Lyapunov exponents of polynomials with respect to certain weighted Lyubich's measures

Atma Ram Tiwari; Shrihari Sridharan

REVIEW 4 major objections 5 minor 15 references

Lyapunov exponents of polynomials with respect to certain weighted Lyubich's measures

T0 review · 4 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read For hyperbolic polynomials of any degree, the Lyapunov exponent's second derivatives in real and imaginary coefficient directions are opposite, under every weighted Lyubich measure.

desk verdict The paper computes a degree-d expansion for Lyapunov exponents and proves a sign symmetry only by differentiating a p_j→1 limit, and that limit exchange is false—their own equidistributed case contradicts it. read the letter →

arxiv 1908.07711 v1 pith:C6EW3AHU submitted 2019-08-21 math.DS

Shrihari Sridharan , Atma Ram Tiwari This is my paper · ORCID

classification math.DS MSC 37B2537F1537F10

keywords LyapunovexponentsweightedLyubich'smeasuresJuliasetshyperbolicpolynomialspressurefunctionsecondderivativescoefficientdependencecomplexdynamics

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 sets out to prove that the Lyapunov exponent of a monic, centred, hyperbolic polynomial—computed on its Julia set with respect to a weighted Lyubich's measure—has a rigid second-order symmetry: perturbing any coefficient's real part bends the exponent exactly as much as perturbing its imaginary part bends it in the opposite direction. The setting is $P(z)=z^d+\sum_{r=0}^{d-2}(\alpha_r+i\beta_r)z^r$ with $\alpha_r^2+\beta_r^2<1$ and bounded critical orbit, and the weighted Lyubich measure assigns probabilities $p_1,\dots,p_d$ to the $d$ preimage branches of a generic point. By expanding the topological conjugacy between $P$ and $z^d$ in powers of the coefficients and evaluating the resulting integrals in the limit $p_j\uparrow 1$, the paper obtains an explicit second-order expansion whose cross-derivatives satisfy $\partial^2\Lambda/\partial\alpha_r\partial\alpha_s=-\partial^2\Lambda/\partial\beta_r\partial\beta_s$. If correct, this quantifies a well-behaved pattern inside a quantity that usually measures instability, and it extends a phenomenon previously seen only for quadratic and cubic polynomials to all degrees.

What carries the argument

The load-bearing object is the topological conjugacy $\Phi_P\colon S^1\to J_P$ between the map $Q(z)=z^d$ on the unit circle and the hyperbolic polynomial $P$, defined by $P\circ\Phi_P=\Phi_P\circ Q$. Because $P$ is hyperbolic and all its coefficients have modulus below 1, this conjugacy depends analytically on the coefficients, and the paper expands it as $$\Phi_P(z)=z+\sum_{\xi_{d-2}+\cdots+\xi_0\ge1}\varphi_{(\xi_{d-2},\dots,\xi_0)}(z)A_{d-2}^{\xi_{d-2}}\cdots $A_0^{{\xi_0}}$,$$ then solves for the coefficient functions $\varphi_r,\varphi_{r^2},\varphi_{rs}$ by matching powers in the defining equation. The Lyapunov exponent is re-expressed through the pressure identity as $-\log d-\int_{S^1}\log|\Phi_P(z)|\,d\nu$, so the whole computation reduces to integrals of these coefficient functions against the weighted Lyubich measure; those integrals are evaluated in the limit $p_j\uparrow 1$, yielding the second-order expansion (5.3) that carries the theorem.

What would settle it

Compute the Lyapunov exponent numerically for a concrete cubic, say $z^3+(\alpha_1+i\beta_1)z+(\alpha_0+i\beta_0)$ with small real coefficients, using the weighted Lyubich measure approximated by $\sum_{\eta}p_{\eta_1}\cdots p_{\eta_n}\delta_\omega$ over the $d^n$ preimages of a generic point, with weights such as $(1/2,1/4,1/4)$ rather than any weight near 1. Estimate $\partial^2\Lambda/\partial\alpha_1^2$ and $\partial^2\Lambda/\partial\beta_1^2$ by finite differences and check whether they equal $+1/4$ and $-1/4$, with the sum exactly zero; if the sum is nonzero or depends on the chosen weights, the theorem's 'irrespective of the probability vector' clause fails.

Watch

Extended reading notes

Core claim

The central claim is an identity of mixed partial derivatives. For the polynomial $P$ above and the weighted Lyubich measure $\mu_{\vec p}$ built from any strictly positive probability vector $\vec p$, the Lyapunov exponent $\Lambda_{\mu_{\vec p}}(P)=-\int_{J_P}\log|P'|\,d\mu_{\vec p}$ satisfies $$\frac{\$partial^{2}$\Lambda_{\mu_{\vec p}}}{\partial\alpha_r\partial\alpha_s}=-\frac{\$partial^{2}$\Lambda_{\mu_{\vec p}}}{\partial\beta_r\partial\beta_s}$$ for all $0\le r,s\le d-2$. The proof computes, as one weight $p_j$ approaches 1, the expansion $$\Lambda\to -\log d+\sum_{r=0}^{d-2}\frac{\alpha_r}{d-1}+\sum_{r=0}^{d-2}\frac{d-2r+1}{2(d-1)^2}(\$alpha_r^{2}$-\$beta_r^{2}$)+\sum_{0\le r<s\le d-2}\frac{d-r-s+1}{(d-1)^2}(\alpha_r\alpha_s-\beta_r\beta_s),$$ and the identity is read off term by term. The paper also shows that using only real coefficients leaves every first and second $\alpha$-derivative unchanged, so complex coefficients enter the quadratic part solely through the $\beta^2$ and $\beta_r\beta_s$ terms, with signs opposite to their $\alpha$ counterparts.

Load-bearing premise

The proof assumes that the second-order expansion computed in the limit $p_j\to1$ can be differentiated to give the exact second derivatives of the Lyapunov exponent for every strictly positive probability vector and for every coefficient in the domain $|A_r|^2<1$, meaning the discarded higher-order coefficient terms in (5.1) do not contribute to those derivatives.

Editorial extensions

If this is right

For a quadratic polynomial $z^2+\alpha+i\beta$, the expansion reduces to $-\log 2+\alpha+\frac32(\alpha^2-\beta^2)$, so the second $\alpha$-derivative is $3$ and the second $\beta$-derivative is $-3$.
For a cubic polynomial $z^3+(\alpha_1+i\beta_1)z+(\alpha_0+i\beta_0)$, the identity holds simultaneously for both coefficients, with diagonal contributions $\frac12(\alpha_0^2-\beta_0^2)$ and $\frac14(\alpha_1^2-\beta_1^2)$ and a cross contribution $\frac34(\alpha_1\alpha_0-\beta_1\beta_0)$.
The statement is made for every strictly positive probability vector, so the sign-flip identity is presented as a feature of the whole family of weighted Lyubich measures, not of a special choice of weights.
Real–complex agreement of first and second $\alpha$-derivatives means that, up to second order, the imaginary part of each coefficient affects the Lyapunov exponent only through terms of the opposite sign.

Reading between the lines

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

If the second-order expansion is exact, the Hessian of $\Lambda$ on the $2(d-1)$-dimensional real coefficient space has block form with real block $H$ and imaginary block $-H$, so the quadratic part is the real part of a holomorphic quadratic form $\sum_{r\le s}c_{rs}A_rA_s$; the paper does not state this geometric reading explicitly.
A numerical finite-difference check of the second derivatives for weights far from the vertex (for example uniform weights) would test whether the 'every strictly positive probability vector' clause holds beyond the $p_j\uparrow 1$ limit in which the expansion is derived.
The same coefficient-expansion machinery could be applied to third derivatives; if the sign-flip pattern persists in the next order, the symmetry may reflect a general real–imaginary structure of the Lyapunov exponent as a function of holomorphic coefficients.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Quick take: the paper extends the authors' earlier quadratic and cubic computation to arbitrary degree d. The explicit coefficient functions φr, φr2, φrs and the second-order expansion for the Lyapunov exponent are real computations, and the sign symmetry ∂²Λ/∂α_r∂α_s = −∂²Λ/∂β_r∂β_s is a clean observation. If the goal was just to document that computation, this is mostly fine.

The problem is the theorem as stated. Section 5 explicitly evaluates the integrals only "when one of the p_j ↑ 1" and gives formulas (5.2)–(5.3) with an arrow. Section 6 differentiates those limiting formulas and asserts the second derivatives for every strictly positive probability vector p. That inference needs uniformity in p and control of the O(≥3) remainder; neither is supplied. The weighted measure itself depends on p, so there is no reason the coefficient of |A|² in a p-dependent expansion can be read off from the p→degenerate limit.

And it is not just a missing technical hypothesis—the paper's own statements show the claim is false as written. For d=2 and p=(1/2,1/2), the weighted Lyubich measure is the equidistributed Lyubich measure. Section 5 notes that every integral in (5.1) vanishes for that measure, so Λ = −log2 + O(|A|³) and ∂²Λ/∂α² at A=0 is 0. Differentiating (5.2) gives 3/2. The same issue appears for cubic.

The integral evaluations in Section 5 are also listed without derivation, and the O(≥3) terms in (5.1) are never controlled, so even in the limiting case the expansion is not rigorously established beyond formal power series. That said, the coefficient functions from the conjugacy equation are derived cleanly and appear correct as formal expansions.

Bottom line: as a research claim, Theorem 1.1 is not supported; the p-independence in particular is contradicted by the equidistributed case. There may be a salvageable restricted result—for example, a version holding only in the p_j→1 limit, or with a careful statement about which derivatives survive. The computation itself could be useful to someone who wants explicit expansions for Lyapunov exponents of polynomials.

For peer review: I would send it out—the computation is nontrivial and the flaw is the kind a referee could catch and fix. But I would not cite the theorem in its present form. My recommendation: engage with the computation, not the theorem; expect major revision.

Referee Report

4 major / 5 minor

Summary. The manuscript studies a monic, centred hyperbolic polynomial P(z)=z^d+Σ_{r=0}^{d-2}(α_r+iβ_r)z^r with |α_r+iβ_r|<1 and considers its Lyapunov exponent Λ_{μ_p}(P) = -∫_{J_P} log|P'| dμ_p, where μ_p is the weighted Lyubich measure associated with a strictly positive probability vector p=(p_1,...,p_d). Using the analytic conjugacy Φ between z^d and P, the paper derives an expansion of Λ_{μ_p} in the coefficients A_r and states two theorems: Theorem 1.1 claims that ∂²Λ/∂α_r∂α_s = -∂²Λ/∂β_r∂β_s for every r,s and every p, and Theorem 1.2 compares first and second derivatives for real versus complex coefficient polynomials. The proofs in Section 6 differentiate the limiting expansions (5.2) and (5.3), which are obtained only as p_j↑1, and apply the resulting identities to all p.

Significance. If correct, the p-independence of the second-order mixed derivatives in Theorem 1.1 would be a clean and checkable rigidity statement for weighted equilibrium measures on polynomial Julia sets, and it would naturally extend the d=2,3 observations of the authors' earlier paper [12]. The computation is self-contained from the conjugacy equation (2.2) and contains no fitted parameters, so the claim is in principle falsifiable. The obstacle is that the proof as written derives exact coefficient identities from a one-sided limit p_j↑1 and never controls the remainder in the expansion; because μ_p itself depends on p, the passage from the limiting expansion to fixed-p second derivatives is the crux and is not justified.

major comments (4)

[Section 5, Eqs. (5.2)-(5.3); Section 6] Equations (5.2) and (5.3) are explicitly obtained only 'as p_j ↑ 1', yet the proof of Theorems 1.1 and 1.2 differentiates these expressions and concludes identities for every strictly positive probability vector. Since the measure μ_p depends on p, the second derivatives at a fixed p are not obtained by differentiating the p_j→1 limit unless one proves that the limit is uniform in the coefficients and commutes with differentiation in A_r. No such uniformity or interchange argument is given. This is the central gap of the paper.
[Section 5, Eq. (5.1)] The proof never controls the remainder O(Σξ_r≥3). Theorem 1.1 concerns exact second derivatives on the full domain |A_r|²<1, so the remainder must be shown to be twice differentiable with second derivatives converging to 0 as A→0. The functions φ_r, φ_{r²}, φ_{rs} in (4.3)-(4.5) are infinite series, and the paper does not justify their convergence or their termwise differentiability; therefore the 'direct differentiation' in Section 6 is not a valid derivation.
[Section 5, paragraph after Eq. (5.1)] The paper states that every integral in (5.1) vanishes when the measure is the equidistributed Lyubich measure. For d=2 and p=(1/2,1/2), the weighted Lyubich measure coincides with the equidistributed measure, so the expansion gives Λ_{μ_p}(P)=-log 2+O(|A_0|^3) at A_0=0. The second-order coefficient obtained by differentiating (5.2) is 3/2 in that case. This does not by itself disprove the identity in Theorem 1.1, but it shows that the p-independent coefficient values used in the proof are not available at p=(1/2,1/2), and it makes the p-dependence of the coefficients concrete rather than a mere formal obstruction.
[Section 5, displayed limiting values] The limiting values of the integrals are asserted without derivation, for both P_R and P_C. These limits are the only quantitative input that produces (5.2) and (5.3), and they involve the complicated infinite series φ_r, φ_{r²}, and φ_{rs}. No calculation, no interchange of sums and integrals, and no convergence argument is presented, so the central computation cannot be checked from the manuscript.

minor comments (5)

[Section 6, displayed first derivative] The sum over s≠r has coefficient (d−r−s)/(d−1)^2, but differentiating the cross term in (5.2) gives (d−r−s+1)/(d−1)^2; the displayed formula should be corrected.
[Equation (5.1)] The summation notation '∑_{r=0}^{d−3}∑_{r<s=1}^{d−2}' is malformed; this should be written with an explicit sum over 0≤r<s≤d−2.
[Theorem 1.1] The theorem uses the notation Λ_μ and Λ_{μ_p} inconsistently; the measure should be named once and the p-dependence made explicit throughout.
[Abstract and Introduction] The text contains numerous typographical errors (for example, 'ponit s', 'senstivity', 'exponen ts', 'co eﬃcient'); the manuscript needs a careful proofreading pass.
[Equation (3.4)] The integrand uses Φ(z) although the conjugacy was denoted Φ_P; the notation should specify which conjugacy is integrated and with respect to which variable.

Circularity Check

0 steps flagged · score 0.0 of 10

Self-contained conjugacy computation; no circular reduction found.

full rationale

I find no circular step in the paper's derivation. The coefficient functions φr, φr2, and φrs are obtained by solving the conjugacy equation (2.2)/(4.1), not by assuming the target Lyapunov identities. The expansion (5.1) is written with an explicit remainder, and the limiting formulas (5.2) and (5.3) are computed from those coefficient functions together with the stated integral limits for the weighted Lyubich measure. Theorem 1.1 is then obtained by differentiating (5.3), and the constants that appear are computed quantities rather than fitted parameters. The target identity ∂²Λ/∂αr∂αs = −∂²Λ/∂βr∂βs is not assumed anywhere in the derivation of the expansion. The authors' earlier paper [12] is cited as motivation and for an interpretation of analytic dependence of the conjugacy, but that analyticity is itself attributed to the external sources [15,8,4], so the self-citation is not load-bearing. The skeptical concern that passing from the limit pj → 1 to every fixed p and differentiating through the remainder may require uniform estimates is a rigor/correctness gap, not a reduction of the conclusion to its inputs. Since no equation is reused as its own conclusion and no fitted input is relabeled as a prediction, the appropriate circularity score is 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The derivation relies on standard hyperbolicity and conjugacy results, on an unproved convergence statement for the weighted measures, and on an unstated truncation assumption. There are no fitted parameters and no new physical entities.

assumptions (4)

domain assumption P is hyperbolic with bounded critical orbit, and the conjugacy Φ_P depends analytically on the coefficients A_r.
Section 2 invokes structural stability (Lyubich [8]) and analytic dependence (theorems 4.1-4.2 of [12]) to justify the power series in coefficients.
domain assumption The sequence of weighted preimage measures (3.2) converges in the weak-* topology to a P-invariant measure μ_p independent of the base point ζ.
Section 3 asserts convergence without proof or citation; the paper needs this to define the integrals it evaluates.
ad hoc to paper The interchange of the logarithm expansion and integration in (5.1) is valid, and the O(Σξ_r ≥ 3) remainder may be neglected when differentiating second derivatives over the whole parameter domain.
This truncation is the load-bearing step in the proof of Theorem 1.1; no error bound is given, and the theorem as stated covers all admissible coefficients, not just a neighborhood where remainder derivatives vanish.
ad hoc to paper The listed limits of the six integrals in Section 5 are valid uniformly and are branch-independent.
Section 5 states these limits with no computation; the main formulas (5.2) and (5.3) depend on them.

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Pith. "Pith review of Lyapunov exponents of polynomials with respect to certain weighted Lyubich's measures." pith.science (2026). https://pith.science/paper/C6EW3AHU

@misc{pith2026190807711,
  author       = {Pith},
  title        = {Pith review of: Lyapunov exponents of polynomials with respect to certain weighted Lyubich's measures},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/C6EW3AHU}},
  note         = {Machine review of arXiv:1908.07711}
}

abstract

In this paper, we consider a monic, centred, hyperbolic polynomial of degree $d \ge 2$, restricted on its Julia set and compute its Lyapunov exponents with respect to certain weighted Lyubich's measures. In particular, we show a certain well-behavedness of some coefficients of the Lyapunov exponents, that quantifies the non-well-behavedness in a system.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [12]

The dependence of Lyapunov exponents of poly- nomials on their coeﬃcients

Sridharan, S. and Tiwari, A.,R., “The dependence of Lyapunov exponents of poly- nomials on their coeﬃcients”, J. Comput. Dyn. , 6, (2019), 95 - 109

work page 2019

[2] [1]

Inﬂu- ence of parametric perturbations on Lyapunov exponents of disc rete linear time-varying systems

Barabanov, E. , Czornik, A. , Niezabitowski, M. and V aidzelevich, A., “Inﬂu- ence of parametric perturbations on Lyapunov exponents of disc rete linear time-varying systems”, Systems Control Lett. , 122, (2018), 54 - 59

work page 2018

[3] [2]

Beardon, A. F. , Iteration of rational functions: Complex analytic dynamic al systems, Graduate Texts in Mathematics, 132, Springer-Verlag, New York, (1991)

work page 1991

[4] [3]

Kolmogorov entropy and nu- merical experiments

Benettin, G. , Galgani, L. and Strelcyn, J.-M. , “Kolmogorov entropy and nu- merical experiments” (1976), Physical Review A , 14 (1976), 2338 - 2345

work page 1976

[5] [4]

and Gamelin, T.W

Carleson, L. and Gamelin, T.W. , Complex Dynamics , Universitext: Tracts in Mathematics, Springer-Verlag, New York, 1993

work page 1993

[6] [5]

A link between topological entropy and Lyapunov exponents

Catalan, T., “A link between topological entropy and Lyapunov exponents”, Ergod. Th. Dynam. Sys. , 39, (2019), 620 - 637

work page 2019

[7] [6]

Central limit asymptotics for shifts of ﬁnite type

Coelho, Z. and Parry, W., “Central limit asymptotics for shifts of ﬁnite type”, Israel J. Math. , 69, (1990), 235 - 249

work page 1990

[8] [7]

Lyapunov exponents, path-integrals and forms

Gozzi, E. and Reuter, M. , “Lyapunov exponents, path-integrals and forms”, Chaos, solitons and fractals , 4, (1994), 1117 - 1139

work page 1994