 215006, P .R. China E-mail:ytzang@suda.edu.cn 56

    work page arXiv 2025