arXiv:2608.10655v1

Absolutely continuous invariant measures and the Collet-Eckmann condition for long-branched Sturmian unimodal maps

Jorge Olivares-Vinales

math.DS37E0537A0537D2537B10

Paper metadata

Original submission: August 11, 2026
Claimed MSC: 37E05, 37A05, 37D25, 37B10
arXiv categories: math.DS
Paper license: arXiv non-exclusive distribution license ↗
Comments: 29 pages

Abstract

We study long-branched unimodal maps for which the sequence of values of the kneading map form a Sturmian word. We identify these kneading classes with those arising from the stunted Lorenz construction of Anušić, Bruin, and Činč, and their quadratic representatives with the maps constructed by Blé from rigid rotations. We compute the cutting and co-cutting times explicitly and establish criteria for the existence and nonexistence of absolutely continuous invariant probability measures. We also show that, for Lebesgue-almost every rotation angle, no finite-order $S$ -unimodal realization satisfies the Collet-Eckmann condition.

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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.

Current report

Detailed mathematical audit

Generated August 18, 2026

01Statements4 reported findingsCorrect

The Sturmian kneading characterization, the arithmetic sufficient conditions for existence and nonexistence of an absolutely continuous invariant probability measure, and the universal failure of the Collet–Eckmann condition are correct. One notation typo in the admissibility proof is reported in Part 2 and does not affect any conclusion.

Theorem 1.1Correct

Characterization by the lower mechanical word

Pages 2 and 9–10 · Theorem 1.1 and its proof · arXiv:2608.10655v1

When $Q(k)\leq1$ , admissibility and $Q(1)=0$ imply that the binary word $Q(1)Q(2)\cdots$ is lexicographically no larger than each of its shifts. Lemma 2.8 then shows that a Sturmian word with this property is the unique lower mechanical word of its slope, so $Q(k)=\lfloor k\omega\rfloor-\lfloor(k-1)\omega\rfloor$ . Conversely, that lower mechanical word is Sturmian and Proposition 3.1 verifies admissibility. Summing the kneading recursion gives $S_k=1+k+\lfloor k\omega\rfloor$ .

Theorem 1.2Correct

Existence of an absolutely continuous invariant probability measure

Pages 3 and 16–19 · Theorem 1.2 and Section 4 · arXiv:2608.10655v1

The fixed-order uniform decay-of-geometry estimate supplies $\mu_{m(k)}\leq\exp(-b_\ell k)$ . Lemma 4.2 controls each central pullback by $\mu_{n+1}\leq K_\ell\mu_n^{1/\ell}$ , and Lemma 4.1 bounds the cascade length $L_k$ by the appropriate odd continued-fraction coefficient. The hypothesis $\limsup_n\ell^{a_{2n+1}}/n<\eta_\ell$ therefore makes $M_\ell\exp(-b_\ell k/\ell^{L_k})$ smaller than the universal threshold for all large $k$ . Lemma 4.3 propagates this through every cascade, and the Bruin–Shen–van Strien criterion then yields the claimed invariant probability measure.

Full paper, version 1 ↗

Theorem 1.3Correct

Arithmetic sufficient condition for nonexistence of an acip

Pages 3 and 20–22 · Theorem 1.3 and Section 5 · arXiv:2608.10655v1

The co-cutting formula $R(q_{2n})=q_{2n+1}$ places the interval from $c$ to $f^{q_{2n}}(c)$ inside the relevant kneading neighborhood. Since consecutive block indices differ by $a_{2n+1}(q_{2n}-p_{2n})$ , divergence of the resulting weighted recurrence series forces divergence of the total lengths of the kneading neighborhoods, which is equivalent to nonexistence of an acip. The Nowicki–Przytycki trajectory-recurrence estimate supplies $|f^m(c)-c|\geq C_f\Lambda_f^{-m}$ , giving exactly the sufficient series in the theorem.

Theorem 1.4Correct

Failure of the Collet–Eckmann condition

Pages 3 and 24–28 · Theorem 1.4 and Section 6 · arXiv:2608.10655v1

If a realization were Collet–Eckmann, the interval joining $c$ to $f^m(c)$ would be a monotonicity interval for $f^{R(m)}$ . Exponential shrinking and logarithmic recurrence would then bound the Cesaro means of the first-disagreement times $R(m)$ . For the long-branched Sturmian kneading sequence, mechanical-word occurrences have lower frequencies bounded by the one-sided closest-return gaps $u_n(\omega)$ , and $\sum_nu_n(\omega)=\infty$ . Proposition 6.7 consequently forces the same Cesaro means to diverge, which is the required contradiction.

02Proofs5 reported findingsCorrect

The proofs of all four central theorems and their material combinatorial, geometric, inducing, and recurrence inputs are correct and complete. A subscript in Proposition 3.1 is a uniquely repairable notation typo.

Theorem 1.1 and Theorem 3.8Correct and complete

Kneading characterization and co-cutting formulas

Pages 8–13 · Lemma 2.8, Propositions 3.1–3.2, and Theorem 3.8 · arXiv:2608.10655v1

Balance and lexicographic minimality force the prefix of length $n$ to contain $\lfloor n\omega\rfloor$ symbols $1$ , proving the mechanical formula. The cutting-time recursion then telescopes. For co-cutting times, the lower intermediate convergents are precisely the successive one-sided record returns of the rotation; inserting their denominators into the continued-fraction recursions gives $S_{q_{2m+1}-p_{2m+1}}=q_{2m+1}$ and $R(q_{2m})=q_{2m+1}$ .

Proof of Theorem 1.2Correct and complete

Uniform geometry and central-cascade induction

Pages 16–19 · Section 4 · arXiv:2608.10655v1

The distortion estimate is transported through the nonflat normal form with constants depending only on $\ell$ , yielding the central-step recursion. The induction in Lemma 4.3 keeps every scaling factor below the required threshold through a cascade. The two cases $a_1=2$ and $a_1>2$ use respectively $a_{2k+1}$ and the shifted sequence $a_{2k-1}$ ; the latter has the same limiting upper bound after reindexing. Thus the stated arithmetic hypothesis covers every sufficiently deep level.

Proof of Theorem 1.3Correct and complete

Inducing summability and quantitative recurrence

Pages 20–22 · Section 5 · arXiv:2608.10655v1

Summation by parts proves that finiteness of the long-branched inducing lift is equivalent to $\sum_k|J_k|<\infty$ . The nested-neighborhood estimate and the exact co-cutting return times give the lower series in Proposition 5.4. Nonflatness supplies the global derivative bound required for the cited trajectory-recurrence theorem, so Lemma 5.6 converts this lower series into the one printed in Theorem 1.3.

Proof of Theorem 1.4Correct and complete

Contradictory mean first-disagreement bounds

Pages 24–28 · Section 6 · arXiv:2608.10655v1

Lemma 6.1 verifies monotonicity up to the first itinerary disagreement. The cited Collet–Eckmann shrinking and recurrence estimates then give a uniform upper bound for $N^{-1}\sum_{m\leq N}R(m)$ . On the combinatorial side, unique ergodicity gives the stated lower occurrence frequencies, the record-return argument proves $\sum_nu_n(\omega)=\infty$ , and the distinct shifts $m_j<2j$ transfer those occurrences to a divergent lower bound for the same Cesaro means. The contradiction covers every irrational angle and finite critical order.

Proposition 3.1Typo

The displayed order relation has the wrong subscript

Page 10 · final display in the proof of Proposition 3.1 · arXiv:2608.10655v1

Printed: $\bigl(Q_\gamma(m+j)\bigr)_{j\geq1}\succeq_{\mathrm{pl}}\bigl(Q_\gamma(j)\bigr)_{j\geq1}$ . Correction: replace the subscript $\mathrm{pl}$ by $\mathrm{lex}$ . The paragraph proves a lexicographic comparison, Theorem 2.1 is stated with $\succeq_{\mathrm{lex}}$ , and the proof of Theorem 1.1 uses the corrected notation. The intended correction is unique and changes no inference.

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Paper: arXiv:2608.10655v1
Authors listed: Jorge Olivares-Vinales
Audit date: August 18, 2026

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Absolutely continuous invariant measures and the Collet-Eckmann condition for long-branched Sturmian unimodal maps

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Characterization by the lower mechanical word

Existence of an absolutely continuous invariant probability measure

Arithmetic sufficient condition for nonexistence of an acip

Failure of the Collet–Eckmann condition

Kneading characterization and co-cutting formulas

Uniform geometry and central-cascade induction

Inducing summability and quantitative recurrence

Contradictory mean first-disagreement bounds

The displayed order relation has the wrong subscript

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