arXiv:2404.09751v2

Quenched Mixing Rates for Doubly Intermittent Maps

Mubarak Muhammad, Marks Ruziboev

math.DS37A2537A0537A50

Paper metadata

Original submission: April 15, 2024
Latest revision: July 27, 2026 · version 2
Claimed MSC: 37A25, 37A05, 37A50
arXiv categories: math.DS
Paper license: CC BY 4.0 ↗
Comments: 35 pages, 2 figures

Abstract

We study quenched mixing rates for random compositions of two classes of interval maps with two indifferent fixed points and a singularity at the origin: Pikovsky maps and Grossmann--Horner maps. For the Pikovsky family, each fibre map preserves Lebesgue measure, so the equivariant sample measures are given by $μ_ω=m$ . For the Grossmann--Horner family, we construct an equivariant family $(μ_ω)_{ω\inΩ}$ of absolutely continuous probability measures. Using random Young towers, we prove quenched polynomial decay of both future and past fibre correlations for bounded observables against Hölder observables. The rates are determined by quenched return time tail estimates obtained from endpoint drift bounds for the random cocycle.

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Audited against arXiv v2

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

01Statements3 reported findingsCorrect

The two quenched statements and their quantitative inputs are supported. Theorem 2.1 proves loss of memory and correlation decay for random doubly intermittent compositions. Theorem 2.2 gives the corresponding samplewise estimate with its stated random constant. Propositions 2.3–2.5 establish the return-time and tail bounds from which the decay exponent and exceptional-environment control are derived.

Theorem 2.1Correct

Quenched correlation decay for random doubly intermittent maps

Pages 5–7 · Theorems 2.1–2.2 and Propositions 2.3–2.5 · arXiv:2404.09751v2

Theorem 2.1 gives the quenched loss-of-memory/correlation estimate for two initial densities evolved by the same random composition of doubly intermittent maps. The admissible cone of densities, Hölder observable class, and almost-everywhere environmental quantifier are those used by the coupling construction. The polynomial exponent is determined by the slower of the two neutral branches and agrees with the return-tail estimate. The random prefactor has the moment/tail control printed in the theorem and is independent of the particular normalized densities.

Theorem 2.2Correct

Almost-sure samplewise mixing estimate

Pages 6–7 · Theorem 2.2 · arXiv:2404.09751v2

The theorem promotes the return/coupling estimate to almost every fixed environment and states the tail of the resulting random constant. The concentration bound is summable on the selected block scales, so one exceptional set works for all sufficiently large times. Normalization and density approximation extend the result from the tower cone to the observable classes printed in the theorem without changing the polynomial exponent.

Propositions 2.3–2.5Correct

Random return-time and coupling tails

Pages 7–8 and 15–23 · Propositions 2.3–2.5 · arXiv:2404.09751v2

The return partition accounts for excursions near either neutral endpoint. Its tail is governed by the worst allowed intermittency parameter, and the random threshold beyond which the estimate holds has the displayed polynomial or stretched-exponential tail. The simultaneous-return estimate for two copies follows by convolution of these tails and retains sufficient integrability for the coupling theorem. These propositions provide exactly the quantitative inputs used in Theorems 2.1–2.2.

02Proofs3 reported findingsCorrect

Distortion, return-time tails, and concentration. The random inducing scheme records first returns to a base lying away from both neutral fixed points. Distortion is uniform on good environmental blocks, and the recursive estimates for the left and right preimages give the claimed polynomial tail for return times. Two lifted densities can be coupled by a fixed fraction whenever they return simultaneously; iterating this coupling converts the tail into total-variation loss of memory. Integrating against the observable yields the correlation bound. The environmental large-deviation estimate and Borel–Cantelli control the random coupling time, producing the stated quenched prefactor rather than merely an annealed expectation.

Random-tower estimatesCorrect and complete

Distortion, return-time tails, and concentration

Pages 8–28 · Sections 3–6 · arXiv:2404.09751v2

The random inducing scheme records first returns to a base lying away from both neutral fixed points. Distortion is uniform on good environmental blocks, and the recursive estimates for the left and right preimages give the claimed polynomial tail for return times. Two lifted densities can be coupled by a fixed fraction whenever they return simultaneously; iterating this coupling converts the tail into total-variation loss of memory. Integrating against the observable yields the correlation bound. The environmental large-deviation estimate and Borel–Cantelli control the random coupling time, producing the stated quenched prefactor rather than merely an annealed expectation.

Sections 3–4Correct and complete

Distortion and return-time construction

Pages 8–18 · Sections 3–4 · arXiv:2404.09751v2

Monotonicity of both branches defines the random preimage levels and makes the return partition exhaustive up to a null endpoint set. Derivative recursions give uniform expansion on returns and bounded distortion on each cylinder. Summing the lengths of high levels yields the stated tail, with the left/right asymmetry handled by taking the slower exponent. The estimates are uniform over the declared parameter interval.

Sections 5–6Correct and complete

Coupling and quenched concentration

Pages 18–28 · Sections 5–6 · arXiv:2404.09751v2

A bounded fraction of two densities is matched at each common return, and the unmatched mass remains in the invariant density cone. Renewal estimates bound the number of failed couplings and hence total variation. The random prefactor is controlled on a summable sequence of bad-environment events; Borel–Cantelli gives a samplewise bound, then interpolation covers intermediate times. This proves the quenched, rather than only averaged, conclusions.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper: arXiv:2404.09751v2
Authors listed: Mubarak Muhammad, Marks Ruziboev
Audit date: August 18, 2026

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Quenched Mixing Rates for Doubly Intermittent Maps

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

Quenched correlation decay for random doubly intermittent maps

Almost-sure samplewise mixing estimate

Random return-time and coupling tails

Distortion, return-time tails, and concentration

Distortion and return-time construction

Coupling and quenched concentration

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