Positive-Measure Acceleration Strata for Prevalent Perturbations of N-Fold Pullbacks of Supercritical Almost Mathieu Potentials
Abstract
Fix a strip width ρ>0, a Diophantine frequency α, an integer N≥2, and λ>1. For the family Vε,δ(x) =2λcos(2πN x)+εδ(x), we prove that there is an open prevalent set 𝒫strat of real-analytic strip-ρ perturbations such that every δ∈𝒫strat admits ε0(δ)>0 for which Leb{ E∈Σ(Vε,δ,α): ω(α,E;Vε,δ)=s }>0, 0<ε<ε0(δ), s=1,…,N. The proof relates acceleration to the total multiplicity of real zeros of analytic overlap coefficients for first-return cocycles. Under quantitative control of the return products, Poisson integration and Kac normalization identify acceleration with half this multiplicity. A positive analytic dual state gives the finite-volume spectral gaps and Schur curvature needed at the upper spectral edge. Schur variations separate the perturbed maxima, and parameter exclusion preserves s pairs of simple zeros on a positive-measure spectral set.
Related
- Normalized solutions to mass supercritical Schrödinger equations with radial potentials
- Normalized solutions of L2-supercritical NLS equations with periodic potentials
- Siebert's formula for virtual pullbacks
- Poincaré compactifications of pullbacks of parallel vector fields under counterexamples to the real Jacobian conjecture
- Finite holomorphic pullbacks, Segre numbers, and generic Lê numbers
BibTeX
@misc{liang2026positivemeasure,
title = {{Positive-Measure Acceleration Strata for Prevalent Perturbations of N-Fold Pullbacks of Supercritical Almost Mathieu Potentials}},
author = {Jinhao Liang and Yiqian Wang and Jiahao Xu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.05243},
eprint = {2610.05243},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.05243},
}
RIS
TY - UNPB
TI - Positive-Measure Acceleration Strata for Prevalent Perturbations of N-Fold Pullbacks of Supercritical Almost Mathieu Potentials
AU - Jinhao Liang
AU - Yiqian Wang
AU - Jiahao Xu
PY - 2026
DA - 2026/10/04
PB - arXiv
DO - 10.48550/arxiv.2610.05243
UR - https://arxiv.org/abs/2610.05243
AB - Fix a strip width ρ>0, a Diophantine frequency α, an integer N≥2, and λ>1. For the family V_ε,δ(x) =2λcos(2πN x)+εδ(x), we prove that there is an open prevalent set 𝒫_strat of real-analytic strip-ρ perturbations such that every δ∈𝒫_strat admits ε_0(δ)>0 for which Leb{ E∈Σ(V_ε,δ,α): ω(α,E;V_ε,δ)=s }>0, 0<ε<ε_0(δ), s=1,…,N. The proof relates acceleration to the total multiplicity of real zeros of analytic overlap coefficients for first-return cocycles. Under quantitative control of the return products, Poisson integration and Kac normalization identify acceleration with half this multiplicity. A positive analytic dual state gives the finite-volume spectral gaps and Schur curvature needed at the upper spectral edge. Schur variations separate the perturbed maxima, and parameter exclusion preserves s pairs of simple zeros on a positive-measure spectral set.
ER -