发表机构
Hohai University; School of Mathematics, Nanjing University（河海大学; 南京大学数学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明在超临界几乎Mathieu势的N重拉回中，对常见的实解析扰动，存在正测度的谱集，其上加速等于整数s（1≤s≤N），通过首次返回余循环的零点重数、Poisson积分与Kac归一化及Schur变分实现。
AI 中文摘要
固定条带宽度\\(\rho>0\\)、Diophantine频率\\(\alpha\\)、整数\\(N\ge2\\)及\\(\lambda>1\\)。对于族\\( V_{\varepsilon,\delta}(x) =2\lambda\cos(2\pi N x)+\varepsilon\delta(x) \\)，我们证明存在一个开且常见的实解析条带-\\(\rho\\)扰动集合\\(\mathcal P_{\rm strat}\\)，使得每个\\(\delta\in\mathcal P_{\rm strat}\\)都允许\\(\varepsilon_0(\delta)>0\\)，满足\\[ \operatorname{Leb}\left\{ E\in\Sigma(V_{\varepsilon,\delta},\alpha): \omega(\alpha,E;V_{\varepsilon,\delta})=s \right\}>0, \quad 0<\varepsilon<\varepsilon_0(\delta), \quad s=1,\ldots,N. \\] 证明将加速与首次返回余循环的解析重叠系数的实零点总重数联系起来。在返回乘积的定量控制下，Poisson积分和Kac归一化将加速等同于该重数的一半。一个正解析对偶态提供了上谱边缘所需的有限体积谱隙和Schur曲率。Schur变分分离了扰动的极大值，参数排除在正测度谱集上保留了\\(s\\)对简单零点。
英文摘要
Fix a strip width \(ρ>0\), a Diophantine frequency \(α\), an integer \(N\ge2\), and \(λ>1\). For the family \[ V_{\varepsilon,δ}(x) =2λ\cos(2πN x)+\varepsilonδ(x), \] we prove that there is an open prevalent set \(\mathcal P_{\rm strat}\) of real-analytic strip-\(ρ\) perturbations such that every \(δ\in\mathcal P_{\rm strat}\) admits \(\varepsilon_0(δ)>0\) for which \[ \begin{gathered} \operatorname{Leb}\left\{ E\inΣ(V_{\varepsilon,δ},α): ω(α,E;V_{\varepsilon,δ})=s \right\}>0, 0<\varepsilon<\varepsilon_0(δ), \qquad s=1,\ldots,N. \end{gathered} \] 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.
