arXiv:2607.04187v3

Existence of the Lyapunov exponent for $S$ -unimodal maps

Yuya Arima

math.DS37A0537D2537E05

Paper metadata

Original submission: July 5, 2026
Latest revision: July 26, 2026 · version 3
Claimed MSC: 37A05, 37D25, 37E05
arXiv categories: math.DS
Paper license: arXiv non-exclusive distribution license ↗
Comments: 26 pages

Abstract

In this paper, we show that for any $S$ -unimodal map $T$ on $[0,1]$ with a non-flat critical point the Lyapunov exponent exists for Lebesgue almost every point and is equal to a constant $λ_T\in\mathbb{R}$ . Moreover, $λ_T=0$ if and only if $T$ admits neither an absolutely continuous $T$ -invariant probability measure with positive entropy nor a strictly stable periodic orbit. Consequently, if an $S$ -unimodal map with a non-flat critical point is infinitely renormalizable or non-statistical then for Lebesgue almost every $x\in [0,1]$ the Lyapunov exponent along the orbit of $x$ exists and is equal to $0$ . A key ingredient is the following result of independent interest. If an $S$ -unimodal map with a non-flat critical point has no periodic attractor then for Lebesgue almost every $x\in [0,1]$ the lower Lyapunov exponent along the orbit of $x$ is non-negative. This shows that, in the absence of periodic attractors, exponential contraction cannot occur along the orbit of Lebesgue almost every point.

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Audited against arXiv v3

Not a correctness certificate. A “Correct” result may include yellow typos or minor formal corrections that do not affect substantive soundness. It means this audit found no unresolved substantive error under the stated criteria; it does not replace expert scrutiny or formal verification.

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Generated August 18, 2026

01Statements3 reported findingsCorrect

Both main theorems are supported. Theorem 1.1 proves existence of a common almost-everywhere Lyapunov exponent for every non-flat S-unimodal map and characterizes its positive and negative signs. Theorem 1.2 constructs uncountably many logistic parameters with non-statistical empirical behavior while the critical orbit has zero lower exponent and strictly larger subsequential derivative growth.

Theorem 1.1Correct

Almost-everywhere Lyapunov exponent and its sign classification

Pages 2–3 · Theorem 1.1 · arXiv:2607.04187v3

Every S-unimodal map with a non-flat critical point has a constant $\lambda_T$ such that the full Lyapunov exponent exists and equals $\lambda_T$ at Lebesgue-almost every point. The theorem correctly characterizes $\lambda_T>0$ by an absolutely continuous invariant probability of positive entropy and $\lambda_T<0$ by a strictly stable periodic orbit. In the remaining maps, including infinitely renormalizable and conservative cases without either object, the exponent exists and is zero. The non-flatness hypothesis is retained in the recurrence and integrability steps.

Theorem 1.2(1)–(2)Correct

Uncountable family with nonconvergent empirical measures

Pages 3–4 and 30–36 · Theorem 1.2(1)–(2) · arXiv:2607.04187v3

The nested kneading construction yields uncountably many distinct logistic parameters. For each, the empirical measures of the critical value, Lebesgue measure, and Lebesgue-almost every orbit have the same accumulation set, while alternating blocks force that set to contain at least two measures. The parameter intervals shrink and have disjoint binary choices at infinitely many stages, which proves uncountability rather than merely existence.

Theorem 1.2(3)Correct

Subsequential derivative growth above the lower exponent

Pages 4 and 33–36 · Theorem 1.2(3) · arXiv:2607.04187v3

Selected return times $n_k$ end after the expanding blocks of the kneading itinerary. Distortion estimates give convergence of $n_k^{-1}\log |(f_a^{n_k})'(f_a(c))|$ to the integral for an invariant accumulation measure, and that limit is positive relative to the zero lower exponent. Intervening neutral blocks make the overall liminf zero. The inequalities in the statement therefore follow from two compatible subsequences of the same orbit.

02Proofs3 reported findingsCorrect

Proposition 5.1 and the proofs of Theorems 1.1–1.2. The classification divides the map into an attracting-cycle basin, an absolutely continuous invariant-measure case, and the residual conservative or renormalizable alternatives. Proposition 5.1 controls visits of typical orbits to the non-flat critical point, making the negative tail of $\log|T'|$ uniformly negligible. Birkhoff's theorem applies to bounded truncations, and the tail estimate lets their limits pass to the full logarithm. The cited interval-map classification then identifies the negative and positive cases; in the residual case both signs are excluded and the limit is zero. This proves existence as well as the sign alternatives.

Slow recurrence and parameter constructionCorrect and complete

Proposition 5.1 and the proofs of Theorems 1.1–1.2

Pages 20–36 · Sections 5–6 · arXiv:2607.04187v3

The classification divides the map into an attracting-cycle basin, an absolutely continuous invariant-measure case, and the residual conservative or renormalizable alternatives. Proposition 5.1 controls visits of typical orbits to the non-flat critical point, making the negative tail of $\log|T'|$ uniformly negligible. Birkhoff's theorem applies to bounded truncations, and the tail estimate lets their limits pass to the full logarithm. The cited interval-map classification then identifies the negative and positive cases; in the residual case both signs are excluded and the limit is zero. This proves existence as well as the sign alternatives.

Proposition 5.1Correct and complete

Critical-recurrence control for the unbounded logarithm

Pages 20–29 · Proposition 5.1 · arXiv:2607.04187v3

Non-flatness compares $-\log|T'|$ near the critical point with minus the logarithm of the distance to that point. Pullback estimates and the interval nesting bound the measure of points making exceptionally deep returns, and the resulting series is summable. Borel–Cantelli makes those returns negligible for almost every orbit. This is exactly the uniform-integrability input needed to pass from truncated Birkhoff averages to the true Lyapunov exponent.

Section 6Correct and complete

Kneading realization and parameter limit

Pages 29–36 · Section 6 · arXiv:2607.04187v3

Finite kneading prefixes define nested logistic-parameter intervals, and the monotonicity theorem realizes the infinite itinerary at their unique intersection points. Alternating choices create a Cantor family. Hofbauer–Keller estimates transfer the prescribed symbolic block frequencies to empirical measures of typical points, while bounded distortion controls derivatives on the expanding blocks. Diagonal selection makes every announced limit hold for the final parameter, not merely along approximating maps.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2607.04187v3
Authors listed: Yuya Arima
Audit date: August 18, 2026

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Existence of the Lyapunov exponent for SSS-unimodal maps

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Almost-everywhere Lyapunov exponent and its sign classification

Uncountable family with nonconvergent empirical measures

Subsequential derivative growth above the lower exponent

Proposition 5.1 and the proofs of Theorems 1.1–1.2

Critical-recurrence control for the unbounded logarithm

Kneading realization and parameter limit

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Existence of the Lyapunov exponent for $S$ -unimodal maps